NORMAL MODE THEORY 



35 



Since equation (143) must satisfy ^(L) = 0, 



L L 1 2 3 



sin ztt- =0; - = -, -, -, ■ • • 

 \ \ 2 2 2' 



That is, 



(144) 



This means that under the given boundary condi- 

 tions there cannot be a stationary state of the type 

 of (137) imless the wavelength X has one of a number 

 of definite ratios to the depth L. The ratio \/L must 

 be either 2/1, 2/2, 2/3, 2/4, or in general 2/j. A set 

 of wavelengths Xj can be defined by 



Xy=-L,j = 1,2,3,- 



(145) 



Then, if the actual wavelength in the problem is 

 equal to one of these \y, the expression (143) will 

 satisfy equation (141) ; and the stationary state equa- 

 tion (137), with this value of ^, will satisfy both the 

 wave equation (138) and the boundary conditions 

 (139a). If the actual wavelength is not equal to one 

 of the \j, then there can be no stationary state in the 

 given medium in which the wave planes are parallel 

 to the interfaces. 



Mathematically, all this means that the total 

 problem defined by equations (138) and (139a) can 

 be solved only if the coefficient of yp in equation 

 (140) has certain definite values ay defined by 



4x2 



(146) 



These values, ay, are called characteristic values. The 

 solutions (143) corresponding to them, namely 



27r , — 



i/j{y) =Asm.—y = A sin Vai V (147) 

 Xy 



are called characteristic functions of the problem. 

 Clearly, every characteristic value ay corresponds to 

 a possible frequency /y and wavelength Xy; by pos- 

 sible is meant that it gives rise to a stationary state, 

 or normal mode of vibration. The characteristic func- 

 tion \pj determines the distribution of acoustic pres- 

 sure within this normal mode of vibration. 



If the boundary conditions are changed, the char- 

 acteristic values and characteristic functions change 

 also, although the differential equation which y// must 

 satisfy remains equation (140). If the boundary con- 

 ditions (139b), which correspond most closely to 

 actual conditions in the sea, are assumed, it is found 

 that, by methods similar to those described pre- 

 viously, a normal mode can arise only if 



L 

 X 



13 5 

 4' 4' 4'' 



Xy = — where i 



J 



1,3,5,-- 



(149) 



(148) 



We notice that the characteristic wavelengths Xyare 

 different from the first case. The characteristic values 

 and functions can be calculated by using equations 

 (146) and (147) and by remembering that the Xymust 

 now be taken from equation (149). 



Clearly a sum of two normal modes also satisfies 

 the differential equation (138) and the imposed 

 boundary conditions. 



Suppose we have the general case where the 

 boundary conditions determine an infinite number 

 of normal modes 



p = c 4' Ay) cos 2T/y(< - ey), j = 1, 2, 3, ■ • • . (150) 



Suppose we have initial conditions on y at t = 0, 

 of the nature 



p(y,0) = D,p,{y), (151) 



where D is some constant. For equation (150) 

 to satisfy the initial conditions (151), we must 

 choose J = K, e. = (l/27r/,) arc cos D/c. Since this can 

 always be done, we have the result that if the initial 

 pressure disturbance is a multiple of one of the char- 

 acteristic functions, then one of the solutions of the 

 boimdary problem will also satisfy the initial condi- 

 tions. 



This result can be generalized. If the initial distri- 

 bution of the pressure is a linear combination of 

 several of the characteristic fimctions ^,(2/), then we 

 shall show that these initial conditions can be satis- 

 fied by a corresponding sum of normal modes. Sup- 

 pose 



p{y,0) = T,DjMy) (152) 



where the Dj are any constants. Then the distribu- 

 tion of pressure given by 



p(y,t) = Z Ay cos 27rMt - ei)Uy) (153) 

 J 



will satisfy the initial conditions (152) if only the Ay 

 and ey are chosen so that 



Ay cos 2wfj€j = Dj. (154) 



The expression (153) also satisfies the wave equation 

 and the boundary conditions since it is a sum of 

 normal modes; hence, it is the solution to the problem 

 when the initial pressure disturbance can be expressed 

 as a finite sum of characteristic functions. 



Now suppose the initial disturbance cannot be ex- 

 pressed in the form of (152), but is a general function 



