NORMAL MODE THEORY 



37 



G is thus the sum of two terms, one increasing ex- 

 ponentially with the distance s from the .source, the 

 other decreasing exponentially. The only solutions of 

 this character which have physical significance are 

 those for which the first (increasing) term is zero. If 

 bi were not zero, the sound intensity would increase 

 rapidly with the distance from the source and that is 

 physically impossible. 



Since the greatest possible value of X„ is 2L, or 

 twice the depth, it follows that sound of wavelength 

 greater than 2L will have an exponential rather than 

 a trigonometric solution; because 61 must be zero, 

 such sound will suffer an exponential pressure decay 

 with increasing range. It is clear that the longer the 

 wavelength, the more rapid will be the decay. A more 

 detailed discussion of this type of transmission is 

 given in a report by the Naval Research Laboratory 

 [NRL], where the detailed properties of the bottom 

 are taken into account.* 



The angle d in the solution (164a) may be chosen 

 arbitrarily. Thus to any value of j in the equation 

 (163) belongs an infinite set of characteristic func- 

 tions. These characteristic functions can be com- 

 bined to satisfy particular initial conditions; how- 

 ever, the rules for their combination are too involved 

 to be presented here. 



The derivation of the wave equation (27) was based 

 partly on the assumption that the velocity of propa- 

 gation was everywhere the same, in other words, that 

 the medium was homogeneous. Let c be an arbitrary 

 function of {x,y,z). In the ocean, the variation in 

 sound velocity is due mainly to the variation in water 

 temperature with depth. 



To assume that the velocity is variable, amounts 

 for most practical purposes to assuming that c in 

 equation (27) is now a function of position. The 

 method of normal modes can be appUed to find a 

 solution in that case just as in the case of constant 

 sound velocity. As before, we assume a stationary 

 state of form (160). Substituting equation (160) into 

 the wave equation, we get as the time-independent 

 differential equation the following: 



d^ aV av 4^P 



— I- + — I- + -I- -I ^—4, = 



dx- dy^ dz^ (?{xyz) 



(165) 



which differs from equation (161) only in that c is 

 now variable. The solution of equation (165) satis- 

 fying the imposed initial and boundary conditions 

 can be found as before by the superposition of an 

 infinite number of normal modes; in this case of vari- 

 able c, however, the computation of the characteristic 



values and functions is more trouble.some. An ap- 

 plication of this type of analysis is discussed in 

 Section 3.7. 



2.7.3 Intensity as a Function of 

 Phase Distribution 



Whenever the sound source is harmonic, the pres- 

 sure distribution resulting from given initial and 

 boundary conditions can be written in the form 



p = a(x,2/,2)e^''-^'^'-*<^'"'^'^ (166) 



a and « being real functions of x,y, and z. For some 

 purposes it is convenient to set 



V{x,y,z) 



e(x,y,z) = 



Co 



(167) 



(168) 



(169) 



so that equation (166) becomes 



p = a(a;,z/,2)e^"^C'-<''^">^ 

 or, explicitly for the real pressure, 



/ v\ 



p = a{x,y,z) cos27r/l < )• 



Since we know from Section 2.4.2 that I = pu, we 

 must derive an expression for pu. This is done by 

 making use of equation (44), relating the derivatives 

 of p and u^: 



dUx 1 dp 



dt p dx 



From equation (169), then 



dp .. „.2wfdV da 



— = a(sm H) 1 cos « 



dx Co dx dx 



(170) 



where 



H = 2Tf(t--\ (171) 



Therefore, from equation (170), 



dUx a . „2TrfdV 1 ^da 



— = — sm H cos H— • 



dt p Co dx p dx 



Integrating this, we get 



M, = — cos H- —-7- sm H—- (172) 



pCo ax 2irfp ox 



From equations (172) and (169), we obtain 



pux = — cos^ H — — - sm H cos H— • (173) 



pco dx 2-Kjp dx 



The average energy flow in the x direction Ix, at the 

 point x,y,z, is just the time average of equation (173) 

 over a complete period. The time average of the 



