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**Unformatted text preview: **A table of their values (for the case c = 1) can be found in the preceding section. The Bessel roots do not follow any easily discernible pattern, and are not rational multiples of each other. This result, known as Bourget’s hypothesis, [ 142 ; p. 484], was rigorously proved by the German pure mathematician Carl Ludwig Siegel in 1929, [ 127 ]. Thus, the vibrations of a circular drum are also truly quasi-periodic, thereby providing a mathematical explanation of why drums sound percussive. The frequencies ω ,n = c ζ ,n correspond to simple eigenvalues, with a single radially symmetric eigenfunction J ( ζ ,n r ), while the “angular modes” ω m,n , for m > 0, are dou- ble, each possessing two linearly independent eigenfunctions (12.147). According to the general formula (12.133), each eigenfunction engenders two independent normal modes of vibration, having the explicit forms cos( c ζ ,n t ) J ( ζ ,n r ) , sin( c ζ ,n t ) J ( ζ ,n r ) , cos( c ζ m,n t ) J m ( ζ m,n r ) cos mθ, sin( c ζ m,n t ) J m ( ζ m,n r ) cos mθ, cos( c ζ m,n t ) J m ( ζ m,n r ) sin mθ, sin( c ζ m,n t ) J m ( ζ m,n r ) sin mθ. (12 . 149) The general solution to (12.144–145) is then expressed as a Fourier–Bessel series: u ( t, r, θ ) = 1 2 ∞ summationdisplay n = 1 bracketleftbig...

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