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


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:


 * (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).



Thunder generation
"Thunder can be defined as the acoustic emission associated with a lightning discharge".

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
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). 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.



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. 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.

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) 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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