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Unformatted text preview: Vibration of a Rectangular Drum Let us first consider the vibrations of a membrane in the shape of a rectangle R = braceleftbig < x < a, < y < b bracerightbig , with side lengths a and b , whose edges are fixed to the ( x, y )plane. Thus, we seek to solve the wave equation u tt = c 2 u = c 2 ( u xx + u yy ) , < x < a, < y < b, (12 . 136) subject to the initial and boundary conditions u ( t, , y ) = u ( t, a, y ) = 0 = u ( t, x, 0) = u ( t, x, b ) , u (0 , x, y ) = f ( x, y ) , u t (0 , x, y ) = g ( x, y ) , < x < a, < y < b. (12 . 137) As we saw in Section 12.2 the eigenfunctions and eigenvalues for the associated Helmholtz equation on a rectangle, c 2 ( v xx + v yy ) + v = 0 , ( x, y ) R, (12 . 138) when subject to the homogeneous Dirichlet boundary conditions v (0 , y ) = v ( a, y ) = 0 = v ( x, 0) = v ( x, b ) , < x < a, < y < b, (12 . 139) are v m,n ( x, y ) = sin mx a sin ny b , where m,n = 2 c 2 parenleftbigg m 2 a 2 + n 2 b 2 parenrightbigg , (12 . 140) with m, n = 1 , 2 , . . . . The fundamental frequencies of vibration are the square roots of the eigenvalues, so m,n = radicalBig m,n = c radicalbigg m 2 a 2 + n 2 b 2 , m, n = 1 , 2 , . . . . (12 . 141) The frequencies will depend upon the underlying geometry meaning the side lengths of the rectangle, as well as the wave speed c , which is turn is a function of the membranes...
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