# 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.:

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

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

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

!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 $\theta_{max}$), the wave will eventually hit the
bottom or the surface, otherwise it will bend continuously towards the
horizontal until it passes the critical angle($\theta_c$) and then will
be completely reflected back.

$\theta_{max}=(2\Delta c/c_{max}))^{1/2}$

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

$sin\theta_c = c_1/c_2$

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{width="800"}

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 $\theta_o$ with the
horizontal would be trapped.

$\theta_o=\theta_{max}(z_s/D_s)^{1/2}$

where $z_s$ is the depth of the source and $D_s$ 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.

$R = (r_2/r_1 - cos\theta_t/cos\theta_i)/(r_2/r_1 + cos\theta_t/cos\theta_i)$

where $r_1$ is the characteristic acoustic impedance of water and $r_2$
is the characteristic acoustic impedance of the other body, $\theta_i$
is the incident angle and $\theta_t$ 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
$\theta_i=\theta_t=0$

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

$TL=10log[I(1)/I(r)]$

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

$TL=TL(geom)+TL(losses)$

If the sound is trapped between two perfect reflecting surfaces

$TL=10log r + ar$

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

$SL - TL >= NL - DI + DT_N$

where *SL* is the sound emitted by the target, *NL* is the noise level,
*DI* is the directivity index and $DT_N$ 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

$SL - 2TL + TS >= NL - DI + DT_N$

where *SL* is the sound emitted by the source, *NL* is the noise level,
*DI* is the directivity index and $DT_N$ is the detection threshold for
noise-limited performace and *TL* is the transmission loss and*TS* is
the target strength, a measure of how good acoustic reflector the target
is.

# References

`3. Lawrence E. Kinsler, Austin R. Frey, Alan B. Coppens, James V. Sanders(2000) ,`*`Fundamentals of Acoustics 4th ed`*`, Wiley`

[^1]:
    
[^2]:
    