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Partial Differentials Notes_Part_16

# Partial Differentials Notes_Part_16 - Vibration of a...

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Vibration of a Rectangular Drum Let us first consider the vibrations of a membrane in the shape of a rectangle R = braceleftbig 0 < x < a, 0 < 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 Δ u = c 2 ( u xx + u yy ) , 0 < x < a, 0 < y < b, (12 . 136) subject to the initial and boundary conditions u ( t, 0 , 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 ) , 0 < x < a, 0 < 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 ) + λ 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 ) , 0 < x < a, 0 < y < b, (12 . 139) are v m,n ( x, y ) = sin mπx a sin nπy b , where λ m,n = π 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 ω m,n = radicalBig λ m,n = π 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 membrane’s density and stiffness. The higher the wave speed, or the smaller the rectangle, the faster the vibrations. In layman’s terms, (12.141) quantifies the observation that smaller, stiffer

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Partial Differentials Notes_Part_16 - Vibration of a...

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