Spacetime triangle diagram technique

In physics and mathematics, the spacetime triangle diagram (STTD) technique, 
also known as the Smirnov method of incomplete separation of variables, is the direct space-time domain method for electromagnetic and scalar wave motion.
STTD versus Green's function technique.
The STTD technique belongs to the second among the two principal ansätze for theoretical treatment of waves — the frequency domain and the direct spacetime domain. 
The most well-established method for the inhomogeneous (source-related) descriptive equations of wave motion is one based on the Green's function technique. For the circumstances described in Section 6.4 and Chapter 14 of Jackson's "Classical Electrodynamics", it can be reduced to calculation of the wave field via retarded potentials (in particular, the Liénard–Wiechert potentials).
Despite certain similarity between Green's and Riemann–Volterra methods (in some literature the Riemann function is called the Riemann–Green function ), their application to the problems of wave motion results in distinct situations: 
and it was the Riemann–Volterra representation that Smirnov used in his "Course of Higher Mathematics" to prove the uniqueness of the solution to the above problem (see, item 143).
are invoked. The Riemann-Volterra approach presents the same or even more serious difficulties, especially when one deals with the bounded-support sources: here the actual limits of integration must be defined from the system of inequalities involving the space-time variables and parameters of the source term. However, this definition can be strictly formalized using the spacetime triangle diagrams. Playing the same role as the Feynman diagrams in particle physics, STTDs provide a strict and illustrative procedure for definition of areas with the same analytic representation of the integration domain in the 2D space spanned by the non-separated spatial variable and time.
Most important concretizations.
General considerations.
Several efficient methods for scalarizing electromagnetic problems in the orthogonal coordinates formula_2 were discussed by Borisov in Ref. 
The most important conditions of their applicability are formula_3 and 
formula_4, where formula_5 are the 
metric (Lamé) coefficients (so that the squared length element is formula_6). Remarkably, this condition is met for the majority of practically important coordinate systems, including the Cartesian, general-type cylindrical and spherical ones.
For the problems of wave motion is free space, the basic method of separating spatial variables is the application of integral transforms, while for the problems of wave generation and propagation in the guiding systems the variables are usually separated using expansions in terms of the basic functions (modes) meeting the required boundary conditions at the surface of the guiding system.
Cartesian and cylindrical coordinates.
In the Cartesian formula_7 and general-type cylindrical coordinates 
formula_8 separation of the spatial variables result in the initial value problem for a hyperbolic PDE known as the 1D Klein–Gordon equation (KGE)
Here formula_10 is the time variable expressed in units of length using some characteristic velocity (e.g., speed of light or sound),
formula_11 is a constant originated from the separation of variables, and formula_12 represents a part of the source
term in the initial wave equation that remains after application of the variable-separation procedures (a series coefficient or a result of
an integral transform).
The above problem possesses known Riemann function
where formula_14 is the Bessel function of the first kind of order zero.
Passing to the canonical variables formula_15 one gets the simplest STTD diagram reflecting straightforward application of the Riemann–Volterra method, with the fundamental integration domain represented by spacetime triangle "MPQ" (in dark grey).
Rotation of the STTD 45° counter clockwise yields more common form of the STTD in 
the conventional spacetime formula_16.
For the homogeneous initial conditions the (unique) solution of the problem is given by the Riemann formula
Evolution of the wave process can be traced using a fixed observation point (formula_18) successively increasing the triangle height (formula_10) or, alternatively, taking "momentary picture" of the wavefunction formula_20 by shifting the spacetime triangle along the formula_21 axis (formula_22).
More useful and sophisticated STTDs correspond to pulsed sources whose support is limited in spacetime. Each limitation produce specific modifications in the STTD, resulting to smaller and more complicated integration domains in which the
integrand is essentially non-zero. Examples of most common modifications and their combined actions are illustrated below.
Spherical coordinates.
In the spherical coordinate system — which in view of the General considerations must be represented in the sequence 
formula_23, assuring formula_3 — one can scalarize problems for the transverse electric (TE) or transverse magnetic (TM) waves using the Borgnis functions, Debye potentials or Hertz vectors. Subsequent separation of the angular variables formula_25 via expansion of the initial wavefunction formula_26 and the source
where formula_29 is the associated Legendre polynomial of degree formula_30 and order 
formula_31, results in the initial value problem for the hyperbolic 
Euler–Poisson–Darboux equation
known to have the Riemann function
where formula_34 is the (ordinary) Legendre polynomial of degree formula_30.
Equivalence of the STTD (Riemann) and Green's function solutions.
The STTD technique represents an alternative to the classical Green's function method. Due to uniqueness of the solution to the initial value problem in question, in the particular case of zero initial conditions the Riemann solution provided by the STTD technique must coincide with the convolution of the causal Green's function and the source term.
The two methods provide apparently different descriptions of the wavefunction: e.g., the Riemann function to the Klein–Gordon problem is a Bessel function (which must be integrated, together with the source term, over the restricted area represented by the fundamental triangle "MPQ") while the retarded Green's function to the Klein–Gordon equation is a Fourier transform of the imaginary exponential term (to be integrated over the entire plane formula_36, see, for example, Sec. 3.1. of Ref.
) reducible to
Extending integration with respect to formula_38 to the complex domain, using the residue theorem 
("with the poles formula_39 chosen as 
formula_40
to satisfy the causality conditions") one gets
Using formula 3.876-1 of Gradshteyn and Ryzhik,
the last Green's function representation reduces to the expression
in which 1/2 is the scaling factor of the Riemann formula and formula_14 the Riemann function, while the Heaviside step function formula_45 reduces, for formula_46, the area of integration to the fundamental triangle "MPQ", making the Green's function solution equal to that provided by the STTD technique.