# Engineering Acoustics/Thunder acoustics

**Thunder** is defined as the sound signature associated to the shock
wave produced after a lightning discharge. Thunder has intrigued and
frightened humans for millennia. Early explanations of this phenomenon
included fights between Zeus and his subordinates in Greek mythology, or
the collision of clouds in the sky by Aristotle. It was only in the late
1800s that the true physical causes were discovered by the scientific
community i.e. the heating of a narrow channel to \~24000K. The air
molecules within ionize to generate a highly powerful shock wave that
can be heard over distances up to 25 km, depending on the wind.

## Lightning fundamentals

!CGtypes\|400px\|Figure 2. The 4 groups of cloud-to-ground (CG)
discharges (adapted from *Lightning: Physics and Effects*
(2003)). discharges (adapted from Lightning: Physics and Effects (2003)).")

There are two main types of lightning discharges: cloud-to-ground (CG)
and intra-cloud (IC), the latter accounting for about 3/4 of all
discharges (there are however other types of discharges that are less
commonly encountered such as ball
lightning).

The CGs can be categorized into 4 groups:[^1]

:   \(a\) downward negative (90% of all CGs are of this group),
:   \(b\) upward positive,
:   \(c\) downward positive, and
:   \(d\) upward negative.

The charges have three main modes in which they can be sent to the
ground from the cloud (see diagram):

1.  Dart-leader-return stroke sequences.
2.  Continuing currents (that last for hundreds of milliseconds) which
    are long quasi-stationary arcs.
3.  M-components (named after D.F. Malan who first studied these
    processes in the 1930s) which are transient processes during a
    continuing current (2nd mode).

!modesoftransfer\|400px\|Figure 3. The three main modes of charge
transfer to ground (adapted from *Lightning: Physics and Effects*
(2003)).).")

## Thunder generation

\"Thunder can be defined as the acoustic emission associated with a
lightning discharge\".[^2]


### Types

All processes in CG and IC discharges produce thunder which can be
divided into 2 categories:

-   audible (frequencies greater than 20 Hz) which come from a series of
    decaying shockwaves produced by the expansion of various portions of
    the nearly instantaneous heated lightning channel which is filled
    with ionized air molecules
    (plasma)). Of these
    there are: a) rumbling thunder which is a long low series of
    growling-like sounds and b) clapping thunder that is usually loud
    and quick.
-   non-audible or infrasonic thunder (frequencies lower than 20 Hz)
    which is thought to originate from ICs where large volumes of air
    are displaced by the rapid removal of electrons or protons from the
    cloud itself. This category of thunder has only recently received
    the attention of the scientific community.

### Maximum amplitude frequency

!Figure 4. The frequency spectrum of thunder (and unfiltered rain
sounds) obtained using Audacity 1.3
(Beta). obtained using Audacity 1.3 (Beta)."){width="300"}
It has been empirically found that the loudest frequency in thunder is:

$$f_{\rm max} = c_0 \left( \frac{p_0}{E_0} \right)^\frac{1}{2}$$ where
$c_0$ is the speed of sound, $p_0$ is the ambient pressure, and $E_0$ is
the energy per unit length of lightning channel which is defined as:

$E_0 = \frac{1}{\pi {R_0}^2} \int\limits_{0}^{t_{\rm disch}}\rho I^2\,dt$

where $R_0$ is the initial channel radius, $\rho$ is the resistivity of
the plasma and $t_{\rm disch}$ is the discharge duration. The values of
$E_0$ have been found to vary around 50 kJ/m.

### A.A. Few\'s Model of thunder generation

It is widely accepted that audible thunder is generated by the lightning
channel and the subsequent shock wave that travels extremely rapidly
(\~3000 m/s).[^3] A.A. Few provides a experimentally-proved thunder
generation mechanism.

#### Assuming perfectly cylindrical/spherical expansion

The shock wave time history can be cut into three intervals: the first
consists of a strong shock with an extremely high pressure ratio across
the boundary. The second section is a weak shock that travels at a
relatively slower pace. And finally the third section of the shockwave
is the acoustic wave that propagates at 343 m/s i.e. the speed of sound
at 293K.

The distance traveled by the strong shock wave before it turns into a
weak shock can be found by performing a work-energy balance on the fluid
that has been compressed by the strong shock (i.e. work is done on the
fluid by volume and pressure changes). A so-called relaxation radius,
$R_s$ (for spherical shock waves or $R_c$ for cylindrical shock waves)
can thus be defined to account for the distance traveled by the strong
shock:

Spherical relaxation radius:

$$R_s=\left(\frac{3E_t}{4 \pi p_0}\right)^\frac{1}{3}$$

Cylindrical relaxation radius:

$$R_c=\left(\frac{E_0}{\pi p_0}\right)^\frac{1}{2}$$

where $E_t$ is the total energy released by the spherical shock wave. It
is in this last section of the shock wave that the thunder is heard.

Most studies that were done on thunder could not analyze from close
range a naturally produced sound. This is because of the impossibility
to predict exactly where lightning will strike and thus to place a
microphone near it. The most common way of artificially generating
lightning is with a rocket, that is connected to a steel wire and fired
into a thundercloud to create a short circuit near ground. This
\"forces\" an electrical current from the electrically-charged
thundercloud to the ground via the attached electrical wire.

This type of discharge is commonly called artificially triggered
lightning, and was used, among others, by Depasse in 1986, 1990 and 1991
at Saint-Privat D\'Allier, France, where the pressure profile behind a
lighting-generated shockwave was matched to the theoretical profile
obtained from cylindrical shockwave theory developped by Few in
1969.[^4]

!Figure 5. The different sizes of tortuous segments in a lightning
rod.-3.png "Figure 5. The different sizes of tortuous segments in a lightning rod."){width="400"}

#### Effect of tortuosity on rumbling or clapping thunder

Lightning channels are not straight channels with perfectly circular gas
dynamic expansion and hence the tortuosity of these must be accounted
for. It was for this reason that A. A. Few derived three different
levels of tortuosity in lightning rods. Macro, meso and micro-tortuous
segments can be qualitatively observed in any CG discharge. It was found
that macro- and meso-tortuous segments are very important in organizing
the pulses and acoustic behaviour of a CG discharge.[^5] Through
computational studies, it was found that 80% of the acoustic energy is
released at a 30 degree angle of the plane perpendicular to the main axe
of a macro tortuous discharge. It is in this region that an observer
will hear a loud clap sort of thunder whereas one placed outside this
region will hear a rumbling sort of thunder.

## Thunder propagation

!Thunder propagation
schematics.{width="400"}

### Attenuation

There are two types of attenuation effects. The first is due to the
finite amplitude of the propagating acoustic waves which cause a
non-negligeable amount of stretching to the wave. There is a so-called
\"eroding\" effect which tries to break down the sudden pressure jump at
the wave front into a more rounded profile. This is a form of
dissipation due to visco-thermal losses that affects the higher
frequencies and so explains why only lower frequencies are heard when
lightning strikes 1 or more kilometres away. The second form of
attenuation is due to the scattering and aerosol effect left by the rain
drops and thunderclouds (filled with water vapour) commonly found in
most lightning conditions. These micro particles also attenuate the
higher frequencies of a thunder clap or rumble. See
shockwave and
detonation wikis for more
information about the decay of a strong shock.

### Environment

Recalling that the speed of sound, c, is dependent on the density of the
medium it is thus likely that, depending on the conditions surrounding
the lightning rod such as the air composition, atmospheric pressure, the
thunder will travel at a unique velocity, pitch, frequency band and
duration depending on the characteristics of the lightning rod. Indeed,
as shown in the study by Blanco et al. (2009) [^6] the geometry plays a
vital role in the perceived resulting sound. Furthermore, there is a
level of attenuation that must be accounted for as the sound travels
through the atmosphere and ground obstacles (such as trees, buildings,
bridges, land).

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## References


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