28 



WAVE ACOUSTICS 



The corresponding solution of equation (94) is there- 

 fore 



We can write this expression for p in the form 



J, = Ae "'"e 2'''/['-(^/=']. (99) 



For n = Q, a vanishes, and equation (99) reduces 



to 



_ A ^i^ifU-{x/c)1 



which is just the solution for plane waves propagated 

 harmonically into a perfect fluid. By comparing 

 equations (99) and (100), we see that the effect of 

 viscosity is to cause the amplitude of the pressure 

 vibration to decay exponentially with distance, by 

 the factor e"*", where a is the positive real niunber 

 defined by equation (97). A vibration of the type 

 of (99) is referred to as a damped vibration, and e'"^ 

 is called the damping factor. 



To see whether this energy loss due to shear 

 viscosity is the cause of the attenuation observed in 

 the sea, one can first calculate a for sea water, by 

 using the known values of po,c,m for sea water, and 

 the known frequency / of the sound source. The in- 

 tensity loss is measured between two points so far 

 from the sound source that the wave propagation be- 

 tween those two points approximates plane wave 

 propagation. Then this observed intensity loss is 

 compared with the theoretical intensity loss calcu- 

 lated from equations (97) and (99) . It is found that 

 only for very great frequencies (much higher than 

 100 kc) can an appreciable fraction of the observed 

 attenuation be ascribed to shear viscosity; at lower 

 frequencies, the theoretical loss from viscosity makes 

 up only a very small part of the observed attenua- 

 tion. Thus other causes must be sought for the extinc- 

 tion of sound energy in the sea. The sound transmis- 

 tion studies of Section 6.1 of NDRC have had as one 

 of their primary objectives the discovery of the 

 factors governing the intensity loss of sound in the 

 sea. Although some progress has been made, the 

 problems of attenuation in the sea have by no means 

 been completely solved (see Chapters 5 to 10). 



Since the observed attenuation of sound in the sea 

 is much greater than the value indicated by equation 

 (99), it appears possible that there is another type of 

 viscosity, in addition to the classical shear viscosity, 

 which may be responsible for part or all of the re- 

 maining attenuation. The classical theory of the 

 flow of viscous fluids is based on Stokes' hypothesis 

 that f rictional forces within a fluid arise only from a 



change in the shape of a volume element; in other 

 words, that a change in the size of a volume element, 

 if its shape remained unaltered, would meet no re- 

 sistance. The concept of a compression viscosity has 

 been suggested to represent the resistance of the 

 fluid to pure volume dilatation. Such a compression 

 viscosity would not be discovered in a stationary flow 

 of the type employed to measure shear viscosity be- 

 cause in these experiments the fluid acts essentially 

 as an incompressible fluid. But in the transmission 

 of sound this conjectural compression viscosity would 

 contribute a term to the expression for a which would 

 also be proportional to the square of the frequency/. 

 Actual determinations of the constant a at many 

 different frequencies show that between and 100 kc 

 the attenuation increases less rapidly than the 

 square of the frequency. There are no theoretical 

 grounds for assuming any power law for the depend- 

 ence of attenuation on frequency. If a power law is as- 

 sumed, the empirical curve is best fitted by a 1.4th 

 power dependence, but even this best fit is poor. It 

 thus appears that factors other than viscosity must 

 account for much of the attenuation of sound ob- 

 served in the sea. 



2.6 EFFECT OF A BOUNDARY 



2.6.1 Conditions of Transition and 

 Boundary Conditions 



We shall now return to the assumption of a perfect 

 fluid and turn our attention to the effects of bounda- 

 ries. Consequently, we now drop the assumption that 

 waves are propagated in a single homogeneous infi- 

 nite medium. Instead, we shall consider the case that 

 all space is filled up by two different homogeneous 

 media separated by a plane, which we choose as the 

 plane y = 0. For the one medium (the sea), at ?/ < 0, 

 we denote the density, excess pressure, bulk modulus, 

 and sound velocity by p, p, k, and c respectively; for 

 the other medium, air, for example, a,ty > 0, we call 

 these quantities pi, pi, ki, and Ci. 



It is necessary, from a physical point of view, to 

 assume that the pressure in both media is the same 

 at the boundary. Otherwise, the force per unit mass 

 at the interface would become infinite. We have, 

 thus, 



p = piSity = 0. (101) 



Also, if the two media are to remain in contact with 

 each other at all times, the displacements normal to 



