36 



WAVE ACOUSTICS 



f{y). It is a remarkable fact of mathematics that if 

 we allow the sum to be an infinite one, then any func- 

 tion of y can be expressed in that form. That is, it is 

 possible to express f{y) as 



/(2/) = H^i^Ay) 



3=1 



= 2^cy sm —y 

 y=i Ay 



(155) 



where the c, depend on the law of formation of the Xy 

 [whether (145) or (149), etc.], and usually become 

 small very rapidly as j increases. 



If the boundary conditions are (139a), then the Xy 

 are given by equation (145), and we have 



f(y) = T.^j sin -y. 

 j=i ^ 



(156) 



In equation (156) the coefficients cy are given, ac- 

 cording to the usual laws of Fourier analysis, by 



2 f^ TTKV 



c. = 7 f(y) sin -rdy. (157) 



The c, are called Fourier coefficients of fiy). Once 

 we know these Fourier coefficients, we can solve our 

 problem as in the finite case. We have 



CO 



p(y,o) = T,cjUy) (158) 



as our initial condition; and 



CO 



P(y,t) = Z ^J cos 2;r/y(< - €y)l^y(2/) 



y=i (159) 



Aj cos 2irfj€j = Cj 



as the set of solutions to the total problem. 



While each term in the sum (159) represents a sta- 

 tionary state of vibration, the infinite sum is not sta- 

 tionary, in view of the fact that the terms have 

 different frequencies /y. 



2.7.2 General Waves in a Medium 

 with Parallel Plane Boundaries 



Section 2.7.1 showed how the assumption of a 

 stationary state led to a possible solution which satis- 

 fied the wave equation, the boundary conditions, and 

 the initial conditions. However, the treatment in 

 Section 2.7.1 was restricted to the case of plane waves 

 moving perpendicular to two enclosing plane bound- 

 aries. This section explains how this method may be 

 generalized for the case of general waves in a medium 

 with two parallel plane boundaries. 



We assume again a stationary state in the medium, 

 of the form 



P{x,y,z,t) = cos2Tft-\l/(x,y,z). (160) 



If this solution is substituted into the wave equation 

 (27), we find that ^ satisfies a partial differential 

 equation of the form 



ay av 



dV 47r2 







(161) 



which is time independent. In addition, ^ must 

 satisfy the boundary conditions imposed at the 

 bounding planes y = and y = L. The boundary 

 conditions may be of the form (139a), (139b), or 

 (139c). 



We shall treat only the case characterized by the 

 conditions (139a). The treatment of the other cases 

 is completely analogous. We attempt to find a solu- 

 tion of equation (161) of the form 



,p(x,y,z) = sin ^Gix,z) (162) 



in which the constant Xj, must have one of the values 



X„ = 



2L 



J = 1, 2, 3, 



(163) 



dz^ \X2 xy 



to satisfy the boundary conditions. For the G{x,z) 

 we have the equation 



^ + ^ + Mf,-.-^JG = 0. (.164) 



Any solution of this equation when multiplied by 

 cos 27r/< sin 2jr!//Xa is a solution of the wave equation 

 (27) and also satisfies the boundary conditions (139a). 

 Equation (164) will be satisfied by any plane wave 

 solution in the xz plane with the wavelength X* 

 given by 



X* y X 



x^ x^ 



Such a solution will be of the form 



27r . 2ir 



G = ai cos -^s -\- a' sm — s; s = x cos d + zsind- 

 X* X* 



(164a) 



It is easily verified that this function G satisfies (164). 

 If X„ < X, X* is imaginary instead of being real. In 

 that case, the solution should be written in the form 



G = hxe 



(2)r/X')s 



+ hi.e 



-(2ir/\')s 



in which X' is the real constant given by 



X' \ \l x= 



