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Unformatted text preview: so that the general solutions are given by (Proposition 1 ) G mn ( t ) = B mn cos( mn t ) + B mn sin( mn t ) , B mn , B mn R , (271) for m, n N . The functions u mn := F mn G mn , u mn ( x, y,t ) = ( B mn cos( mn t ) + B mn sin( mn t )) sin parenleftBig m a x parenrightBig sin parenleftBig n b y parenrightBig , (272) m, n N , are eigenfunctions of the second-order differential operators in both space and time: 2 t u mn = F mn G mn = 2 mn F mn G mn = 2 mn u mn , (273) c 2 u mn = c 2 F mn G mn = 2 mn F mn G mn = 2 mn u mn , (274) and therefore they are solutions of the two-dimensional wave equation ( 247 ) which satisfy the boundary condition ( 248 ). The frequency of u mn is mn 2 = c 2 radicalbigg m 2 a 2 + n 2 b 2 = c m 2 b 2 + n 2 a 2 2 ab [Hz] , m,n N . (275) Compare this value with the frequency cn/ (2 L ) of the n-th normal mode of a vibrating string (Section 12.3 ). In the 2D case, it is possible that there are several modes with the same frequency, depending on the values of a and b ! Example: Consider a square membrane of area 1, i. e. a = b = 1. We obtain the modes u mn ( x, y,t ) = ( B mn cos( mn t ) + B mn sin( mn t )) sin ( mx ) sin ( ny ) , (276) with eigenvalues mn = c m 2 + n 2 , (277) for m, n N . Because mn = nm , The modes u mn and u nm have the same frequencies. However, these are two different functions if m negationslash = n . This can be seen from the location of the nodal lines, i. e. points ( x, y ) with F mn ( x, y ) = sin ( mx ) sin ( ny ) = 0: x = k m , k = 1 , .. . ,m 1 or y = n , = 1 , .. . ,n 1 . (278) For example, the modes u 12 and u 21 both have the same frequency c 5 / 2. Their nodal lines are located at y = 1 / 2 and x = 1 / 2, respectively. Any 49 linear combination of u 12 and u 21 is again a solution of the two-dimensional wave equation ( 247 ) which satisfies the boundary condition ( 248 ), and it has the same frequency c 5 / 2. For example, by choosing B 12 = 1, B 21 = p R , B 12 = B 21 = 0, we obtain the vibration cos( c 5 t ) ( F 12 ( x, y ) + pF 21 ( x, y )) , p R . (279) The nodal lines coincide with the zeros of F 12 + pF 21 , and so they are defined by the equation sin( x ) sin(2 y ) + p sin(2 x ) sin( y ) = 0 . (280) We find that solutions ( x, y ) of ( 280 ) satisfy cos( y ) + p cos( x ) = 0 . (281) For every value of p R , we obtain different nodal lines....
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This note was uploaded on 11/17/2011 for the course MATH 529 taught by Professor Staff during the Spring '08 term at UNC.
- Spring '08