Question

Please follow the hint (picture 2)

+ 12 302 27. A drumhead is a circular membrane of radius a. When it is struck, waves propagate across the drumhead. The membrane vibrates with displacement 5, where $(r, 0, 1) = n(r, e)e-ie and n(t,0) satisfies the Helmholtz equation 1 a an 1 aan +k?n=0 rar where k = w2/u2 and v is the speed with which waves propagate across the drumhead. (The speed v depends on the tension in the drumhead, among other things.) The bound- ary condition is that n= 0 at r= a. Separate variables, and find the eigenfunctions. Determine the first three allowable frequencies w in terms of the drum parameters v and a.

Answer 1

Please comment if you need any clarification.

If you find my answer helpful please put thumbs up.thank you..

Let » - RO Then (3*)et er ho (3*) + KRE-0 292 +42 + 1 de -0 re 202 Multiply through by to get: ( ) + x202 + best + a2e -0 Once again we choose cosmo beton--m= 0 - sin me and then the equation for Ris Bessel's equation: (*) + (4202 – m2)R - 0 We need a solution that is finite at the origin (the center of the drumhead), i.e. Im(kr). Then we choose the eigenvalues to make n zero atra. Thus ka - X the nth zero of Im(x). The solutions are of the form Im(imná ) (Am. cosm8 + Bm sinme) and the allowed frequencies are The zeros are on -2.4048. 5.5201 and 8.654. II. - 3.8317, 7.0156 X2. - 5.1356 Thus the first three frequencies are w - 2.4048v/a, 3.83177/a and 5.1356v/a. CS Scanned with CamScanner