12 



WAVE ACOUSTICS 



causes the fluid to deviate so slightly from its state 

 of equilibrium that the change in pressure is propor- 

 tional to the fractional change in density. Second, it 

 is assumed that the changes caused by the passing 

 of the sound wave take place so rapidly that there is 

 practically no conduction of heat. We shall denote 

 the fractional change in density as heretofore by a; 

 the change in pressure will be called excess pressure 

 and denoted by p. 



Thus we assume that the fractional change in 

 density and the excess pressure caused by the passing 

 sound wave are both small and that they are propor- 

 tional to each other: 



P = Kff. 



(18) 



The constant of proportionality k is called the bulk 

 modulus. It depends not only on the chemical nature 

 of the fluid (such as the concentration of salts in the 

 sea), but also on the equilibrium temperature T, 

 the equilibrium pressure Fo, and the equiUbrium 

 density p. 



The temperature in the ocean varies from point to 

 point, usually decreasing with increasing depth. The 

 equiUbrium pressure increases rapidly with the depth, 

 and the density increases very slightly with depth. 

 As a result, the bulk modulus k is itself a function of 

 all three coordinates x, y, and z, although its greatest 

 changes take place in a vertical direction. To the ex- 

 tent that the temperature distribution of the ocean is 

 subject to diurnal and seasonal changes, k is also a 

 function of the time t. However, in the following 

 sections we shall usually simplify matters by disre- 

 garding these variations in space and time and by 

 treating k as a constant. 



2.2 WAVE EQUATION IN A PERFECT 

 FLUID 



2.2.1 



Derivation 



If in a certain region of a fluid in equilibrium the 

 pressure is changed from its equilibrium value, the 

 fluid immediately produces forces which aim toward 

 restoring the equilibrium state. Vibrations result, 

 which are propagated as waves through the fluid. 

 These waves are sound waves, and the fundamental 

 differential equation governing their propagation will 

 now be developed by using the basic equations de- 

 rived in the preceding sections. The particular equa- 

 tions used are the equation of continuity (4), the 



equations of motion (15), the law of forces (16), and 

 the equation of state (18). 

 From equation (18) we have 



dp 

 dx 



d<T 



K — : 

 dx ' 



d(T 



dp 



dz 



d(T 



K 



dz 



dp 



dy dy' 



By putting these values for d<j/dx, da/dy, and 

 da/dz in the law of forces (16) we obtain 



fx = —K 



da 



fy = 



da 



. da 



h = -«^- 

 dz 



(19) 



dx ' " dy' 



After these values for the components of the force 

 on a small box are substituted into the equations of 

 motion (15), we obtain the following relations. 



dUx 



da 



PTT = 



PTT = 



(20) 



dt 



dz 



Since we assume that density changes and velocity 

 changes are all comparatively small, the expressions 

 pdux/dt and podUx/dt will differ by a second-order 

 term, and hence can be regarded as equal. Then 

 equations (20) become 



Po' 



dUx 



m 



dUy 



Pa- 



da 



+ K- = 



dx 



da 



(21) 



Po- 



= 0. 



dt ' dy 

 dUz da 



~dt '% 



In order to apply the equation of continuity (4), we 

 differentiate the first equation of (21) with respect to 

 X, the second with respect to y, and the third with 

 respect to z. The equations are added, leading to 



d /dUx dUy dUz\ fd^a d^a d^a\ 



"'dtKdx + "^ + ^y^ '^w + ^ + ^y = ^■ 



(22) 



From the equation of continuity, the first parenthesis 

 is —da/dt; and equation (22) reduces to 



dt^ ~ po\ 



k5x^ dy'^ 



(23) 



Since a = p/k, from equation (18), where k is con- 

 stant, equation (23) implies 



d^p _ K (d^P^ d^p d^p\ 



df 



Po 



+ 7:1 + 



iVdx^ dy 



dz^j 



(24) 



