Underwater Acoustics
The following is a very brief overview of the basic principles of underwater acoustics.
What is Sound?
A propagating sound wave consists of alternating compressions and rarefaction's which are detected by a receiver as changes in pressure. Structures in our ears, and also most manmade receptors, are sensitive to these changes in sound pressure (Richardson et al.1995, Gordon and Moscrop 1996).
The basic components of a sound wave are amplitude, wavelength, and frequency:
The amplitude of a sound wave is proportional to the maximum distance a vibrating particle is displaced from rest. Small variations in amplitude produce weak or quiet sounds, while large variations produce strong or loud sounds. The wavelength of a wave is the distance between two successive compressions or the distance the wave travels in one cycle of vibration. The frequency of a sound wave is the rate of oscillation or vibration of the wave particles (i.e. the rate amplitude cycles from high to low to high, etc.). Frequency is measured in cycles/sec or Hertz (Hz). To the human ear, an increase in frequency is perceived as a higher pitched sound, while an increase in amplitude is perceived as a louder sound. Below are examples of sound waves that vary in frequency and amplitude.
These two waves have the same frequency but different amplitudes.  
These two waves have the same amplitude but different frequencies. 
Note that increasing the frequency of a sound in equal steps will lead to
perceived increases in pitch that seem to grow smaller with each step. For example, click
on the sound frequencies below, and you'll see a more noticeable difference between 200 Hz
and 225 Hz than 400 Hz and 425 Hz.
200Hz 225Hz 250Hz 275Hz 300Hz 325Hz 350Hz 375Hz 400Hz 425Hz 450Hz 475Hz
Humans generally hear sound waves whose frequencies are between 20 and 20,000 Hz. Below 20 Hz, sounds are referred to as infrasonic, and above 20,000 Hz as ultrasonic.
infrasonic (about 20 Hz) < human hearing < ultrasonic (about 20,000 Hz)
The Decibel Scale
If the amplitude of a sound is increased in a series of equal steps, the loudness of the sound will increase in steps which are perceived as successively smaller. Sound intensity is generally described using logarithmic units called decibels (dB). On the decibel scale, everything refers to power, which is (amplitude)^{2} ; 0.0 dB corresponds to about the normal threshold of hearing and 130 dB to the point where sound becomes painful to humans.
common sounds 
dB in air 
threshold of hearing  0 dB 
whisper at 1 meter  20 dB 
normal conversation  60 dB 
jet engine  140 dB 
painful to human  130 dB 
Warning: noise levels cited in air do not equal underwater levels for reasons that will be described in the following sections.
Why use the decibel scale? Because sound "loudness" varies exponentially, we'd have to deal with a lot of zeros when doing computations involving the parameters of sound, and we'd have to multiply numbers rather than simply add and subtract them. By using the decibel scale, calculations are simplified and relative values relate more closely to perception.
Phase
A fourth property of sound, its phase, is less directly related to perceived sound intensity. Phase is important in describing how complex sounds can be constructed from the simple sinusoidal waves. Below is an example of two sound waves with the same frequency and amplitude  only their alignment with respect to time differs. By specifying amplitude, wavelength, and phase, any sinusoid can be exactly described. By describing these parameters for all frequency components, any complex signal can be described exactly.
Sound Speed
The speed of a wave is the rate at which vibrations propagate through the medium. Wavelength and frequency are related by:
l = c/f
where lambda = wavelength, c = speed of sound in the medium, and f = frequency. The speed of sound in water is approximately 1500 m/s while the speed of sound in air is approximately 340 m/s. Therefore, a 20 Hz sound in the water is 75m long whereas a 20 Hz sound in air is 17m long.
Sound Pressure
Sound pressure is sound force per unit area, and is usually cited in micropascals (µPa), where 1 Pa is the pressure resulting from a force of one Newton exerted over an area of one square meter. The instantaneous pressure p(t) that a vibrating object exerts on an area is directly proportional to the vibrating object's velocity and acoustic impedance (rc):
p(t) = rcu
where r = density
c = sound speed
u = particle velocity
Pressure can also be defined in terms of force:
p = F/A
where p = pressure, F = force, and A = area
Sound Intensity
A sound’s acoustic intensity is defined as the acoustical power per unit area in the direction of propagation:
Sound Intensity (I) (W/m^{2}) = p_{e} /(rc) = rcu
where p_{e} or the "effective pressure" is equal to p/Ö 2
r is the density of water
and c is the speed of sound
[NOTE: rc is referred to as acoustic impedance;
rc in water is 1.5 x 10^{5} (Pa ·s)/m ;
rc in dry 20ºC air is 415 (Pa ·s)/m]
Sound Pressure Level and Sound Intensity Level
Sound levels extend over many orders of magnitude and, for this reason, it is convenient to use a logarithmic scale when measuring sound. Both Sound Pressure Level (SPL) and Sound Intensity Level (SIL) are measured in decibels (dB) and are usually expressed as ratios of a measured and a reference level:
Sound Pressure Level (dB) = 20 log (p/p_{ref}) where p_{ref} is the reference pressure
Sound Intensity Level (dB) = 10 log (I/I_{ref}) where I_{ref} is the reference intensity
In other words, the decibel is 10 times the log of the ratio of two intensities, and 20 times the log of the ratio of two pressures. The units for both SPL and SIL are dB relative to the reference intensity (often abbreviated as dB re 1µPa or dB//1µPa). Whenever "level" is added to the terms sound intensity or sound pressure, it usually implies that the measurement is in dBs. Because decibels implies a ratio of two values (and therefore a dimensionless measurement), SPL and SIL are equivalent when measured in dB.
Because the dB scale is relative, reference levels must be included with dB values if they are to be meaningful. The reference levels for SPL and SIL are equivalent but are reported in different units. The commonly used reference pressure level in underwater acoustics is 1 µPa while 20 µPa (which is roughly the human hearing threshold at 1000 Hz) is used as the reference level in air. The reference intensity in water is
I_{ref} = p^{2} _{ref} / (D_{water} c_{water}) = 6.7 x 10^{19} W/m^{2}
where reference pressure in water (p_{ref}) is 1µPa rms,
and the density of water (D_{water}) is about 1000kg/m^{3},
and the speed of sound in water (c_{water}) is about 1500 m/s.
Historically, the reference intensity in air was the sound intensity barely audible to humans, 1 10^{12}watts/m^{2} or 1 pW/m^{2}. (A painful (airborne) sound to humans = 10 watts/m^{2}).
In addition to the reference level, the distance from the source for that reference level must also be cited; typically the units of SIL are dB relative to the reference intensity at 1 meter (e.g. 20 dB re 1µPa @ 1m) (i.e. how intense the sound would be were it measured only 1m from the source). In practice, one can rarely measure source level at the standard 1m reference, so that source levels are usually estimated by measuring SPL at some known range from the source (assumed to be a single point), and the attenuation effects predicted and subtracted from the measured value to estimate the level at the reference range.
Ideally, it is Sound Intensity Levels that we’d like to measure. However, it’s easier to measure sound pressure than sound intensity, so we measure pressure, and from that infer intensity. Within the same medium, sound intensity or power is proportional to the average of the squared pressure:
I µ p^{2}
therefore
SIL(dB) = 10 log (I/I_{r)} = 10 log (p^{2} _{water} / p^{2} _{refwater}) = 20 log (p_{water}/1µPa)
In other words, once we start using the decibel scale, SIL and SPL are pretty much the same thing.
Conversion of dB from air to water (and vice versa)
Based on the above discussion, it should now be obvious that 120 dB in air is not the same as 120 dB in water, primarily because of the differences in reference measurements. How do we make meaningful comparisons between a ship's engine underwater and a jet engine? In air, the sound pressure level is referenced to 20 µPa, while in water the sound pressure level is referenced to 1 µPa. Given the above equation for dB's, the conversion factor for dB air è water
dB = 20 log (p_{water}/1µPa) = 20 log (20) = + 26 dB
Therefore a pressure comparison between air and water differs by 26 dB.
The characteristic impedance of water is about 3600 times that of air; the conversion factor for a sound intensity in air vs water is 63 dB.
10 log (3600) = 36 dB
36+26 = 62 dB
A simplified example....
If a jet engine is 140 dB re 20µPa @ 1m, then underwater this would be equivalent to
SIL_{water} = SIL_{air} + 62 = 202 dB re1µPa
To convert from water to air, simply subtract the 62 dB from the SL in water. A supertanker generating a 190 dB sound level would be roughly equivalent to a 127 dB sound in air. (Note that these are gross generalizations because the source level often changes with the frequency component of the sound.)
Nosie Source  Maxiumum Source Level  Remarks  Reference 
Undersea Earthquake  272 dB  Magnitude 4.0 on Richter scale (energy integrated over 50 Hz bandwidth)  Wenz, 1962. 
Seafloor Volcano Eruption  255+ dB  Massive steam explosions  Deitz and Sheehy, 1954; Kibblewhite, 1965; Northrop, 1974; Shepard and Robson, 1967; Nishimura, NRLDC, pers. comm., 1995. 
Airgun Array (Seismic)  255 dB  Compressed air discharged into piston assembly  Johnston and Cain, 1981; Barger and Hamblen, 1980; Kramer et al., 1968. 
Lightning Stike on Water Surface  250 dB  Random events during storms at sea  Hill, 1985; Nishimura, NRLDC, pers. com., 1995. 
Seismic Exploration Devices  212230 dB  Includes vibroseis, sparker, gas sleeve, exploder, water gun and boomer seismic profiling methods.  Johnston and Cain, 1981; Holiday et al., 1984. 
Container Ship  198 dB  Length 274 meters; Speed 23 knots  Buck and Chalfant, 1972; Ross, 1976; Brown, 1982b; Thiele and Ødegaard, 1983. 
Supertanker  190 dB  Length 340 meters; Speed 20 knots  Buck and Chalfant, 1972; Ross, 1976; Brown, 1982b; Thiele and Ødegaard, 1983. 
Blue Whale  190 dB (avg. 145172)  Vocalizations: Low frequency moans  Cummings and Thompson, 1971a; Edds, 1982. 
Fin Whale  188 dB (avg. 155186)  Vocalizations: Pulses, moans  Watkins, 1981b; Cummings et al., 1986; Edds, 1988. 
Offshore Drill Rig  185 dB  Motor Vessel KULLUK; oil/gas exploration  Greene, 1987b. 
Offshore Dredge  185 dB  Motor Vessel AQUARIUS  Greene, 1987b. 
Humpback Whale  180 dB (avg. 175180)  Fluke and flipper slaps  Thompson et al., 1986. 
Bowhead Whale  180 dB (avg. 152180)  Vocalizations: Songs  Cummings and Holiday, 1987. 
Right Whale  175 dB (avg. 172175)  Vocalizations: Pulsive signal  Cummings et al., 1972; Clark 1983. 
Gray Whale  175 dB (avg. 175)  Vocalizations: moans  Cummings et al., 1968; Fish et al., 1974; Swartz and Cummings, 1978. 
Open Ocean Ambient Noise  74100 dB (7197 dB in deep sound channel)  Estimate for offshore central Calif. sea state 35; expected to be higher (= or > 120 dB) when vessels present.  Urick, 1983, 1986. 
Note: Except where noted, all the above are nominal total broadband power levels in 201000 Hz band. These are the levels that would be measured by a single hydrophone (reference 1 µPa @ 1 m) in the water.
SOURCE: Ocean Acoustic Observatories Alternate Source Test (AST)Source, path receiver model of sound
One of the more popular models used to describe the propagation of sound through water or air is the "source, path, receiver" model (Richardson 1995). The basic parameters (there are many we will not discuss) in this model are:
source: source level (SL)
path or medium: transmission loss (TL), ambient noise level (NL)
receiver: signal to noise ratio (SNR), sound intensity level (SIL), detection threshold (DT)
A simple model of sound propagation is:
SIL = SL  TL
where TL = 10 log (Intensity at 1 meter/Intensity at r_{2} meters away from the source)
Transmission loss can also be estimated by adding the effects of geometrical spreading, absorption and scattering. For our purposes we'll deal only with spreading (TL_{g}) and absorption loss (TL_{a}):
TL = TL_{g }+ TL_{a}
where
TL_{g }= 20 log r_{2}
(for geometrical spherical spreading; r_{2} is in meters)
TL_{a }= a r_{2} x 10^{3} (units are dB/km)
where a is the attenuation coefficient and a function of frequency, r_{2} is in meters, and 10^{3} is a conversion factor for m to km
Note: The rate at which sound is absorbed by water is related to the square of frequency (a µ f^{ 2}); lower frequency sounds have low absorption coefficients and therefore propagate long distances. If you know the frequency of the sound you're dealing with, the attenuation coefficient (a) can be looked up in the appropriate table or graph in any acoustics textbook.
An example.....
What is a humpback whale's sound intensity level at 1 km in deep water (assume spherical spreading)?
source level = 150 dB re 1µPa @ 1 meter, frequency = 120 Hz therefore a ~ .003
transmission loss = TL_{g }+ TL_{a} = 20 log (1 km) + .003(50) = 60 +.15 = 60.15 dB re 1µPa @ 1 meter.
SIL = SL  TL
SIL = 150  60.15
SIL @ 90 dB
Signal to Noise Ratio
Finally, whether or not a particular acoustic signal can be detected in the ocean is a factor of the level of the signal of interest relative to the background noise level of the ocean, or ambient noise. This is normally expressed as a "signal to noise ratio" (SNR), where any value greater than 1 implies that the signal is detectable above the noise, while a number below 1 implies that the signal is "buried" in the noise. For rough, "back of the envelope" calculations of SNR, ambient noise level (NL) is subtracted from the sound intensity level:
SNR = SIL  NL
A number greater than 0 dB implies we could detect the signal from background noise, while a number less than 0 dB would imply we could not hear the signal. In the above example of the vocalizing humpback, could we hear this animal above background noise at this distance ? (assume NL at 120 Hz is about 70 dB)
SNR = 90  70
SNR = 20 dB
This whale vocalization is about 20 dB above ambient noise level, and we are likely to hear it!
In practice, this basic concept becomes much more complicated. First, the ambient noise field of the ocean is quite variable with respect to time, location, and frequency. Effects can be seasonal, for example the presence of absence of a storm track that introduces loud wave noise, or hourly, such as the passing of a ship. Also, the propagation properties of the water column vary widely with location, depending on the physical oceanographic properties, local bathymetry, and bottom properties. Sophisticated numerical models have been developed over the last several decades to provide improved prediction of acoustic environmental properties. Finally, natural sound sources such as marine mammals and earthquakes, may have significant variability in their source level making the calculation of signaltonoise ratio even more difficult.
quantity 
unit 
symbol 
relation 
length 
meter 
m 

mass 
kilogram 
kg 

time 
second 
s 

frequency 
hertz 
Hz 

force 
newton 
N 
kg x m/s^{2} 
pressure 
pascal 
Pa 
N/m^{2} 
energy 
joule 
J 
N x m 
power 
watt 
W 
J/s 
intensity 
W/m^{2} 

density 
kg/m^{3} 

speed 
m/s 
References
 Adapted from AN INTRODUCTION TO UNDERWATER ACOUSTICS
 SiD Sonar Control a Sound Library collecting and exhibiting sound data of Submarines and other Naval/Maritime issues!
 Introduction to Naval Weapons Engineering
 Fundamentals of Naval Weapons Systems
 Environmetal Support Package Near Real Time Analysis
 MODAS Products
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