Acoustic Radiation

Definition

Acoustic. One branch of physics which studies sound.

The word acoustic comes from the Greek word akoustikos. “which is related to hearing.”

Sound. It comes from the Latin word sonum: “which is related to the hearing sensation created by perturbation of the material medium (elastic, fluid, solid).” In physics, it is a vibration, generally in a gas, created by expansion and compression of gas molecules. Sound waves propagate in the fluid medium and do not propagate in the vacuum. Sounds can be produced in the atmosphere and oceans by living animals or by structures through interaction with the wind, as, for example, trees murmuring, mountains roaring, river sounds, and waves breaking and can be created by various instruments such as music instruments, microphones, speakers, and transducers and also by instruments developed for remote sensing such as SONAR (Sound Navigation and Ranging), ADCP (Acoustic Doppler Current Profiler), and SODAR (Sound Detection and Ranging or what are called echo sounders for atmosphere and ocean). A sound propagating in a medium is characterized by its speed c:

$$ {c^2} = {{{\partial P}} \left/ {{\partial \rho }} \right.} $$

(1)

where P is the pressure and ρ the density, and ∂ is derived.

In a gas,

$$ c\ = {{\left({{{\gamma P}} \left/ {\rho } \right.} \right)}^{1/2}} $$

(2)

where γ is the heat capacity ratio.

Notice that sound speed in the air for standard conditions of temperature and pressure near the surface is close to 340 m/s, while at the ocean surface it is close to 1,500 m/s, which is faster. This will have an incidence on different ways for acoustic signal processing to be done in the ocean and atmosphere.

Sound or rather a sound wave is a mechanical pressure oscillation, which is generally longitudinally propagating.Period T. It is the signal duration corresponding to the time when the sound wave is reproduced identically.Frequency. f = 1/T (T in s and f in hertz). Frequency audio spectrum (distribution of acoustic energy as function of frequencies can be divided in four zones related to human hearing power: 0–20 Hz infrasound (not audible), 20–300 Hz is low-pitched, 300–6,000 Hz is medium range, 6,000–20,000 Hz is high-pitched, more than 20,000 Hz are ultrasonic sounds (not audible).

Sound amplitude. It corresponds to acoustic pressure fluctuation of the medium Δp (amount of energy in the sound wave) measured at one point of a surface S. It is the ratio of pressure P by the surface element S.

$I = {P \left/ {S} \right.}$ in W/m²

For a spherical acoustic source, the intensity at distance r is

$$ Ir = P{{{\,(\mathrm{ sound}\,\mathrm{ power}\,\mathrm{ of}\,\mathrm{ the}\,\mathrm{ source})}} \left/ {{4\pi\,{r^2}}} \right.} $$

Radiation. It is the way acoustic wave energy radiates and concerns acoustic rays from the acoustic sources through the concerned medium. For example, from a microphone radiating along different directions, we are interested in the radiation diagram corresponding to the knowledge of rays (analogy with optical rays) along different directions.

Introduction

We shall begin to analyze (sound) or (acoustic) waves and the wave equation from which we are able to describe energy propagation and rays. If one compares them to electromagnetic waves, acoustic waves are simpler and can be described by the velocity potential, which is a scalar. We shall apply the principles of acoustic wave radiation to different types of acoustic sources used such as monopolar, dipolar sources and the response of the medium at different distances showing that the acoustic field will change of as the characteristics do so. These considerations are really fundamental for remote sensing with acoustic sounders and they are similar to electromagnetic waves, though propagation equations are different. They give information about distances when field characteristics will present some useful organization properties.

Wave equation and acoustic waves in flows

It corresponds to the study of a plane wave transmitted in a flow by a vibrating plane. It can be, for example, the diaphragm of a microphone vibrating along an axis x.

A plane acoustic wave is a general concept from the physics of waves. It corresponds to a wave where wave fronts (surfaces of the same phase) are infinite planes, perpendicular to the same direction of propagation.

The equation of pressure variation p (or wave propagation equation) is

$$ {{{{\partial^2}p}} \left/ {{\partial {x^2}}} \right.} = \left[ {{1 \left/ {{{c^2}}} \right.}} \right]\,{{{{\partial^2}p}} \left/ {{\partial {t^2}}} \right.} $$

(3)

where, as already said, c is the sound speed.

For a sinusoidal wave, the solution of the equation is

$$ p(x,t) = {p_0}\,\cos \left[ ({{{{2\pi }} \left/ {{T)\left( {t-{x \left/ {c} \right.}} \right)}} \right.}} \right] $$

(4)

with the wave number $k={{{2\pi }} \left/ {{{T_c}={{{2\pi }} \left/ {\lambda } \right.}}} \right.}$ and λ is the wave length.

We get

$$ p(x,t) = {p_0}\,\cos \left( {{{{2\pi\,t}} \left/ {T-kx } \right.}} \right) $$

(5)

where 2π/T is the pulsation ω.

Generally, one uses the complex notation:

$$ p(x,t) = {p_0}\,\exp (\omega t - kx) $$

(6)

A linear relation between the displacement gradient (compressibility) gives p = −κ ∂ζ/∂x (where κ is a coefficient of compressibility) from which we have, solving the sound equation:

$$ {{\partial \zeta }} \left/ {{\partial t = \left( {1 \left/{({\rho_0}c)\exp \left(\,\,j\left(\omega t - kx\right) \right.}\right.}\right)}} \right.$$

(7)

This means that particle velocity is in phase with acoustic pressure.

Other derived definitions are useful parameters for remote sensing techniques as the acoustic impedance Z, which is the ratio between pressure and the complex amplitude of the particle.

To illustrate the interest in using acoustic impedance, in a project for the Titan satellite sounding (Weill and Blanc, 1987), it was suggested to use acoustic impedance from the satellite’s surface to discriminate, by acoustic remote sensing, between solid and liquid surface just before a possible crash of the rocket at the satellite’s surface.

For the flow, it is equal to Zc = ρ₀c.

For the plane wave (or progressive wave) ∀ x, we have Z(x) = Zc.

The condensation of the wave is the spatial derivative −∂ζ/∂x, which corresponds to a relative change of density.

Monopoles, dipoles, and pulsing sphere

If wave propagation Eq. 3 is satisfied, we can work with harmonic solutions of the equation and use wave superposition in the Fourier space. Let us consider one source S of strength q radiating at radial distance r and the solution of the equation for particles radial velocities is

$$ v = \mathrm{ A}\,{{{(\exp {(-jkr) \left/ {{{r^2} + jk\,\exp\,(-jkr))}} \right.}}} \left/ {r} \right.} $$

(8)

A is a constant and the boundary conditions are such that the solution vanishes at infinity. It is important to notice that radiation behavior is different as function of distance.

The acoustic flux Fa of the radial velocity v across a sphere of radius is id v*4πr² (sphere considered at the distance r) is

$$ Fa = 4\pi \mathrm{ A}\,(\exp (-jkr) + jkr\,\exp (-jkr)) $$

(9)

Therefore, when kr is small, the first term of (Eq. 9) predominates and the conditions correspond to what is called the near field (velocity in phase with the source), and when kr is large, we are in the far field conditions (velocity 90° in advance with the source). These very simple statements are very important in acoustic remote sensing, if (for example) active sea foam, which is an acoustic transmitter at the sea surface, has to be modeled; see Vagle and Farmer (1992) to understand acoustic noise below the surface and bubble sound emission.

A more general representation of acoustic sources corresponds to dipolar sources constituting two radiating sources of strengths or magnitudes q+ and q− separated by a distance a and such that m = qa is the dipolar momentum (as considered in electromagnetism).

Solving the wave equation for the two monopoles with θ the angle between r and the direction of the dipole gives two velocity components: (a) one for the near field (small r): ${{{m(1+jkr)\,\exp (-jkr)\,\cos (\theta )}} \left/ {{4\pi {r^3}}} \right.}$, to which is added a transverse component varying as sin θ (orthogonal component), and (b) one for the far field (large r): $-{k^2}m\,\cos (\theta )\,{{{\exp (-jkr)}} \left/ {{4\pi r}} \right.}$, which is typically a radial component. Dipolar sources or a combination of dipolar sources are very useful to simulate acoustic antennas and acoustic sources. Moreover, field analysis considering the distance where the field can be considered as far is very important to interpret, for example, signals coming from active systems (which transmit acoustic waves) or passive systems (which only receive acoustic waves as acoustic radiometers by analogy with electromagnetic and optical radiometers.

Close to monopoles and dipoles, which are theoretical concepts, is the pulsating sphere of radius a. With the condition that ka << 1, it is shown (for low displacement velocities) that it is equivalent to the radiation of a dipole source at a sphere’s center, with radial velocity at the surface equal to $m\,\,\,\cos (\theta )\,2\pi {a^3}$ with m = 2πa³U₀, where U₀ is the amplitude of the velocity. This analogy between dipoles and pulsing spheres is very useful to solve a large amount of questions relative to acoustic environment such as bubble acoustic emission.

A main question in the domain of acoustic radiation and for remote sensing purposes using transmitters and receivers corresponds to antenna design. Acoustic radiation does not concern very thin beams as in the optical domain but rather larger beams, and it is necessary to know precisely beam angles to determine the observation volumes. Once the acoustic source has been chosen as microphones, loudspeakers, transducers, or compression chambers, which convert electricity in pressure fluctuations and acoustic waves, we next choose an antenna to transmit and receive acoustic waves through the considered medium.

Different types of antennas are used (as multi-beam antennas, synthetic antennas combining different elementary sources, horn and parabolic antennas), from which several properties of transmitted and received signals must be reached: directivity, which corresponds to being able to get a signal in a preferential direction, antenna beam control, which concerns the knowledge and design of antenna beam angle and to limit secondary beams to a very low level, as, for example, for atmospheric sounding with SODAR.

For the first acoustic sounders, parabolic antennas have been mainly used; see Neff and Coulter (1986). Though it is easy to theoretically and analytically model such antennas, it often needs heavy computation; see Rocard (1951). A validation of antenna design in an anechoic chamber (a shielded room designed to attenuate wave echoes caused by reflections from the internal surfaces of the room), at least to test far-field behavior, is always necessary to qualify the antenna response. This makes necessary the use of large anechoic chambers to realize far-field conditions. Whatever is the antenna design, a perfect knowledge of antenna beam is necessary and will limit unpleasant surprises as important secondary beams or a beam larger than what was wanted could occur.

In Figure 1, we show experimental results obtained from the design of an offset antenna for a 6 kHz acoustic sounder controlled in an anechoic chamber. An offset dish antenna is a type of satellite dish. It is so-called offset because the antenna feed is offset to the side of the reflector, in contrast to a typical circular parabolic antenna where the feed is in front of the center of the reflector.

Figure 1

Realization of a three offset antennas for a 6 kHz minisounder. One antenna is vertical and two others are slanting. One distinguishes compression chambers as acoustic sources and the horns and antennas parts (parabolic portion + circular aperture covered by acoustic foam).

The antenna’s characteristics are shown on Figure 2. Notice that if directivity is good and if secondary beams are relatively low, the antenna beam always differs from theory since it is not easy to model all the elements of the antenna and anyway we have always to validate the modeling. However, a beam width of 13° and a receiver gain of 116–126 dB were obtained, which were in fact the objectives of the design.

Figure 2

At the top, antenna beam characteristics as measured in an anechoic chamber. At the bottom, a simple schema of the antenna characteristics showing an exponential horn transmitting sound in a portion of parabola.

Conclusion

We have presented elementary elements about acoustic radiation. Of course, acoustic radiation study in the atmosphere and ocean requires deep specific studies, but considering acoustic remote sensing, questions must be mainly summed up to in regard to radiation pattern of acoustic sources and to antenna design. However, monopole and dipole behavior and also more complicated concepts such as quadripoles and higher acoustic poles systems, which are not presented here, suggest what kind of questions have generally to be solved.

Cross-references

Acoustic Waves, Propagation

Acoustic Waves, Scattering