# Engineering Acoustics/Basic Concepts of Underwater Acoustics

The study of Underwater Acoustics was always of great importance, when it comes to navigation instruments, for those who depend on the sea. Methods to find the position on earth according to the stars were available since long ago, but the first apparatus that track what is underwater are relatively recent. One of these instruments, which improved the safety of navigation is the fathometer. It has the simple concept of measuring how much time a sound wave generated at the ship takes to reach the bottom and a reflected wave returns. If one knows the speed of sound in the medium, the depth could be easily determined. Another mechanism consists of underwater bells on lightships or lighthouses and hydrophones on ships that are to find the distance between them. These could be considered the forerunners of the SONAR (SOund Navigation And Ranging). There are a lot of animals that also take advantage of underwater sound propagation to communicate.

# Speed of Sound

In 1841, Jean-Daniel Colladon [1] was able to measure the speed of sound underwater for the first time. He conducted experiments in Lake Geneva, where he was able to transmit sound waves from Nyon to Montreux (50 km). The idea of the experiment was to propagate a sound wave making use of a hammer and an anvil to generate the wave and a parabolic antenna to capture the sound at distance. A flash of light is emitted at the same time that the hammer hit the anvil and the delay between the light and sound is used to determine the speed of sound.

The difference in speed of sound with depth is much more significant than its difference along the surface.

The equation for speed of sound(m/s) in water developed by Del Grosso,[1] applicable in Neptunian[2] waters, depends on the Temperature(T) in Celsius, Salinity(S) in ppt(part per thousand) and gauge Pressure(P) in atmospheres.:

${\displaystyle c(T,S,P)=1449.08+4.57Te^{-(T/86.9+(T/360)^{2})}+1.33(S-35)e^{-T/120}+0.1522Pe^{T/1200+(S-35)/400}+1.4610^{-5}P^{2}e^{-T/20+(S-35)/10}}$

where the pressure is a function of depth[Km] and the Latitude, given by:

${\displaystyle P=99.5(1-0.00263cos2\phi )Z+0.239Z^{2}}$

Figure 1: Speed of sound profile at low latitudes. Salinity gradient was not taken into account.

The Axis passing by the region where the speed of sound is minimum is known as Deep Sound Channel Axis.

The speed of sound is very sensible to the temperature, which changes considerably on the thermocline [2]. Beyond 1000 m meters in depth, the pressure governs the equation, increasing slowly the speed with depth. The salinity has very low effect on the equation unless in very specific situations, such as heavy rain or the encounter between a river and the sea. The shape of the curve may change drastically from one place to another, e.g., the curve would be more like a linear function of depth in very cold places.

# Refraction

The gradient of speed of sound within the water causes a phenomena similar to a mirage, in which rays of light are bent. If we divide the water in multiple layers parallel to surface, we should get various medium with different speed of sound i.e., different specific characteristic impedance. Considering a source of sound pressure underwater and making use of Snell's Law [3] we can see the path the wave will follow. Snell's Law tells us that the sound bends towards the lower sound speed layer. If the sound wave angle with the horizontal is too high (higher than ${\displaystyle \theta _{max}}$ ), the wave will eventually hit the bottom or the surface, otherwise it will bend continuously towards the horizontal until it passes the critical angle(${\displaystyle \theta _{c}}$ ) and then will be completely reflected back.

${\displaystyle \theta _{max}=(2\Delta c/c_{max}))^{1/2}}$

(${\displaystyle c_{max}}$  is the maximum speed found in the SOFAR channel.)

${\displaystyle sin\theta _{c}=c_{1}/c_{2}}$

This process happens over and over again causing the sound to be trapped in certain depth range known as SOFAR (Sound Fixing and Ranging) channel [4]. As the sound cannot reach the bottom nor the surface, the losses are small and no sound is transmitted to the air nor the seabed, helping sound propagate through large distances. Signals have been detected in ranges that exceed 3000 km.

This channel can be used for communication successfully by some species of cetacea.

We can see that the sound concentrates at some depths and is much less present in others, causing some regions to be more noisy than others.

Sound trapping in the SOFAR channel

Note that if the surface temperature is very low this phenomena may no longer occur. The wave would be bouncing on the surface and being reflected back just like we can see on the graph for 15.19° angle. The same effect occurs on the mixed layer which is the layer affected by the agitation of waves, causing the speed of sound to be only dependent on pressure. This effect may cause shadow zones.

If you have a source that is between the deep sound channel axis and the surface, only the rays making an angle less than ${\displaystyle \theta _{o}}$  with the horizontal would be trapped.

${\displaystyle \theta _{o}=\theta _{max}(z_{s}/D_{s})^{1/2}}$

where ${\displaystyle z_{s}}$  is the depth of the source and ${\displaystyle D_{s}}$  is the depth of the sound axis

# Reflection

Reflection also occurs when the sound wave hits a another body, such as the seabed, the surface, animals, ships and submarines.

${\displaystyle R=(r_{2}/r_{1}-cos\theta _{t}/cos\theta _{i})/(r_{2}/r_{1}+cos\theta _{t}/cos\theta _{i})}$

where ${\displaystyle r_{1}}$  is the characteristic acoustic impedance of water and ${\displaystyle r_{2}}$  is the characteristic acoustic impedance of the other body, ${\displaystyle \theta _{i}}$  is the incident angle and ${\displaystyle \theta _{t}}$  is the angle of the transmitted wave, which can be obtained via Snell's Law. The formula is for 2D case, but we can easily recall the 1D case by setting ${\displaystyle \theta _{i}=\theta _{t}=0}$

If we can measure the reflected wave, we can determine the reflection factor and with it we are able to determine the characteristic acoustic impedance of the body that the wave hit to then have an idea of what the body might be.

# Transmission Loss

The transmission loss is defined as

${\displaystyle TL=10log[I(1)/I(r)]}$

where ${\displaystyle I(r)}$  is the intensity of sound measured at a distance ${\displaystyle r}$  Sometimes it is useful to separate ${\displaystyle TL}$  in loss due to geometrical spreading and loss due to absorption

${\displaystyle TL=TL(geom)+TL(losses)}$

If the sound is trapped between two perfect reflecting surfaces

${\displaystyle TL=10logr+ar}$

where a is the absorption coefficient in dB/m.

# Sonar Equations

The passive sonar measures the incoming waves and is able to determine the position of the target if there is more than one device, by triangulation. Its equation determines that the Sound Level coming from the source reduced by the Transmission Loss has to be higher than the background noise (generated by waves, wind, animals, ships and others) in order to get any measurement.

Passive Sonar Equation

${\displaystyle SL-TL>=NL-DI+DT_{N}}$

where SL is the sound emitted by the target, NL is the noise level, DI is the directivity index and ${\displaystyle DT_{N}}$  is the detection threshold for noise-limited performace and TL is the transmission loss.

The active sonar emits a wave and measures the reflected sound waves. Since the wave will propagate the double distance. The transmission loss term is multiplied by two. The equation determines the conditions to get valid measures (higher than background noise).

Active Sonar Equation

${\displaystyle SL-2TL+TS>=NL-DI+DT_{N}}$

where SL is the sound emitted by the source, NL is the noise level, DI is the directivity index and ${\displaystyle DT_{N}}$  is the detection threshold for noise-limited performace and TL is the transmission loss andTS is the target strength, a measure of how good acoustic reflector the target is.

# References

1. Del Grosso (1974), J. Acoust. Soc. Am.,56,1084(1974)., p. 56,1084
2. Leroy (1969), J. Acoust. Soc. Am.,56,1084(1974)., p. 46,216
3. Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, James V. Sanders(2000) ,Fundamentals of Acoustics 4th ed, Wiley