142 DYNAMICAL THEOKY OF SOUND 



53. Square Membrane. Normal Modes. 

 To ascertain the normal modes of a limited membrane we 

 assume as usual that f varies as cos (nt 4- e), so that 



where k* = n*p/P ......................... (2) 



At a fixed boundary we must have f = 0. It is found that 

 the solution of (1) subject to this condition is possible only 

 for a series -of definite values of k, which determine, by (2), the 

 corresponding frequencies. 



In the case of a rectangular membrane, we take the origin 

 at a corner, and the axes of x, y along the edges which meet 

 there. The equations of the remaining edges being, say, 

 x = a, y = b, the equation (1) and the boundary condition 

 are satisfied by 



f=0sin sin^cos(ri$ + e), ......... (3) 



CL (J 



where s, s' are integers, provided 



It may be shewn, by an easy extension of Fourier's theorem, 

 that (3) is the only admissible type of solution in the present 

 case ; it was given by Poisson in 1829. 



In any normal mode for which s or s' > 1, we have nodal 

 lines parallel to the edges. It appears from (4) that if the 

 ratio a? : b 2 is not equal to that of two integers, the frequencies 

 are all distinct, and the nodal lines are restricted to these 

 forms. But if a 2 : 6 2 is commensurable, some of the periods 

 coincide, and the corresponding modes may be superposed 

 in arbitrary proportions ( 16). The nodal lines may then 

 assume a great variety of forms. The simplest instance is 

 that of the square membrane (a = 6), when 



) ...................... (5) 



