Partial Differentials Notes_Part_17

Partial Differentials Notes_Part_17 - A table of their...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

Page1 / 3

Partial Differentials Notes_Part_17 - A table of their...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online