Underwater Acoustic Propagation 
Toby Haynes, B.A.Sc. Electrical Engineering 
Introduction 
This paper is a summary of sonar propagation, the acoustic properties of water, and constants and equations required to calculate propagation delay and attenuation for underwater acoustic channels. Typical properties are given for sea water near the coast of British Columbia where I have done some work on acoustic communication devices for scuba divers. 
Salinity 
Salinity, S, is measured in parts per thousand (ppt, or grams of salt per kilogram of sea water). Sea salt includes boric acid and MgSO_{4} as well as NaCl. The average salinity of sea water is 35 ppt (page 14 of [2]). 
A map on page 44 of [2] shows the surface salinity on the British Columbia coast to be about 31 ppt. Ocean surface salinity is higher near the equator than near the poles and higher in the Atlantic than the Pacific, although the range is only 33 to 37 ppt. The Mediterranean Sea reaches 39 ppt and the Red Sea 41 ppt. 
Temperature 
Sea surface temperature varies greatly with location and season. Maps on page 36 and 37 of [2] give the surface temperature off of southern British Columbia as 7 C in February and 15 C in August, for an average of 11 C. 
Surface temperatures range from zero near the poles to 25 C near the equator yearround. At a depth of several kilometers, the sea temperature is near 2 C (page 42 of [2]). 
A thermocline (region of temperature gradient) separates cold deep water from warmer surface water. The thermocline's depth and gradient varies through the year. In the eastern North Pacific, the depth of the thermocline ranges from 100 m (January) to 20 m (August). There, deep water remains near 5 C and shallow water ranges from 5 C in March to 14 C in August (see page 39 of [2]). 
Speed of Sound 
Fresh Water 
For distilled water, page 107 of [1] gives an empirical formula for the speed of sound that is accurate to within 0.05% for 0<=T<=100 C and 0<=P<=200 bar: 
c(P,t) = 1402.7 + 488t  482t^{2} + 135t^{3} + (15.9 + 2.8t + 2.4t^{2}) P /100 
where 
Page 397 of [1] gives c=1403 m/s for fresh water at 0 C and one atmosphere. 
Sea Water 
Page 397 of [1] gives c=1449 m/s for sea water at 0 C, one atmosphere, and 35 ppt salinity. The typical value over the continental shelves at mid latitudes is given as 1500 m/s. An empirical formula for the speed of sound is: 
c(D, S, t) = 1449.05 + 45.7t  5.21t^{2} + 0.23t^{3}+ (1.333  0.126t + 0.009t^{2}) (S  35) + D(D) 
D(D) @ 16.3D + 0.18D^{2} 
at latitude 45
degrees in the oceans, where 
For other latitudes f in degrees, replace D with D (1  0.0026 cos 2f). This gives c with a standard deviation of 0.06 m/s down to a depth D = 4 km in oceanic waters. 
A more complicated correction gives a standard deviation of 0.02 m/s: 
D (D,S) = (16.23 + 0.253t) D + (0.213  0.1t) D^{2} + [0.016 + 0.0002 (S  35)] (S  35) t D 
For conditions off the British Columbia coast near the surface (11 C, 31 ppt salinity), this gives c = 1489 m/s. 
Doppler Shift 
Because ultrasonic wavelengths are small, Doppler shift is significant in underwater acoustic channels where waves, currents, or vessel velocity cause relative motion between the transmitter and receiver or reflector. The bandwidth of sounding equipment and carrier and symbol tracking of communication equipment must allow for this frequency shift. 
Doppler shift observed at the receiver increases with frequency and with the relative radial velocity between the transmitter and receiver. Shift is positive if the transmitter and receiver are moving toward each other and negative if they are moving apart. 
In the case of sounding equipment, where the transmitter and receiver are collocated, it is the radial velocity of the reflector that determines Doppler shift. The shift is twice that for a 1way path since the reflected image moves at twice the reflector velocity. 
Doppler shift is calculated as follows: 
f_{T} = v / l = v f_{C} / c for 1way path 
f_{T} = 2 v / l = 2 v f_{C} / c for echo path 
where 
Acoustic Absorption 
For a plane wave, the pressure amplitude drops exponentially with distance due to absorption by the medium (page 144 of [1]). 
P(x) = P_{0} e ^{ }^{a}^{ x} 
where 
P(x) = Pressure
amplitude at distance x 
Acoustic intensity (power) is proportional to the square of the pressure amplitude. 
I(x) = I_{0} e ^{2}^{a}^{ x} 
where 
This can be expressed in decibels: 
IdB(0)  IdB(x) = 10 log10 e ^{2}^{a}^{ x} 
IdB(0)  IdB(x) = 8.7ax 
IdB(0)  IdB(x) = ax 
where 
Acoustic attenuation increases with frequency and is approximately proportional to f^{2}. 
Fresh Water 
A measured value for acoustic absorption is given on page 148 of [1] for fresh water at 20 C and one atmosphere: 
a = f^{2} 2.50x10^{14} 
a = f^{2} 2.175x10^{13} 
where 

Figure 1 Absorption Coefficient of Fresh Water (20 C, 1 Atmosphere) 

Sea Water 
Absorption in sea water is greater than in fresh water due to chemical relaxation absorption by boric acid and MgSO_{4}. This occurs when acoustic pressure associates and dissociates the ions of a compound. 
A general empirical formula for the absorption coefficient of sea water is given below, from page 158 of [1], that is valid from 0 to 30 C and 1 to 400 atm. Salinity is 35 ppt. 
a = Af_{1}f^{2} / (f_{1}^{2} + f^{2}) + Bf_{2}f^{2} / (f_{2}^{2} + f^{2}) + Cf^{2} 
f_{1} = 1.32x10^{3} (T + 273) e^{1700/(T+273)} relaxation frequency of boric acid, Hz 
f_{2} = 1.55x10^{7} (T + 273) e^{3052/(T+273)} relaxation frequency of MgSO_{4}, Hz 
A = 8.85x10^{8} (1 + 2.3x10^{2} T  5.1x10^{4} T^{2}) 
B = 4.88x10^{7} (1 + 1.3x10^{2} T) (1  0.9x10^{3} P_{0}) 
C = 4.76x10^{13} (1  4.0x10^{2} T + 5.9x10^{4} T^{2}) (1  3.8x10^{4} P_{0}) 
where 
The following simplified version is given on page 398 of [1]. This applies at 5 C, 1 atmosphere, and 35 ppt salinity. 
a = f^{2} ( 8x105 / (0.7+f^{2}) + 0.04 / (6000+f^{2}) + 4x10^{7} ) 
where 

Figure 2 Absorption Coefficient of Sea Water 
Spherical Spreading 
Acoustic waves spread spherically from the source in the absence of reflection or refraction, so that the pressure amplitude is inversely proportional to the distance. When absorption is also considered, the total attenuation is: 
IdB(r_{2})  IdB(r_{1}) = 10 log10(r_{1}^{2} / r_{2}^{2})  a (r_{2}  r_{1}) 
IdB(r_{2})  IdB(r_{1}) = 20 log10( r_{1} )  20 log10( r_{2} )  a (r_{2}  r_{1}) 
where 
The following graphs show the combination of spherical spreading and absorption at 50 and 200 kHz for ranges up to 1km. Channel attenuation is much higher at 200 kHz, and higher in sea water than fresh water. 

Figure 3 Spherical Spreading Channel at 50 kHz (5 C, 1 Atm., Sea at 35 ppt) 


Figure 4 Spherical Spreading Channel at 200 kHz (5 C, 1 Atm., Sea at 35 ppt) 

Surface Reflection 
When the transmitter and receiver are below the water surface, acoustic energy arrives at the receiver both directly and by reflection from the surface. The receiver sees both the actual transmitter and its image above the surface. Waves reflected by the water/air boundary are inverted and interfere with the direct waves to give a threedimensional interference pattern. 
The results in the following sections assume that the water is deep enough that reflections from the bottom can be ignored. 
Continuous Waves 
For continuous waves in the far field, the function for the pressure amplitude is given on page 408 of [1] as: 
P(r) is proportional to 2 sin( k h d / r) / r 
where 
The multipath term can be added to the attenuation formula for spherical spreading: 
IdB(r_{2})  IdB(r_{1}) = 20 log_{10}( r_{1} )  20 log_{10}( r_{2} )  a (r_{2}  r_{1}) + 10 log_{10}(4 sin^{2}( k h d / r_{2})) 
where 
The following graphs show the effects of surface reflections at 50 and 200 kHz for ranges to 500 m in sea water. Both the transmitter and receiver are 2 m below the surface. Nulls are closer together at 200 kHz, so signal fading due to motions of the transmitter or receiver will be more of a problem. 

Figure 5 Surface Reflection Channel at 50 kHz (5 C, 1 Atm., 35 ppt, h=d=2 m) 


Figure 6 Surface Reflection Channel at 200 kHz (5 C, 1 Atm., 35 ppt, h=d=2 m) 

Note that the acoustic intensity can reach 6 dB above the spherical spreading level. The "extra" power comes from the image source (3 dB) and from the energy that is cancelled in the nulls (another 3 dB). 
When the water surface is disturbed, the interference pattern is altered. At long range, the effect of surface disturbance is to reduce the amplitude of the reflected wave. This reduces the peak levels and increases the null levels to bring the attenuation closer to that for spherical spreading. 
Very Long Range 
At very long range, where khd/r << 1, so that sin(khd/r) is approximately khd/r, the acoustic power falls off inversely with r^{4} rather than r^{2}. This is the region of the N=0 null in the interference pattern. 
The following graph for received power at 10 kHz with surface reflections shows the effects of the N=0 null at distances over 330 m, where received power remains below that for a spherical spreading channel. This null occurs at smaller ranges as the frequency decreases (wavelength increases). The effect was not evident at the higher frequencies in the previous two graphs. 

Figure 7 Surface Reflection Channel at 10 kHz (5 C, 1 Atm., 35 ppt, h=d=2 m) 

Short CW Pulses 
If a short burst of sine waves is transmitted rather than a continuous wave, the effect of the surface reflection is complicated. 
For a continuous wave, the nulls occur at distances where the following is true: 
k h d / r = N p for N=0,1,2,... 
where N = Difference between reflected and direct paths in wavelengths. 
The null for N=0 is at infinite distance. Other nulls require that the direct and reflected signals overlap by N wavelengths. For short pulses, the first N wavelengths of the direct pulse are received without interference. So are the last N wavelengths of the reflected pulse. It is only during the time of overlap that interference occurs to produce a null. Short pulses will therefore not experience the nulls at close range that correspond to large N. Rather, two distinct pulses, direct and reflected, will be received. 
References 
[1] L. Kinsler, A. Frey, A. Coppens, J. Sanders, "Fundamentals of Acoustics, Third Edition," New York: John Wiley & Sons, 1982. 
[2] G. Pickard, W Emery, "Descriptive Physical Oceanography, An Introduction, 4th Edition," Oxford: Pergamon Press, 1982. 
Disclaimer 
I am not liable for costs due to the use or misuse of this information, or for any errors it contains. The reader is responsible for checking the accuracy of all information presented before it is used. 
(c) Toby Haynes, 2005  2013 