480 VIGOUREUX AND HERSEY [CHAP. 12 



usual to indicate by a coefficient referred to amplitude so that, for plane waves, 

 intensities at points a distance x apart in the direction of propagation are in 

 the ratio of 1 to exp ( — 2aa:;). (For practical purposes the decibel per unit length 

 is often used to express attenuation, which is then 10 times 2a log e or 8.686a.) 

 For sea-water at 10°C the absorption (Horton, 1957) in decibels per kilo- 

 yard at / kc/s is given to a good approximation by the formula 



•^ - + 0.000 275/2, (12) 



4100 +/2 



in which the first term represents the relaxation in dissociation of salts in 

 solution (Liebermann, 1948, 1949) and the second is due to viscosity and to 

 relaxation in compressibility. The salt mostly responsible appears to be mag- 

 nesium sulphate, and dissociation and hydrolysis of both ions are involved. 

 Almost all over the range, the absorption decreases with increase of tempera- 

 ture, the reduction at low frequencies being as much as 50% for 17°C. 



Sound energy can also be scattered by small perfectly reflecting particles 

 uniformly distributed throughout the water. Although in this ideal case the 

 total energy is not thereby decreased, the intensity in the direction of propaga- 

 tion is reduced exponentially as with absorption. The most important scatterer 

 is the air bubble, which also absorbs sound, especially at resonance. Bubbles 

 are sometimes found just below the surface, having been entrained by sea 

 waves ; they are also liberated from air in solution in the sea by the reduction 

 of pressure at propellers of ships. If a shi23 lays a wake across a sonar beam, 

 the sound is often almost completely prevented from going through ; the bubbles 

 in the wake absorb some and reflect the rest, giving rise to a strong sonar echo 

 (Horton, 1957). 



Turbulence and finely divided vegetable and animal matter also scatter, but, 

 as with bubbles, the effect is local and variable. 



E. Reflection at Surface and at Sea- Bed 



The main difference between air and water as carriers of sound is that the 

 characteristic impedance, pc, of water is some 3700 times that of air, which 

 means that the "mismatch" between the two media is very great. For this 

 reason very little sound crosses the free surface either way ; most of it is re- 

 flected. At normal incidence, which is the most favourable direction for trans- 

 mission across the surface, the ratio of transmitted to incident energy is 



^^ 



{pc)w^ /{pc)a 



{pc)a V (pc) 



^ (13) 



1000 



A wave reflected at the surface may reach the sea-bed, at any rate in not too 

 deep water, and if the sea-bed also reflects, the process recurs and propagation 

 takes place in hops between the surface and the bottom. 



In the following development the sea-bed will be regarded as unconsolidated 

 sediment without rigidity, i.e. as being incapable of transmitting shearing 



