title: Numbers station

text: The prelude, introduction, or call-up of a transmission (from which stations' informal nicknames are often derived) includes some kind of identifier, for the station itself, the intended recipient, or both. This can take the form of numeric or radio-alphabet "code names" (e.g. "Charlie India Oscar", "250 250 250", "Six-Niner-Zero-Oblique-Five-Four"), characteristic phrases (e.g. "¡Atención!", "Achtung!", "Ready? Ready?", "1234567890"), and sometimes musical or electronic sounds (e.g. "The Lincolnshire Poacher", "Magnetic Fields"). Sometimes, as in the case of radio-alphabet stations, the prelude can also signify the nature or priority of the message to follow (e.g., it may indicate that no message follows). Often the prelude repeats for a period before the body of the message begins.


title: Numbers station

text: After the prelude, there is usually an announcement of the number of number-groups in the message, the page to be used from the one-time pad, or other pertinent information. The groups are then recited. Groups are usually either four or five digits or radio-alphabet letters. The groups are typically repeated, either by reading each group twice or by repeating the entire message as a whole.


title: Numbers station

text: Some stations send more than one message during a transmission. In this case, some or all of the above process is repeated, with different contents.


title: Numbers station

text: Finally, after all the messages have been sent, the station will sign off in some characteristic fashion. Usually, it will simply be some form of the word "end" in whatever language the station uses (e.g., "End of message; End of transmission", "Ende", "Fini", "Final", "конец"). Some stations, especially those thought to originate from the former Soviet Union, end with a series of zeros, e.g., "00000" "000 000"; others end with music or other sounds.


title: Numbers station

text: Because of the secretive nature of the messages, the cryptographic function employed by particular stations is not publicly known, except in one (or possibly two) cases. It is assumed that most stations use a one-time pad that would make the contents of these number groups indistinguishable from randomly generated numbers or digits. In one confirmed case, West Germany did use a one-time pad for numbers transmissions.


title: Numbers station

text: Although few numbers stations have been tracked down by location, the technology used to transmit the numbers has historically been clear—stock shortwave transmitters using powers from 10kW to 100kW.


title: Numbers station

text: Amplitude modulated (AM) transmitters with optionally–variable frequency, using class-C power output stages with plate modulation, are the workhorses of international shortwave broadcasting, including numbers stations.


title: Numbers station

text: Application of spectrum analysis to numbers station signals has revealed the presence of data bursts, radioteletype-modulated subcarriers, phase-shifted carriers, and other unusual transmitter modulations like polytones. (RTTY-modulated subcarriers were also present on some U.S. commercial radio transmissions during the Cold War.)


title: Numbers station

text: The frequently reported use of high-tech modulations like data bursts, in combination or in sequence with spoken numbers, suggests varying transmissions for differing intelligence operations.


title: Numbers station

text: For spies in the field, low-tech spoken number transmissions continue to have advantages in the 21st century. High-tech data-receiving equipment is difficult to obtain. Even a non-standard civilian shortwave radio can be difficult to obtain in a totalitarian state. Being caught with just a shortwave radio has a degree of plausible deniability that no spying is being conducted.


title: Numbers station

text: The North Korean foreign language service Voice of Korea began to broadcast on the E03 Lincolnshire Poacher's former frequency, 11545 kHz, in 2006, possibly to deliberately interfere with its propagation. However, Lincolnshire Poacher broadcasts on three different frequencies, and the remaining two have not been interfered with. The apparent target zone for the Lincolnshire Poacher signals originating in Cyprus was the Middle East, not the Far East, which is covered by its sister station, E03a Cherry Ripe.


title: Numbers station

text: On 27 September 2006, amateur radio transmissions in the 30 m band were affected by an S06 "Russian Man" numbers station at 17:40 UTC.


title: Numbers station

text: In October 1990, it was reported that a numbers station had been interfering with communications on 6577 kHz, a frequency used by air traffic in the Caribbean. The interference was such that on at least one monitored transmission, it blocked the channel entirely and forced the air traffic controller to switch the pilot to an alternative frequency.


title: Numbers station

text: A BBC frequency, 7325 kHz, has also been used. This prompted a letter to the BBC from a listener in Andorra. She wrote to the World Service "Waveguide" programme in 1983 complaining that her listening had been spoiled by a female voice reading out numbers in English and asked the announcer what this interference was. The BBC presenter laughed at the suggestion of spy activity. He had consulted the experts at Bush House (BBC World Service headquarters), who declared that the voice was reading out nothing more sinister than snowfall figures for the ski slopes near the listener's home. After more research into this case, shortwave enthusiasts are fairly certain that this was a numbers station being broadcast on a random frequency.


title: Numbers station

text: The Cuban numbers station "HM01" has been known to interfere with shortwave broadcaster Voice of Welt on 11530 kHz.


title: Numbers station

text: Numbers station transmissions have often been the target of intentional jamming attempts. Despite this targeting, many numbers stations continue to broadcast unhindered. Historical examples of jamming include the E10 (a station thought to originate from Israel's Mossad intelligence agency) being jammed by the "Chinese Music Station" (thought to originate from the People's Republic of China and usually used to jam "Sound of Hope" radio broadcasts which are anti-PRC in nature).


title: Numbers station

text: Although many numbers stations have nicknames which usually describe some aspect of the station itself, these nicknames have sometimes led to confusion among listeners, particularly when discussing stations with similar traits. M. Gauffman of the ENIGMA numbers stations monitoring group originally assigned a code to each known station.


title: Numbers station

text: Portions of the original ENIGMA group moved on to other interests in 2000 and the classification of numbers stations was continued by the follow-on group ENIGMA 2000. The document containing the description of each station and its code designation was called the "ENIGMA Control List" until 2016, after which it was incorporated into the "ENIGMA 2000 Active Station List"; the latest edition of the list was published in September 2017. This classification scheme takes the form of a letter followed by a number (or, in the case of some "X" stations, more numbers). The letter indicates the language used by the station in question:


title: Numbers station

text: Some stations have also been stripped of their designation when they were discovered not to be a numbers station. This was the case for E22, which was discovered in 2005 to be test transmissions for All India Radio.


title: Numbers station

text: It has been reported that the United States has used numbers stations to communicate encoded information to persons in other countries. There are also claims that State Department-operated stations, such as KKN50 and KKN44, used to broadcast similar "numbers" messages or related traffic.


title: Numbers station

text: The "Atención" station of Cuba became the world's first numbers station to be officially and publicly accused of transmitting to spies. It was the centerpiece of a United States federal court espionage trial, following the arrest of the Wasp Network of Cuban spies in 1998. The U.S. prosecutors claimed the accused were writing down number codes received from Atención, using Sony hand-held shortwave receivers, and typing the numbers into laptop computers to decode spying instructions. The FBI testified that they had entered a spy's apartment in 1995, and copied the computer decryption program for the Atención numbers code. They used it to decode Atención spy messages, which the prosecutors unveiled in court.


title: Numbers station

text: The United States government's evidence included the following three examples of decoded Atención messages. (It was not reported whether the original clear texts were in Spanish, although the phrasing of "Day of the Woman" would indicate so.)


title: Numbers station

text: The moderator of an e-mail list for global numbers station hobbyists claimed that "Someone on the Spooks list had already cracked the code for a repeated transmission [from Havana to Miami] if it was received garbled." Such code-breaking is possible if a one-time pad decoding key is used more than once. If used properly, however, the code cannot be broken.


title: Numbers station

text: In 2001, Ana Belén Montes, a senior US Defense Intelligence Agency analyst, was arrested and charged with espionage. The federal prosecutors alleged that Montes was able to communicate with the Cuban Intelligence Directorate through encoded messages, with instructions being received through "encrypted shortwave transmissions from Cuba".


title: Numbers station

text: In 2006, Carlos Alvarez and his wife, Elsa, were arrested and charged with espionage. The U.S. District Court for the Southern District of Florida stated that "defendants would receive assignments via shortwave radio transmissions".


title: Numbers station

text: In June 2009, the United States similarly charged Walter Kendall Myers with conspiracy to spy for Cuba, and receiving and decoding messages broadcast from a numbers station operated by the Cuban Intelligence Directorate to further that conspiracy. As discovered by the FBI up to 2010, one way that Russian agents of the Illegals Program were receiving instructions was via coded messages on shortwave radio.


title: Numbers station

text: It has been reported that the United States has used numbers stations to communicate encoded information to persons in other countries. There are also claims that State Department-operated stations, such as KKN50 and KKN44, used to broadcast similar "numbers" messages or related traffic, although these radio stations have been off the air for many years.


title: Nick Cassavetes

text: Nicholas David Rowland Cassavetes (born May 21, 1959) is an American actor, director, and writer. He has directed such films as "She's So Lovely" (1997), "John Q." (2002), "The Notebook" (2004), "Alpha Dog" (2006), and "My Sister's Keeper" (2009). His acting credits include an uncredited role in "Husbands" (1970)—which was directed by his father, John Cassavetes—as well as roles in the films "The Wraith" (1986), "Face/Off" (1997), and "Blow" (2001).


title: Nick Cassavetes

text: Cassavetes was born in New York City, the son of Greek-American actor and film director John Cassavetes and actress Gena Rowlands. As a child, he appeared in his father's film "Husbands" (1970). After spending so much of his youth surrounded by the film industry, Cassavetes initially decided he did not want to go into the field. He instead attended Syracuse University on a basketball scholarship, but after an injury effectively ended his athletic career, he decided to rethink his aspirations, ultimately deciding to attend his parents' alma mater, the American Academy of Dramatic Arts in New York.


title: Nick Cassavetes

text: He has appeared in the films "Face/Off", "The Wraith", "Life", "", "Backstreet Dreams" and "The Astronaut's Wife", among others. He has directed several films, including "John Q", "Alpha Dog", "She's So Lovely", "Unhook the Stars", "The Notebook", and "My Sister's Keeper". He also adapted the screenplay for "Blow" and wrote the dialogue for the Justin Timberlake music video "What Goes Around... Comes Around".


title: Nick Cassavetes

text: Cassavetes finished in fifth place in the World Poker Tour (WPT) Invitational Season 5, attempting a bluff. He also appeared on season 5 of The Game Show Network's (GSN) "High Stakes Poker".


title: Nick Cassavetes

text: He also has played himself in the season 7 opener of "Entourage" on HBO and Tattoo Joe as a cameo in "", replacing Liam Neeson from the cast due to scheduling conflicts.


title: Nick Cassavetes

text: In 1985, Cassavetes married Isabelle Rafalovich. They had two daughters together before divorcing. Sasha was born with a heart defect and underwent substantial surgery; Cassavetes' film "John Q." was dedicated to Sasha, and his later adaptation of "My Sister's Keeper" was based in part on Sasha's medical experience.


title: Nick Cassavetes

text: Cassavetes later married Heather "Queenie" Wahlquist, and they have a daughter together. Wahlquist has appeared in several of his films, including a small role in "The Notebook" as Sara, a secondary character and best friend to the female lead Allie Hamilton, portrayed by Rachel McAdams. Cassavetes' own mother, Gena Rowlands, appears as the older, married Allie (Hamilton) Calhoun.


title: First-order logic

text: First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists""" is a quantifier, while "x" is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.


title: First-order logic

text: A theory about a topic is usually a first-order logic together with a specified domain of discourse (over which the quantified variables range), finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold about them. Sometimes, "theory" is understood in a more formal sense as just a set of sentences in first-order logic.


title: First-order logic

text: The adjective "first-order" distinguishes first-order logic from higher-order logic, in which there are predicates having predicates or functions as arguments, or in which quantification over predicates or functions, or both, are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.


title: First-order logic

text: There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.


title: First-order logic

text: First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.


title: First-order logic

text: Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic.


title: First-order logic

text: No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures (that is, categorical axiom systems) can be obtained in stronger logics such as second-order logic.


title: First-order logic

text: The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce. For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001).


title: First-order logic

text: While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification.


title: First-order logic

text: A predicate takes an entity or entities in the domain of discourse and evaluates to true or false. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". In propositional logic, these sentences are viewed as being unrelated, and might be denoted, for example, by variables such as "p" and "q". The predicate "is a philosopher" occurs in both sentences, which have a common structure of ""a" is a philosopher". The variable "a" is instantiated as "Socrates" in the first sentence, and is instantiated as "Plato" in the second sentence. While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not.


title: First-order logic

text: Relationships between predicates can be stated using logical connectives. Consider, for example, the first-order formula "if "a" is a philosopher, then "a" is a scholar". This formula is a conditional statement with ""a" is a philosopher" as its hypothesis, and ""a" is a scholar" as its conclusion. The truth of this formula depends on which object is denoted by "a", and on the interpretations of the predicates "is a philosopher" and "is a scholar".


title: First-order logic

text: Quantifiers can be applied to variables in a formula. The variable "a" in the previous formula can be universally quantified, for instance, with the first-order sentence "For every "a", if "a" is a philosopher, then "a" is a scholar". The universal quantifier "for every" in this sentence expresses the idea that the claim "if "a" is a philosopher, then "a" is a scholar" holds for "all" choices of "a".


title: First-order logic

text: The "negation" of the sentence "For every "a", if "a" is a philosopher, then "a" is a scholar" is logically equivalent to the sentence "There exists "a" such that "a" is a philosopher and "a" is not a scholar". The existential quantifier "there exists" expresses the idea that the claim ""a" is a philosopher and "a" is not a scholar" holds for "some" choice of "a".


title: First-order logic

text: The predicates "is a philosopher" and "is a scholar" each take a single variable. In general, predicates can take several variables. In the first-order sentence "Socrates is the teacher of Plato", the predicate "is the teacher of" takes two variables.


title: First-order logic

text: An interpretation (or model) of a first-order formula specifies what each predicate means, and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, in an interpretation with the domain of discourse consisting of all human beings and the predicate "is a philosopher" understood as "was the author of the "Republic"", the sentence "There exists "a" such that "a" is a philosopher" is seen as being true, as witnessed by Plato.


title: First-order logic

text: There are two key parts of first-order logic. The syntax determines which finite sequences of symbols are well-formed expressions in first-order logic, while the semantics determines the meanings behind these expressions.


title: First-order logic

text: Unlike natural languages, such as English, the language of first-order logic is completely formal, so that it can be mechanically determined whether a given expression is well formed. There are two key types of well-formed expressions: "terms", which intuitively represent objects, and "formulas", which intuitively express statements that can be true or false. The terms and formulas of first-order logic are strings of "symbols", where all the symbols together form the "alphabet" of the language. As with all formal languages, the nature of the symbols themselves is outside the scope of formal logic; they are often regarded simply as letters and punctuation symbols.


title: First-order logic

text: It is common to divide the symbols of the alphabet into "logical symbols", which always have the same meaning, and "non-logical symbols", whose meaning varies by interpretation. For example, the logical symbol formula_1 always represents "and"; it is never interpreted as "or", which is represented by the logical symbol formula_2. However, a non-logical predicate symbol such as Phil("x") could be interpreted to mean ""x" is a philosopher", ""x" is a man named Philip", or any other unary predicate depending on the interpretation at hand.


title: First-order logic

text: Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices.


title: First-order logic

text: Non-logical symbols represent predicates (relations), functions and constants. It used to be standard practice to use a fixed, infinite set of non-logical symbols for all purposes:


title: First-order logic

text: When the arity of a predicate symbol or function symbol is clear from context, the superscript "n" is often omitted.


title: First-order logic

text: In this traditional approach, there is only one language of first-order logic. This approach is still common, especially in philosophically oriented books.


title: First-order logic

text: A more recent practice is to use different non-logical symbols according to the application one has in mind. Therefore, it has become necessary to name the set of all non-logical symbols used in a particular application. This choice is made via a "signature".


title: First-order logic

text: Typical signatures in mathematics are {1, ×} or just {×} for groups, or {0, 1, +, ×, <} for ordered fields. There are no restrictions on the number of non-logical symbols. The signature can be empty, finite, or infinite, even uncountable. Uncountable signatures occur for example in modern proofs of the Löwenheim–Skolem theorem.


title: First-order logic

text: Though signatures might in some cases imply how non-logical symbols are to be interpreted, interpretation of the non-logical symbols in the signature is separate (and not necessarily fixed). Signatures concern syntax rather than semantics.


title: First-order logic

text: The traditional approach can be recovered in the modern approach, by simply specifying the "custom" signature to consist of the traditional sequences of non-logical symbols.


title: First-order logic

text: The formation rules define the terms and formulas of first-order logic. When terms and formulas are represented as strings of symbols, these rules can be used to write a formal grammar for terms and formulas. These rules are generally context-free (each production has a single symbol on the left side), except that the set of symbols may be allowed to be infinite and there may be many start symbols, for example the variables in the case of terms.


title: First-order logic

text: Only expressions which can be obtained by finitely many applications of rules 1 and 2 are terms. For example, no expression involving a predicate symbol is a term.


title: First-order logic

text: The set of "formulas" (also called "well-formed formulas" or "WFFs") is inductively defined by the following rules:


title: First-order logic

text: Only expressions which can be obtained by finitely many applications of rules 1–5 are formulas. The formulas obtained from the first two rules are said to be "atomic formulas".


title: First-order logic

text: is a formula, if "f" is a unary function symbol, "P" a unary predicate symbol, and Q a ternary predicate symbol. However, formula_14 is not a formula, although it is a string of symbols from the alphabet.


title: First-order logic

text: The role of the parentheses in the definition is to ensure that any formula can only be obtained in one way—by following the inductive definition (i.e., there is a unique parse tree for each formula). This property is known as "unique readability" of formulas. There are many conventions for where parentheses are used in formulas. For example, some authors use colons or full stops instead of parentheses, or change the places in which parentheses are inserted. Each author's particular definition must be accompanied by a proof of unique readability.


title: First-order logic

text: This definition of a formula does not support defining an if-then-else function codice_1, where "c" is a condition expressed as a formula, that would return "a" if c is true, and "b" if it is false. This is because both predicates and functions can only accept terms as parameters, but the first parameter is a formula. Some languages built on first-order logic, such as SMT-LIB 2.0, add this.


title: First-order logic

text: For convenience, conventions have been developed about the precedence of the logical operators, to avoid the need to write parentheses in some cases. These rules are similar to the order of operations in arithmetic. A common convention is:


title: First-order logic

text: Moreover, extra punctuation not required by the definition may be inserted—to make formulas easier to read. Thus the formula


title: First-order logic

text: In some fields, it is common to use infix notation for binary relations and functions, instead of the prefix notation defined above. For example, in arithmetic, one typically writes "2 + 2 = 4" instead of "=(+(2,2),4)". It is common to regard formulas in infix notation as abbreviations for the corresponding formulas in prefix notation, cf. also term structure vs. representation.


title: First-order logic

text: The definitions above use infix notation for binary connectives such as formula_18. A less common convention is Polish notation, in which one writes formula_22, formula_23 and so on in front of their arguments rather than between them. This convention is advantageous in that it allows all punctuation symbols to be discarded. As such, Polish notation is compact and elegant, but rarely used in practice because it is hard for humans to read. In Polish notation, the formula


title: First-order logic

text: In a formula, a variable may occur "free" or "bound" (or both). Intuitively, a variable occurrence is free in a formula if it is not quantified: in , the sole occurrence of variable "x" is free while that of "y" is bound. The free and bound variable occurrences in a formula are defined inductively as follows.


title: First-order logic

text: For example, in , "x" and "y" occur only bound, "z" occurs only free, and "w" is neither because it does not occur in the formula.


title: First-order logic

text: Free and bound variables of a formula need not be disjoint sets: in the formula , the first occurrence of "x", as argument of "P", is free while the second one, as argument of "Q", is bound.


title: First-order logic

text: A formula in first-order logic with no free variable occurrences is called a "first-order sentence". These are the formulas that will have well-defined truth values under an interpretation. For example, whether a formula such as Phil("x") is true must depend on what "x" represents. But the sentence will be either true or false in a given interpretation.


title: First-order logic

text: In mathematics, the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤. Then:


title: First-order logic

text: The axioms for ordered abelian groups can be expressed as a set of sentences in the language. For example, the axiom stating that the group is commutative is usually written formula_27


title: First-order logic

text: An interpretation of a first-order language assigns a denotation to each non-logical symbol (predicate symbol, function symbol, or constant symbol) in that language. It also determines a domain of discourse that specifies the range of the quantifiers. The result is that each term is assigned an object that it represents, each predicate is assigned a property of objects, and each sentence is assigned a truth value. In this way, an interpretation provides semantic meaning to the terms, predicates, and formulas of the language. The study of the interpretations of formal languages is called formal semantics. What follows is a description of the standard or Tarskian semantics for first-order logic. (It is also possible to define game semantics for first-order logic, but aside from requiring the axiom of choice, game semantics agree with Tarskian semantics for first-order logic, so game semantics will not be elaborated herein.)


title: First-order logic

text: The most common way of specifying an interpretation (especially in mathematics) is to specify a "structure" (also called a "model"; see below). The structure consists of a domain of discourse "D" and an interpretation function mapping non-logical symbols to predicates, functions, and constants.


title: First-order logic

text: The domain of discourse "D" is a nonempty set of "objects" of some kind. Intuitively, given an interpretation, a first-order formula becomes a statement about these objects; for example, formula_28 states the existence of some object in "D" for which the predicate "P" is true (or, more precisely, for which the predicate assigned to the predicate symbol "P" by the interpretation is true). For example, one can take "D" to be the set of integers.


title: First-order logic

text: A formula evaluates to true or false given an interpretation and a variable assignment μ that associates an element of the domain of discourse with each variable. The reason that a variable assignment is required is to give meanings to formulas with free variables, such as formula_33. The truth value of this formula changes depending on whether "x" and "y" denote the same individual.


title: First-order logic

text: First, the variable assignment μ can be extended to all terms of the language, with the result that each term maps to a single element of the domain of discourse. The following rules are used to make this assignment:


title: First-order logic

text: Next, each formula is assigned a truth value. The inductive definition used to make this assignment is called the T-schema.


title: First-order logic

text: If a formula does not contain free variables, and so is a sentence, then the initial variable assignment does not affect its truth value. In other words, a sentence is true according to "M" and formula_51 if and only if it is true according to "M" and every other variable assignment formula_52.


title: First-order logic

text: There is a second common approach to defining truth values that does not rely on variable assignment functions. Instead, given an interpretation "M", one first adds to the signature a collection of constant symbols, one for each element of the domain of discourse in "M"; say that for each "d" in the domain the constant symbol "c" is fixed. The interpretation is extended so that each new constant symbol is assigned to its corresponding element of the domain. One now defines truth for quantified formulas syntactically, as follows:


title: First-order logic

text: This alternate approach gives exactly the same truth values to all sentences as the approach via variable assignments.


title: First-order logic

text: If a sentence φ evaluates to "true" under a given interpretation "M", one says that "M" "satisfies" φ; this is denoted formula_68. A sentence is "satisfiable" if there is some interpretation under which it is true. This is a bit different from the symbol formula_69 from model theory, where formula_70 denotes satisfiability in a model, i.e. "there is a suitable assignment of values in formula_71's domain to variable symbols of formula_72".


title: First-order logic

text: Satisfiability of formulas with free variables is more complicated, because an interpretation on its own does not determine the truth value of such a formula. The most common convention is that a formula φ with free variables formula_73, ..., formula_74 is said to be satisfied by an interpretation if the formula φ remains true regardless which individuals from the domain of discourse are assigned to its free variables formula_73, ..., formula_74. This has the same effect as saying that a formula φ is satisfied if and only if its universal closure formula_77 is satisfied.


title: First-order logic

text: A formula is "logically valid" (or simply "valid") if it is true in every interpretation. These formulas play a role similar to tautologies in propositional logic.


title: First-order logic

text: A formula φ is a "logical consequence" of a formula ψ if every interpretation that makes ψ true also makes φ true. In this case one says that φ is logically implied by ψ.


title: First-order logic

text: An alternate approach to the semantics of first-order logic proceeds via abstract algebra. This approach generalizes the Lindenbaum–Tarski algebras of propositional logic. There are three ways of eliminating quantified variables from first-order logic that do not involve replacing quantifiers with other variable binding term operators:


title: First-order logic

text: Tarski and Givant (1987) showed that the fragment of first-order logic that has no atomic sentence lying in the scope of more than three quantifiers has the same expressive power as relation algebra. This fragment is of great interest because it suffices for Peano arithmetic and most axiomatic set theory, including the canonical ZFC. They also prove that first-order logic with a primitive ordered pair is equivalent to a relation algebra with two ordered pair projection functions.


title: First-order logic

text: A "first-order theory" of a particular signature is a set of axioms, which are sentences consisting of symbols from that signature. The set of axioms is often finite or recursively enumerable, in which case the theory is called "effective". Some authors require theories to also include all logical consequences of the axioms. The axioms are considered to hold within the theory and from them other sentences that hold within the theory can be derived.


title: First-order logic

text: A first-order structure that satisfies all sentences in a given theory is said to be a "model" of the theory. An "elementary class" is the set of all structures satisfying a particular theory. These classes are a main subject of study in model theory.


title: First-order logic

text: Many theories have an "intended interpretation", a certain model that is kept in mind when studying the theory. For example, the intended interpretation of Peano arithmetic consists of the usual natural numbers with their usual operations. However, the Löwenheim–Skolem theorem shows that most first-order theories will also have other, nonstandard models.


title: First-order logic

text: A theory is "consistent" if it is not possible to prove a contradiction from the axioms of the theory. A theory is "complete" if, for every formula in its signature, either that formula or its negation is a logical consequence of the axioms of the theory. Gödel's incompleteness theorem shows that effective first-order theories that include a sufficient portion of the theory of the natural numbers can never be both consistent and complete.


title: First-order logic

text: The definition above requires that the domain of discourse of any interpretation must be nonempty. There are settings, such as inclusive logic, where empty domains are permitted. Moreover, if a class of algebraic structures includes an empty structure (for example, there is an empty poset), that class can only be an elementary class in first-order logic if empty domains are permitted or the empty structure is removed from the class.


title: First-order logic

text: Thus, when the empty domain is permitted, it must often be treated as a special case. Most authors, however, simply exclude the empty domain by definition.


title: First-order logic

text: A "deductive system" is used to demonstrate, on a purely syntactic basis, that one formula is a logical consequence of another formula. There are many such systems for first-order logic, including Hilbert-style deductive systems, natural deduction, the sequent calculus, the tableaux method, and resolution. These share the common property that a deduction is a finite syntactic object; the format of this object, and the way it is constructed, vary widely. These finite deductions themselves are often called "derivations" in proof theory. They are also often called "proofs", but are completely formalized unlike natural-language mathematical proofs.


title: First-order logic

text: A deductive system is "sound" if any formula that can be derived in the system is logically valid. Conversely, a deductive system is "complete" if every logically valid formula is derivable. All of the systems discussed in this article are both sound and complete. They also share the property that it is possible to effectively verify that a purportedly valid deduction is actually a deduction; such deduction systems are called "effective".


title: First-order logic

text: A key property of deductive systems is that they are purely syntactic, so that derivations can be verified without considering any interpretation. Thus a sound argument is correct in every possible interpretation of the language, regardless of whether that interpretation is about mathematics, economics, or some other area.


title: First-order logic

text: In general, logical consequence in first-order logic is only semidecidable: if a sentence A logically implies a sentence B then this can be discovered (for example, by searching for a proof until one is found, using some effective, sound, complete proof system). However, if A does not logically imply B, this does not mean that A logically implies the negation of B. There is no effective procedure that, given formulas A and B, always correctly decides whether A logically implies B.


title: First-order logic

text: A "rule of inference" states that, given a particular formula (or set of formulas) with a certain property as a hypothesis, another specific formula (or set of formulas) can be derived as a conclusion. The rule is sound (or truth-preserving) if it preserves validity in the sense that whenever any interpretation satisfies the hypothesis, that interpretation also satisfies the conclusion.


title: First-order logic

text: For example, one common rule of inference is the "rule of substitution". If "t" is a term and φ is a formula possibly containing the variable "x", then φ["t"/"x"] is the result of replacing all free instances of "x" by "t" in φ. The substitution rule states that for any φ and any term "t", one can conclude φ["t"/"x"] from φ provided that no free variable of "t" becomes bound during the substitution process. (If some free variable of "t" becomes bound, then to substitute "t" for "x" it is first necessary to change the bound variables of φ to differ from the free variables of "t".)


title: First-order logic

text: To see why the restriction on bound variables is necessary, consider the logically valid formula φ given by formula_81, in the signature of (0,1,+,×,=) of arithmetic. If "t" is the term "x + 1", the formula φ["t"/"y"] is formula_82, which will be false in many interpretations. The problem is that the free variable "x" of "t" became bound during the substitution. The intended replacement can be obtained by renaming the bound variable "x" of φ to something else, say "z", so that the formula after substitution is formula_83, which is again logically valid.


title: First-order logic

text: The substitution rule demonstrates several common aspects of rules of inference. It is entirely syntactical; one can tell whether it was correctly applied without appeal to any interpretation. It has (syntactically defined) limitations on when it can be applied, which must be respected to preserve the correctness of derivations. Moreover, as is often the case, these limitations are necessary because of interactions between free and bound variables that occur during syntactic manipulations of the formulas involved in the inference rule.


title: First-order logic

text: A deduction in a Hilbert-style deductive system is a list of formulas, each of which is a "logical axiom", a hypothesis that has been assumed for the derivation at hand, or follows from previous formulas via a rule of inference. The logical axioms consist of several axiom schemas of logically valid formulas; these encompass a significant amount of propositional logic. The rules of inference enable the manipulation of quantifiers. Typical Hilbert-style systems have a small number of rules of inference, along with several infinite schemas of logical axioms. It is common to have only modus ponens and universal generalization as rules of inference.


title: First-order logic

text: Natural deduction systems resemble Hilbert-style systems in that a deduction is a finite list of formulas. However, natural deduction systems have no logical axioms; they compensate by adding additional rules of inference that can be used to manipulate the logical connectives in formulas in the proof.


title: First-order logic

text: The sequent calculus was developed to study the properties of natural deduction systems. Instead of working with one formula at a time, it uses "sequents", which are expressions of the form


title: First-order logic

text: where A, ..., A, B, ..., B are formulas and the turnstile symbol formula_85 is used as punctuation to separate the two halves. Intuitively, a sequent expresses the idea that formula_86 implies formula_87.


title: First-order logic

text: Unlike the methods just described, the derivations in the tableaux method are not lists of formulas. Instead, a derivation is a tree of formulas. To show that a formula A is provable, the tableaux method attempts to demonstrate that the negation of A is unsatisfiable. The tree of the derivation has formula_88 at its root; the tree branches in a way that reflects the structure of the formula. For example, to show that formula_89 is unsatisfiable requires showing that C and D are each unsatisfiable; this corresponds to a branching point in the tree with parent formula_89 and children C and D.


title: First-order logic

text: The resolution rule is a single rule of inference that, together with unification, is sound and complete for first-order logic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisfiable. Resolution is commonly used in automated theorem proving.


title: First-order logic

text: The resolution method works only with formulas that are disjunctions of atomic formulas; arbitrary formulas must first be converted to this form through Skolemization. The resolution rule states that from the hypotheses formula_91 and formula_92, the conclusion formula_93 can be obtained.


title: First-order logic

text: Many identities can be proved, which establish equivalences between particular formulas. These identities allow for rearranging formulas by moving quantifiers across other connectives, and are useful for putting formulas in prenex normal form. Some provable identities include:


title: First-order logic

text: There are several different conventions for using equality (or identity) in first-order logic. The most common convention, known as first-order logic with equality, includes the equality symbol as a primitive logical symbol which is always interpreted as the real equality relation between members of the domain of discourse, such that the "two" given members are the same member. This approach also adds certain axioms about equality to the deductive system employed. These equality axioms are:


title: First-order logic

text: These are axiom schemas, each of which specifies an infinite set of axioms. The third schema is known as "Leibniz's law", "the principle of substitutivity", "the indiscernibility of identicals", or "the replacement property". The second schema, involving the function symbol "f", is (equivalent to) a special case of the third schema, using the formula


title: First-order logic

text: An alternate approach considers the equality relation to be a non-logical symbol. This convention is known as "first-order logic without equality". If an equality relation is included in the signature, the axioms of equality must now be added to the theories under consideration, if desired, instead of being considered rules of logic. The main difference between this method and first-order logic with equality is that an interpretation may now interpret two distinct individuals as "equal" (although, by Leibniz's law, these will satisfy exactly the same formulas under any interpretation). That is, the equality relation may now be interpreted by an arbitrary equivalence relation on the domain of discourse that is congruent with respect to the functions and relations of the interpretation.


title: First-order logic

text: When this second convention is followed, the term "normal model" is used to refer to an interpretation where no distinct individuals "a" and "b" satisfy "a" = "b". In first-order logic with equality, only normal models are considered, and so there is no term for a model other than a normal model. When first-order logic without equality is studied, it is necessary to amend the statements of results such as the Löwenheim–Skolem theorem so that only normal models are considered.


title: First-order logic

text: First-order logic without equality is often employed in the context of second-order arithmetic and other higher-order theories of arithmetic, where the equality relation between sets of natural numbers is usually omitted.


title: First-order logic

text: If a theory has a binary formula "A"("x","y") which satisfies reflexivity and Leibniz's law, the theory is said to have equality, or to be a theory with equality. The theory may not have all instances of the above schemas as axioms, but rather as derivable theorems. For example, in theories with no function symbols and a finite number of relations, it is possible to define equality in terms of the relations, by defining the two terms "s" and "t" to be equal if any relation is unchanged by changing "s" to "t" in any argument.


title: First-order logic

text: One motivation for the use of first-order logic, rather than higher-order logic, is that first-order logic has many metalogical properties that stronger logics do not have. These results concern general properties of first-order logic itself, rather than properties of individual theories. They provide fundamental tools for the construction of models of first-order theories.


title: First-order logic

text: Gödel's completeness theorem, proved by Kurt Gödel in 1929, establishes that there are sound, complete, effective deductive systems for first-order logic, and thus the first-order logical consequence relation is captured by finite provability. Naively, the statement that a formula φ logically implies a formula ψ depends on every model of φ; these models will in general be of arbitrarily large cardinality, and so logical consequence cannot be effectively verified by checking every model. However, it is possible to enumerate all finite derivations and search for a derivation of ψ from φ. If ψ is logically implied by φ, such a derivation will eventually be found. Thus first-order logical consequence is semidecidable: it is possible to make an effective enumeration of all pairs of sentences (φ,ψ) such that ψ is a logical consequence of φ.


title: First-order logic

text: Unlike propositional logic, first-order logic is undecidable (although semidecidable), provided that the language has at least one predicate of arity at least 2 (other than equality). This means that there is no decision procedure that determines whether arbitrary formulas are logically valid. This result was established independently by Alonzo Church and Alan Turing in 1936 and 1937, respectively, giving a negative answer to the Entscheidungsproblem posed by David Hilbert and Wilhelm Ackermann in 1928. Their proofs demonstrate a connection between the unsolvability of the decision problem for first-order logic and the unsolvability of the halting problem.


title: First-order logic

text: There are systems weaker than full first-order logic for which the logical consequence relation is decidable. These include propositional logic and monadic predicate logic, which is first-order logic restricted to unary predicate symbols and no function symbols. Other logics with no function symbols which are decidable are the guarded fragment of first-order logic, as well as two-variable logic. The Bernays–Schönfinkel class of first-order formulas is also decidable. Decidable subsets of first-order logic are also studied in the framework of description logics.


title: First-order logic

text: The Löwenheim–Skolem theorem shows that if a first-order theory of cardinality λ has an infinite model, then it has models of every infinite cardinality greater than or equal to λ. One of the earliest results in model theory, it implies that it is not possible to characterize countability or uncountability in a first-order language with a countable signature. That is, there is no first-order formula φ("x") such that an arbitrary structure M satisfies φ if and only if the domain of discourse of M is countable (or, in the second case, uncountable).


title: First-order logic

text: The Löwenheim–Skolem theorem implies that infinite structures cannot be categorically axiomatized in first-order logic. For example, there is no first-order theory whose only model is the real line: any first-order theory with an infinite model also has a model of cardinality larger than the continuum. Since the real line is infinite, any theory satisfied by the real line is also satisfied by some nonstandard models. When the Löwenheim–Skolem theorem is applied to first-order set theories, the nonintuitive consequences are known as Skolem's paradox.


title: First-order logic

text: The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This implies that if a formula is a logical consequence of an infinite set of first-order axioms, then it is a logical consequence of some finite number of those axioms. This theorem was proved first by Kurt Gödel as a consequence of the completeness theorem, but many additional proofs have been obtained over time. It is a central tool in model theory, providing a fundamental method for constructing models.


title: First-order logic

text: The compactness theorem has a limiting effect on which collections of first-order structures are elementary classes. For example, the compactness theorem implies that any theory that has arbitrarily large finite models has an infinite model. Thus the class of all finite graphs is not an elementary class (the same holds for many other algebraic structures).


title: First-order logic

text: There are also more subtle limitations of first-order logic that are implied by the compactness theorem. For example, in computer science, many situations can be modeled as a directed graph of states (nodes) and connections (directed edges). Validating such a system may require showing that no "bad" state can be reached from any "good" state. Thus one seeks to determine if the good and bad states are in different connected components of the graph. However, the compactness theorem can be used to show that connected graphs are not an elementary class in first-order logic, and there is no formula φ("x","y") of first-order logic, in the logic of graphs, that expresses the idea that there is a path from "x" to "y". Connectedness can be expressed in second-order logic, however, but not with only existential set quantifiers, as formula_108 also enjoys compactness.


title: First-order logic

text: Per Lindström showed that the metalogical properties just discussed actually characterize first-order logic in the sense that no stronger logic can also have those properties (Ebbinghaus and Flum 1994, Chapter XIII). Lindström defined a class of abstract logical systems, and a rigorous definition of the relative strength of a member of this class. He established two theorems for systems of this type:


title: First-order logic

text: Although first-order logic is sufficient for formalizing much of mathematics, and is commonly used in computer science and other fields, it has certain limitations. These include limitations on its expressiveness and limitations of the fragments of natural languages that it can describe.


title: First-order logic

text: For instance, first-order logic is undecidable, meaning a sound, complete and terminating decision algorithm for provability is impossible. This has led to the study of interesting decidable fragments, such as C: first-order logic with two variables and the counting quantifiers formula_109 and formula_110.


title: First-order logic

text: The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality. In particular, no first-order theory with an infinite model can be categorical. Thus there is no first-order theory whose only model has the set of natural numbers as its domain, or whose only model has the set of real numbers as its domain. Many extensions of first-order logic, including infinitary logics and higher-order logics, are more expressive in the sense that they do permit categorical axiomatizations of the natural numbers or real numbers. This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.


title: First-order logic

text: First-order logic is able to formalize many simple quantifier constructions in natural language, such as "every person who lives in Perth lives in Australia". Hence, first-order logic is used as a basis for knowledge representation languages, such as FO(.).


title: First-order logic

text: Still, there are complicated features of natural language that cannot be expressed in first-order logic. "Any logical system which is appropriate as an instrument for the analysis of natural language needs a much richer structure than first-order predicate logic".


title: First-order logic

text: There are many variations of first-order logic. Some of these are inessential in the sense that they merely change notation without affecting the semantics. Others change the expressive power more significantly, by extending the semantics through additional quantifiers or other new logical symbols. For example, infinitary logics permit formulas of infinite size, and modal logics add symbols for possibility and necessity.


title: First-order logic

text: Restrictions such as these are useful as a technique to reduce the number of inference rules or axiom schemas in deductive systems, which leads to shorter proofs of metalogical results. The cost of the restrictions is that it becomes more difficult to express natural-language statements in the formal system at hand, because the logical connectives used in the natural language statements must be replaced by their (longer) definitions in terms of the restricted collection of logical connectives. Similarly, derivations in the limited systems may be longer than derivations in systems that include additional connectives. There is thus a trade-off between the ease of working within the formal system and the ease of proving results about the formal system.


title: First-order logic

text: It is also possible to restrict the arities of function symbols and predicate symbols, in sufficiently expressive theories. One can in principle dispense entirely with functions of arity greater than 2 and predicates of arity greater than 1 in theories that include a pairing function. This is a function of arity 2 that takes pairs of elements of the domain and returns an ordered pair containing them. It is also sufficient to have two predicate symbols of arity 2 that define projection functions from an ordered pair to its components. In either case it is necessary that the natural axioms for a pairing function and its projections are satisfied.


title: First-order logic

text: Ordinary first-order interpretations have a single domain of discourse over which all quantifiers range. "Many-sorted first-order logic" allows variables to have different "sorts", which have different domains. This is also called "typed first-order logic", and the sorts called "types" (as in data type), but it is not the same as first-order type theory. Many-sorted first-order logic is often used in the study of second-order arithmetic.


title: First-order logic

text: When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic.


title: First-order logic

text: One introduces into the single-sorted theory a unary predicate symbol for each sort in the many-sorted theory, and adds an axiom saying that these unary predicates partition the domain of discourse. For example, if there are two sorts, one adds predicate symbols formula_137 and formula_138 and the axiom


title: First-order logic

text: Then the elements satisfying formula_140 are thought of as elements of the first sort, and elements satisfying formula_141 as elements of the second sort. One can quantify over each sort by using the corresponding predicate symbol to limit the range of quantification. For example, to say there is an element of the first sort satisfying formula φ("x"), one writes


title: First-order logic

text: Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory.


title: First-order logic

text: Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.


title: First-order logic

text: The most commonly studied infinitary logics are denoted "L", where α and β are each either cardinal numbers or the symbol ∞. In this notation, ordinary first-order logic is "L".


title: First-order logic

text: In the logic "L", arbitrary conjunctions or disjunctions are allowed when building formulas, and there is an unlimited supply of variables. More generally, the logic that permits conjunctions or disjunctions with less than κ constituents is known as "L". For example, "L" permits countable conjunctions and disjunctions.


title: First-order logic

text: The set of free variables in a formula of "L" can have any cardinality strictly less than κ, yet only finitely many of them can be in the scope of any quantifier when a formula appears as a subformula of another. In other infinitary logics, a subformula may be in the scope of infinitely many quantifiers. For example, in "L", a single universal or existential quantifier may bind arbitrarily many variables simultaneously. Similarly, the logic "L" permits simultaneous quantification over fewer than λ variables, as well as conjunctions and disjunctions of size less than κ.


title: First-order logic

text: Fixpoint logic extends first-order logic by adding the closure under the least fixed points of positive operators.


title: First-order logic

text: The characteristic feature of first-order logic is that individuals can be quantified, but not predicates. Thus


title: First-order logic

text: is not, in most formalizations of first-order logic. Second-order logic extends first-order logic by adding the latter type of quantification. Other higher-order logics allow quantification over even higher types than second-order logic permits. These higher types include relations between relations, functions from relations to relations between relations, and other higher-type objects. Thus the "first" in first-order logic describes the type of objects that can be quantified.


title: First-order logic

text: Unlike first-order logic, for which only one semantics is studied, there are several possible semantics for second-order logic. The most commonly employed semantics for second-order and higher-order logic is known as "full semantics". The combination of additional quantifiers and the full semantics for these quantifiers makes higher-order logic stronger than first-order logic. In particular, the (semantic) logical consequence relation for second-order and higher-order logic is not semidecidable; there is no effective deduction system for second-order logic that is sound and complete under full semantics.


title: First-order logic

text: Second-order logic with full semantics is more expressive than first-order logic. For example, it is possible to create axiom systems in second-order logic that uniquely characterize the natural numbers and the real line. The cost of this expressiveness is that second-order and higher-order logics have fewer attractive metalogical properties than first-order logic. For example, the Löwenheim–Skolem theorem and compactness theorem of first-order logic become false when generalized to higher-order logics with full semantics.


title: First-order logic

text: Automated theorem proving refers to the development of computer programs that search and find derivations (formal proofs) of mathematical theorems. Finding derivations is a difficult task because the search space can be very large; an exhaustive search of every possible derivation is theoretically possible but computationally infeasible for many systems of interest in mathematics. Thus complicated heuristic functions are developed to attempt to find a derivation in less time than a blind search.


title: First-order logic

text: The related area of automated proof verification uses computer programs to check that human-created proofs are correct. Unlike complicated automated theorem provers, verification systems may be small enough that their correctness can be checked both by hand and through automated software verification. This validation of the proof verifier is needed to give confidence that any derivation labeled as "correct" is actually correct.


title: First-order logic

text: Some proof verifiers, such as Metamath, insist on having a complete derivation as input. Others, such as Mizar and Isabelle, take a well-formatted proof sketch (which may still be very long and detailed) and fill in the missing pieces by doing simple proof searches or applying known decision procedures: the resulting derivation is then verified by a small core "kernel". Many such systems are primarily intended for interactive use by human mathematicians: these are known as proof assistants. They may also use formal logics that are stronger than first-order logic, such as type theory. Because a full derivation of any nontrivial result in a first-order deductive system will be extremely long for a human to write, results are often formalized as a series of lemmas, for which derivations can be constructed separately.


title: First-order logic

text: Automated theorem provers are also used to implement formal verification in computer science. In this setting, theorem provers are used to verify the correctness of programs and of hardware such as processors with respect to a formal specification. Because such analysis is time-consuming and thus expensive, it is usually reserved for projects in which a malfunction would have grave human or financial consequences.


title: First-order logic

text: For the problem of model checking, efficient algorithms are known to decide whether an input finite structure satisfies a first-order formula, in addition to computational complexity bounds: see .


title: Liam Cunningham

text: Liam Cunningham (born 2 June 1961) is an Irish actor. He is known for playing Davos Seaworth in the HBO epic-fantasy series "Game of Thrones".


title: Liam Cunningham

text: Cunningham has been nominated for the London Film Critics' Circle Award, the British Independent Film Award, has won two Irish Film & Television Awards, and shared a BAFTA with Michael Fassbender, for their crime-drama short film "Pitch Black Heist". His film roles include "A Little Princess" (1995), "Jude" (1996), "Dog Soldiers" (2002), "The Crooked Man" (2003), "The Wind That Shakes the Barley" (2006), "Hunger" (2008), "The Escapist" (2008), "Good Vibrations" (2013), "Let Us Prey" (2014) and "The Childhood of a Leader" (2015). In 2020, he was listed at number 36 on "The Irish Times" list of Ireland's greatest film actors.


title: Liam Cunningham

text: Cunningham was born in East Wall, which is an inner city area of the Northside of Dublin. He grew up in Kilmore West with his three sisters and a brother. Cunningham left secondary school at 15 and pursued a career as an electrician. In the 1980s, Cunningham moved to Zimbabwe for three years where he maintained electrical equipment at a safari park and trained Zimbabwean electricians. After returning to Ireland, Cunningham became dissatisfied with his work as an electrician and decided to pursue his interest in acting. He attended acting classes and began to work in local theatre, including the Royal Shakespeare Company. He appeared in a production of "Studs" at The Tricycle Theatre in Kilburn, London.


title: Liam Cunningham

text: Cunningham's debut film role came in "Into the West" (1992), where he played a police officer. His on-screen acting continued with roles in "War of the Buttons" (1994), and "A Little Princess" (1995), before making his role as Phillotson in "Jude" (1996). He continued with character roles in "Falling for a Dancer" (TV, 1998), "RKO 281" (1999), "Shooting the Past" (TV, 1999), "When the Sky Falls" (2000) and "Stranded" (2002). Cunningham came to international prominence with his role as Captain Ryan in the critically acclaimed, independent horror film, "Dog Soldiers" (2002).


title: Liam Cunningham

text: Cunningham starred in well-received films such as "The Wind That Shakes the Barley" (2006) which won the Palme d'Or at the Cannes Film Festival; "Hunger"; "The Escapist" (both 2008); "The Guard"; and "Black Butterflies" (both 2011). He also had roles in many high budget British and American films including "The League of Gentlemen's Apocalypse" (2005), "" (2008), "Harry Brown" (2009), "Clash of the Titans", "Centurion" (both 2010). On television, he appeared as President "Richard Tate" in the BBC programme "Outcasts". Cunningham was producer Philip Segal’s first choice to portray the Eighth Doctor in the TV movie of "Doctor Who" (1996), but was vetoed by Fox executives.


title: Liam Cunningham

text: In 2012, Cunningham joined the main cast for the second season of HBO's "Game of Thrones" portraying former smuggler Davos Seaworth, and in 2013 he starred in "The Numbers Station" alongside John Cusack. He was also cast in season 5 of the BBC series "Merlin" as a sorcerer. He featured in the music video for "High Hopes" by Irish alternative rock band Kodaline from their EP "The High Hopes". In April 2013, he appeared in the seventh series of the BBC One series "Doctor Who" in the episode "Cold War", where he played Captain Zhukov, the commander of a Russian submarine in 1983 facing one of the Ice Warriors. He guest starred in the second season of the VH1 television series "Stay Closer", with Sandra Bullock and Jessica Chastain. In 2015, he played the father in Brady Corbet's directorial debut film, "The Childhood of a Leader".


title: Liam Cunningham

text: In February 2020, it was announced that Cunningham would voice Man-At-Arms in the Netflix animated series "".


title: Liam Cunningham

text: Cunningham resides in Dublin with his wife Colette, with whom he has three children, daughter Ellen and sons Liam Jr. and Sean.


title: Liam Cunningham

text: In 2015, Cunningham was one of over 100 artists who signed a letter to "The Guardian" announcing support for a cultural boycott of Israel.


title: Wars of the Three Kingdoms

text: The Wars of the Three Kingdoms were a series of related conflicts fought between 1639 and 1653 in the kingdoms of England, Scotland and Ireland, then separate entities united in a personal union under Charles I. They include the 1639 to 1640 Bishops' Wars, the First and Second English Civil Wars, the Irish Confederate Wars, the Cromwellian conquest of Ireland and the Anglo-Scottish war (1650–1652). They resulted in victory for the Parliamentarian army, the execution of Charles I, the abolition of monarchy, and founding of the Commonwealth of England, a Unitary state which controlled the British Isles until the Stuart Restoration in 1660.


title: Wars of the Three Kingdoms

text: Political and religious conflict between Charles I and his opponents dated to the early years of his reign. While the vast majority supported the institution of monarchy, they disagreed on who held ultimate authority. Royalists (or 'Cavaliers') generally argued political and religious bodies were subordinate to the king, while most of their Parliamentarian (or 'Roundhead') opponents backed a limited form of constitutional monarchy and opposed the Personal Rule Charles had imposed. This was worsened by differences over religion and religious freedom. Reformed Protestants such as the English Puritans and Scottish Covenanters opposed the changes Charles tried to impose on the Protestant state churches of England and Scotland. In Ireland, the only one with a Catholic majority, the Irish Confederates wanted an end to anti-Catholic discrimination, greater self-governance, and a reversal of land grants to Protestant settlers.


title: Wars of the Three Kingdoms

text: The conflicts began with the Bishops' Wars of 1639–1640, when Scottish Covenanters who opposed Charles' religious reforms gained control of Scotland and briefly occupied northern England. Irish Catholics launched a rebellion in 1641, which developed into ethnic conflict with Protestant settlers. The Irish Catholic Confederation was formed to control the rebellion, and in the ensuing war it held most of Ireland against the Royalists, Parliamentarians and Covenanters. Although all three agreed on the need to quell the rebellion, none trusted the other two with control of an army raised to do so. In August 1642, failure to break the resulting political deadlock sparked the First English Civil War, which pitted Royalists against Parliamentarians and their Covenanter allies in England and Wales.


title: Wars of the Three Kingdoms

text: The war in England ended when Charles surrendered to the Scots in 1646, but divisions among his opponents and his refusal to make significant political concessions caused a renewed outbreak of fighting in 1648. In the Second English Civil War, Parliamentarians again defeated the Royalists and a Covenanter faction called the Engagers. The Parliamentarian New Model Army then purged England's parliament of those who wanted to continue negotiations with the king. The resulting Rump Parliament approved his execution in January 1649 and founded the republican Commonwealth of England. In the Treaty of Breda, the Scots agreed to restore Charles II to the English throne, but were defeated in the 1650–1652 Anglo-Scottish war. Under Oliver Cromwell, the Commonwealth conquered Ireland and most Irish Catholic lands were seized. The British Isles became a united republic ruled by Cromwell and dominated by the army. There were sporadic uprisings until the monarchy was restored in 1660.


title: Wars of the Three Kingdoms

text: The term "Wars of the Three Kingdoms" first appears in "A Brief Chronicle of all the Chief Actions so fatally Falling out in the three Kingdoms" by James Heath, published in 1662, but historian Ian Gentles argues "there is no stable, agreed title for the events...which have been variously labelled the Great Rebellion, the Puritan Revolution, the English Civil War, the English Revolution and… the Wars of the Three Kingdoms." It is generally used by modern historians who see the conflicts in each state as driven by overlapping but often distinct issues, rather than as mere background to the "English Civil War", while others have labelled them the "British Civil Wars".


title: Wars of the Three Kingdoms

text: After 1541, monarchs of England styled their Irish territory as a Kingdom—replacing the —and ruled there with the assistance of a separate Irish Parliament. Also, with the Laws in Wales Acts 1535 and 1542, Henry VIII integrated Wales more closely into the Kingdom of England. Scotland, the third separate kingdom, was governed by the House of Stuart.


title: Wars of the Three Kingdoms

text: By means of the English Reformation, King Henry VIII made himself head of the Protestant Church of England and outlawed Catholicism in England and Wales. In the course of the 16th century Protestantism became intimately associated with national identity in England; Catholicism had come to be seen as the national enemy, particularly as it was embodied in the rivals France and Spain. Catholicism, however, remained the religion of most people in Ireland and for many Irish it was a symbol of native resistance to the Tudor conquest of Ireland.


title: Wars of the Three Kingdoms

text: In the Kingdom of Scotland, the Protestant Reformation was a popular movement led by John Knox. The Scottish Parliament legislated for a national Presbyterian church—namely the Church of Scotland or the "Kirk"—and Mary, Queen of Scots, a Catholic, was forced to abdicate in favour of her son James VI of Scotland. James grew up under a regency disputed between Catholic and Protestant factions; when he took power he aspired to be a "universal King" favouring the English Episcopalian system of bishops appointed by the king. In 1584, he introduced bishops into the Church of Scotland, but met with vigorous opposition, and he had to concede that the General Assembly of the Church of Scotland would continue to run the church.


title: Wars of the Three Kingdoms

text: The personal union of the three kingdoms under one monarch came about when King James VI of Scotland succeeded Elizabeth I to the English throne in 1603, when he also became King James I of England and of Ireland. In 1625, Charles I succeeded his father, and marked three main concerns regarding England and Wales; how to fund his government, how to reform the church, and how to limit (the English) Parliament's interference in his rule. At that time he showed little interest in his other two kingdoms, Scotland and Ireland.


title: Wars of the Three Kingdoms

text: James VI remained Protestant, taking care to maintain his hopes of succession to the English throne. He duly became James I of England in 1603 and moved to London. James concentrated on dealing with the English Court and Parliament, running Scotland through written instructions to the Privy Council of Scotland and controlling the Parliament of Scotland through the Lords of the Articles. He constrained the authority of the General Assembly of the Church of Scotland and stopped it from meeting, then increased the number of bishops in the Church of Scotland. In 1618 he held a General Assembly and pushed through "Five Articles" of Episcopalian practices, which were widely boycotted.


title: Wars of the Three Kingdoms

text: After his death in 1625, James was succeeded by his son Charles I, who was crowned in St Giles' Cathedral, Edinburgh, in 1633, with full Anglican rites. Charles was less skillful and restrained than his father; his attempts to enforce Anglican practices in the Church of Scotland created opposition which reached a flashpoint when he introduced the Anglican Book of Common Prayer. His confrontation with the Scots came to a head in 1639, when he tried and failed to coerce Scotland by military means during the Bishops' Wars.


title: Wars of the Three Kingdoms

text: Charles shared his father's belief in the Divine Right of Kings, and his persistent assertion of this standard seriously disrupted relations between the Crown and the English Parliament. The Church of England remained dominant, but a powerful Puritan minority, represented by about one third of Parliament, began to assert themselves; their religious precepts had much in common with the Presbyterian Scots.


title: Wars of the Three Kingdoms

text: The English Parliament and the king had repeated disputes over taxation, military expenditure, and the role of the Parliament in government. While James I had held much the same opinions as his son regarding Royal Prerogatives, he usually had enough discretion and charisma to persuade Parliamentarians to accept his thinking. Charles had no such skill and, faced with multiple crises during 1639–1642, he failed to prevent his kingdoms from sliding into civil war. When Charles approached Parliament to pay for a campaign against the Scots, they refused. They then declared themselves to be permanently in session—the Long Parliament—and soon presented Charles with a long list of civil and religious grievances requiring his remedy before they would approve any new legislation.


title: Wars of the Three Kingdoms

text: During the English Civil War, the English overseas possessions became highly involved. In the Channel Islands, the island of Jersey and Castle Cornet in Guernsey supported the King until a surrender with honour in December 1651.


title: Wars of the Three Kingdoms

text: Although the newer, Puritan settlements in North America, notably Massachusetts, were dominated by Parliamentarians, the older colonies to the south sided with the Crown. Friction between Royalists, most of whom were Anglican, and Puritans in Maryland came to a head in the Battle of the Severn. The Virginia Company's settlements, Bermuda and Virginia, as well as Antigua and Barbados, were conspicuous in their loyalty to the Crown. Bermuda's Independent Puritans were expelled, settling the Bahamas under William Sayle as the Eleutheran Adventurers. Parliament passed An Act for prohibiting Trade with the Barbadoes, Virginia, Bermuda and Antego in October, 1650, which stated that


title: Wars of the Three Kingdoms

text: The Act also authorised Parliamentary privateers to act against English vessels trading with the rebellious colonies:


title: Wars of the Three Kingdoms

text: Far to the North, Bermuda's regiment of Militia and its coastal batteries prepared to resist an invasion that never came. Built-up inside the natural defence of a nearly impassable barrier reef, to fend off the might of Spain, these defences would have been a formidable obstacle for the Parliamentary fleet sent in 1651 under the command of Admiral Sir George Ayscue to subdue the trans-Atlantic colonies, but after the fall of Barbados, the Bermudians made a separate peace that respected the internal status quo. The Parliament of Bermuda avoided the Parliament of England's fate during The Protectorate, becoming one of the oldest continuous legislatures in the world.


title: Wars of the Three Kingdoms

text: Virginia's population swelled with Cavaliers during and after the English Civil War. Even so, Virginia Puritan Richard Bennett was made Governor answering to Cromwell in 1652, followed by two more nominal "Commonwealth Governors". The loyalty of Virginia's Cavaliers to the Crown was rewarded after the 1660 Restoration of the Monarchy when Charles II dubbed it the "Old Dominion".


title: Wars of the Three Kingdoms

text: Meanwhile, in the Kingdom of Ireland (proclaimed such in 1541 but only fully conquered for the Crown in 1603), tensions had also begun to mount. Thomas Wentworth, Charles I's Lord Deputy of Ireland, angered Catholics by enforcing new taxes while denying them full rights as subjects; he further antagonised wealthy Irish Catholics by repeated initiatives to confiscate and transfer their lands to English colonists. Conditions became explosive in 1639 when Wentworth offered Irish Catholics some reforms in return for their raising and funding an Irish army (led by Protestant officers) to put down the Scottish rebellion. The idea of an Irish Catholic army enforcing what many saw as already tyrannical government horrified both the Scottish and the English Parliaments, which in response threatened to invade Ireland.


title: Wars of the Three Kingdoms

text: Modern historians have emphasised the lack of inevitability of the civil wars, noting that the sides resorted to "violence first" in situations marked by mutual distrust and paranoia. Charles' initial failure to end the Bishops' Wars of 1639 and 1640 quickly persuaded the antagonists that force could serve them better than negotiation. The imposition of Bishops and other Anglican practices to the Scottish Kirk were opposed by most Scots, who supported a Presbyterian system of governance led by a General Assembly, and in individual churches by ministers and committees of elders. The 1638 National Covenant pledged to oppose such imposed "innovations." Signatories were known as Covenanters.


title: Wars of the Three Kingdoms

text: In Ireland, alienated by Church of England domination and frightened by the rhetoric of the English and Scottish Parliaments, a small group of Irish conspirators launched the Irish Rebellion of 1641, ostensibly in support of the "King's Rights." The uprising featured widespread violent assaults on Protestant communities in Ireland, both Anglican and dissenter Protestants in Ulster whose practice was similar to the Scottish Kirk. In England and Scotland, rumours spread that the killings had the king's sanction, which, for many, foreshadowed their own fate if the king's Irish troops landed in Britain. Thus the English Parliament refused to pay for a royal army to put down the rebellion in Ireland; instead Parliament decided to raise its own armed forces. The king did likewise, rallying those Royalists (some of them members of Parliament) who believed their fortunes were best served by loyalty to the king.


title: Wars of the Three Kingdoms

text: The English Civil War ignited in 1642. Scottish Covenanters (as Presbyterians there called themselves) joined forces with the English Parliament in late 1643 and played a major role in the ultimate Parliamentary victory. Over the course of more than two years, the king's forces were ground down by the efficiency of those of Parliament, including the New Model Army, backed as they were by the financial muscle of the City of London. On 5 May 1646, at Southwell, Charles I surrendered to the Scottish army besieging Newark-on-Trent. What remained of the English and Welsh Royalist armies and garrisons surrendered piecemeal over the next few months.


title: Wars of the Three Kingdoms

text: Meanwhile, the rebellious Irish Catholics formed their own government—Confederate Ireland—intending to help the Royalists in return for religious toleration and political autonomy. Troops from England and Scotland fought in Ireland, and Irish Confederate troops mounted an expedition to Scotland in 1644, sparking the Scottish Civil War. There, the Royalists gained a series of victories in 1644–1645, but were crushed after the main Covenanter armies returned to Scotland upon the end of the first English Civil War.


title: Wars of the Three Kingdoms

text: The Scots handed Charles over to the English and returned to Scotland, the English Parliament having paid them a large sum for their expenses in the English campaign. After his surrender, Charles was approached by the Scots, the Presbyterians in the English Parliament, and the "Grandees" of the New Model Army, all attempting to reach an accommodation with him and among themselves which would achieve peace while preserving the crown. But now, a breach between the New Model Army and Parliament widened day by day, until the Puritans in Parliament, with allies among the Scots and the remaining Royalists, saw themselves strong enough to challenge the Army, which began the Second English Civil War.


title: Wars of the Three Kingdoms

text: The New Model Army vanquished the English Royalist and Parliamentarians, as well as their Scottish Engager allies. On account of his secret machinations with the Scottish Engagers, Charles was charged with treason against England. Subsequently, the Grandees and their civilian supporters failed to reconcile with the king or the Puritan majority in Parliament. The Grandees acted and soldiers were used to purge the English Parliament of those who opposed the Army. The resultant Rump Parliament of the Long Parliament then passed enabling legislation for putting Charles I on trial for treason. He was found guilty of treason against the English commons and was executed on 30 January 1649.


title: Wars of the Three Kingdoms

text: After the execution of King Charles I the Rump Parliament passed a series of acts declaring that England was a republic; that the House of Commons—without the House of Lords—would sit as the legislature; and that a Council of State would act as the executive power. In the other two kingdoms the execution of Charles caused the warring parties to unite, and they recognised Charles II as king of Great Britain, France and Ireland, which would lead to a Third English Civil War.


title: Wars of the Three Kingdoms

text: To deal with the threat to the English Commonwealth posed by the two kingdoms (Ireland and Scotland), the Rump Parliament first appointed Cromwell to invade and subdue Ireland. In August 1649, he landed an English army at Rathmines shortly after the Siege of Dublin was abandoned by the Royalists following the Battle of Rathmines. Then, in late May 1650, Cromwell left one army to continue the Irish conquest and returned to England and to take command of a second English army preparing to invade Scotland. On 3 September 1650, he defeated the Scottish Covenanters at the Battle of Dunbar and his forces then occupied Edinburgh and Scotland south of the River Forth. Cromwell was advancing the bulk of his army over the Forth towards Stirling, when Charles II, commanding a Scottish Royalist army, stole a march on the English commander, and invaded England from his base in Scotland. Cromwell divided his forces, leaving part in Scotland to complete the conquest there, then led the rest south in pursuit of Charles II.


title: Wars of the Three Kingdoms

text: The Royalist army failed to gather much support from English Royalists as it moved south into England; so, instead of heading directly towards London and certain defeat, Charles aimed for Worcester hoping that Wales and the West and Midlands of England would rise against the Commonwealth. This did not happen and, one year to the day after the Battle of Dunbar, the New Model Army and the English militia regiments vanquished the last Royalist army of the English Civil War at the Battle of Worcester, on 3 September 1651. It was the last and most decisive battle in the Wars of the Three Kingdoms.


title: Wars of the Three Kingdoms

text: Having defeated all organised opposition, the Grandees of the Parliamentary New Model Army and their civilian supporters dominated the politics of all three nations for the next nine years (see Interregnum (1649–1660)). As for England, the Rump Parliament had already decreed it was a republic and a Commonwealth. Ireland and Scotland were now subjugated and ruled by military governors, and constituent representatives from both nations were seated in the Rump Parliament of the Protectorate, where they were dominated by Oliver Cromwell, the Lord Protector. When Cromwell died in 1658, control of the Commonwealth became unstable. In early 1660, General George Monck, commanding English occupation forces in Scotland, ordered his troops from the Coldstream barracks, marched them south into England, and seized control of London by February 1660. There he accumulated allies and agreements among the English and London establishments, including the newly constituted Convention Parliament, to which he was elected a member. Monck, first a Royalist campaigner, then a Parliamentary soldier, now contrived the Restoration of the monarchy. Monck arranged that the Convention Parliament would invite Charles II to return as king of the three realms—which was done by act of Parliament on 1 May 1660.


title: Wars of the Three Kingdoms

text: The Wars of the Three Kingdoms prefigured many of the changes that ultimately would shape modern Britain but, in the short term, these conflicts in fact resolved little for the kingdoms and peoples. The English Commonwealth did achieve a notable compromise between the monarchy and republic which survived destabilising problems for nearly the next two hundred years. In practice, Oliver Cromwell exercised political power through his control over Parliament's military forces, but his legal position—and provisions for his succession—remained unclear, even after he became Lord Protector. None of the several constitutions proposed during this period was achieved. Thus the Commonwealth and Protectorate of the Parliamentarians—the wars' victors—left no significant new form of government in place after their time.


title: Wars of the Three Kingdoms

text: Still, in the long term, two abiding legacies of British democracy were established during this period:


title: Wars of the Three Kingdoms

text: English Protestants experienced religious freedom during the Interregnum, but there was none for English Catholics. During the term of their control, the Puritan partisans abolished the Church of England and the House of Lords. Cromwell and dissolved it by force, but he failed to establish an acceptable alternative. Nor did he and his supporters move in the direction of popular democracy, as the more radical Parliamentarians (the Levellers) wanted.


title: Wars of the Three Kingdoms

text: During the Interregnum, the New Model Army occupied Ireland and Scotland. In Ireland, the new government confiscated almost all lands belonging to Irish Catholics as punishment for the rebellion of 1641; harsh Penal Laws also restricted this community. Thousands of Parliamentarian soldiers settled in Ireland on confiscated lands. The Commonwealth abolished the Parliaments of Ireland and Scotland. In theory, these countries had representation in the English Parliament, but as this body never held real powers, representation was ineffective. When Cromwell died in 1658 the Commonwealth fell apart—but without major violence. Historians record that adroit politicians of the time, especially George Monck, prevailed over the looming crisis; Monck in particular was deemed the victor "sine sanguine," i.e., "without blood", of the Restoration crisis. And in 1660, Charles II was restored as king of England, Scotland, and Ireland.


title: Wars of the Three Kingdoms

text: Under the English Restoration the political system returned to the prewar constitutional position. Although Charles II's Declaration of Breda, April 1660, offering reconciliation and forgiveness, promising a general pardon for crimes committed during the English Civil War, the new régime executed or imprisoned for life those directly involved in the regicide of Charles I. Royalists dug up Cromwell's corpse and performed a posthumous execution. Those religiously and politically motivated individuals held responsible for the wars suffered harsh repression. Scotland and Ireland regained their Parliaments, some Irish retrieved confiscated lands, and the New Model Army was disbanded. However, the issues which had caused the wars—religion, the powers of Parliament vis-á-vis the king, and the relationships between the three kingdoms—remained unresolved or, more accurately, postponed, only to re-emerge as matters disputed again and leading to the Glorious Revolution of 1688. Only later did the broader features of modern Britain foreshadowed in the civil wars emerge permanently, namely: a Protestant constitutional monarchy and a strong standing army under civilian control.


title: Sonic boom

text: A sonic boom is a sound associated with shock waves created when an object travels through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding similar to an explosion or a thunderclap to the human ear. A decibel is the primary unit measurement of sound. "A thunderclap is incredibly loud, producing levels between 100 and 120 dBA (decibels A)- the equivalent of standing near a jet during take-off."


title: Sonic boom

text: The crack of a supersonic bullet passing overhead or the crack of a bullwhip are examples of a sonic boom in miniature.


title: Sonic boom

text: Sonic booms due to large supersonic aircraft can be particularly loud and startling, tend to awaken people, and may cause minor damage to some structures. This led to prohibition of routine supersonic flight overland. Although they cannot be completely prevented, research suggests that with careful shaping of the vehicle, the nuisance due to the sonic booms may be reduced to the point that overland supersonic flight may become a feasible option.


title: Sonic boom

text: A sonic boom does not occur only at the moment an object crosses the sound barrier and neither is it heard in all directions emanating from the supersonic object. Rather, the boom is a continuous effect that occurs while the object is travelling at supersonic speeds and affects only observers that are positioned at a point that intersects a region in the shape of a geometrical cone behind the object. As the object moves, this conical region also moves behind it and when the cone passes over the observer, they will briefly experience the "boom".


title: Sonic boom

text: When an aircraft passes through the air, it creates a series of pressure waves in front of the aircraft and behind it, similar to the bow and stern waves created by a boat. These waves travel at the speed of sound and, as the speed of the object increases, the waves are forced together, or compressed, because they cannot get out of each other's way quickly enough. Eventually they merge into a single shock wave, which travels at the speed of sound, a critical speed known as "Mach 1", and is approximately at sea level and .


title: Sonic boom

text: In smooth flight, the shock wave starts at the nose of the aircraft and ends at the tail. Because the different radial directions around the aircraft's direction of travel are equivalent (given the "smooth flight" condition), the shock wave forms a Mach cone, similar to a vapour cone, with the aircraft at its tip. The half-angle formula_1 between the direction of flight and the shock wave is given by:


title: Sonic boom

text: where formula_3 is the inverse formula_4 of the plane's Mach number (formula_5). Thus the faster the plane travels, the finer and more pointed the cone is.


title: Sonic boom

text: There is a rise in pressure at the nose, decreasing steadily to a negative pressure at the tail, followed by a sudden return to normal pressure after the object passes. This "overpressure profile" is known as an N-wave because of its shape. The "boom" is experienced when there is a sudden change in pressure; therefore, an N-wave causes two booms – one when the initial pressure-rise reaches an observer, and another when the pressure returns to normal. This leads to a distinctive "double boom" from a supersonic aircraft. When the aircraft is maneuvering, the pressure distribution changes into different forms, with a characteristic U-wave shape.


title: Sonic boom

text: Since the boom is being generated continually as long as the aircraft is supersonic, it fills out a narrow path on the ground following the aircraft's flight path, a bit like an unrolling red carpet, and hence known as the "boom carpet". Its width depends on the altitude of the aircraft. The distance from the point on the ground where the boom is heard to the aircraft depends on its altitude and the angle formula_6.


title: Sonic boom

text: For today's supersonic aircraft in normal operating conditions, the peak overpressure varies from less than 50 to 500 Pa (1 to 10 psf (pound per square foot)) for an N-wave boom. Peak overpressures for U-waves are amplified two to five times the N-wave, but this amplified overpressure impacts only a very small area when compared to the area exposed to the rest of the sonic boom. The strongest sonic boom ever recorded was 7,000 Pa (144 psf) and it did not cause injury to the researchers who were exposed to it. The boom was produced by an F-4 flying just above the speed of sound at an altitude of . In recent tests, the maximum boom measured during more realistic flight conditions was 1,010 Pa (21 psf). There is a probability that some damage—shattered glass, for example—will result from a sonic boom. Buildings in good condition should suffer no damage by pressures of 530 Pa (11 psf) or less. And, typically, community exposure to sonic boom is below 100 Pa (2 psf). Ground motion resulting from sonic boom is rare and is well below structural damage thresholds accepted by the U.S. Bureau of Mines and other agencies.


title: Sonic boom

text: The power, or volume, of the shock wave depends on the quantity of air that is being accelerated, and thus the size and shape of the aircraft. As the aircraft increases speed the shock cone gets "tighter" around the craft and becomes weaker to the point that at very high speeds and altitudes no boom is heard. The "length" of the boom from front to back depends on the length of the aircraft to a power of 3/2. Longer aircraft therefore "spread out" their booms more than smaller ones, which leads to a less powerful boom.


title: Sonic boom

text: Several smaller shock waves can and usually do form at other points on the aircraft, primarily at any convex points, or curves, the leading wing edge, and especially the inlet to engines. These secondary shockwaves are caused by the air being forced to turn around these convex points, which generates a shock wave in supersonic flow.