Question

We've seen that the wave equation takes the form 1 af 2 12 = 0, where V2 is the Laplacian, v is the wave velocity, and f is the function describing the wave. In two Cartesian dimensions the Laplacian reduces to V2 f = 0f + 3f, which is a very convenient form for solving systems with rectangular symmetry (modes in a rectangular box, for instance). However, suppose that you had a circular drumhead, say of radius a. In that case, it would likely make more sense to switch to polar coordinates, r and o, instead, in which the function f = f(x, y) + f(ro). In these coordinates, Eq. (1) takes the form af laf 1 af 1 af ar2 rar * 2262 2242 = 0, (2) as you can check by making the substitutions I = r cos o and y = r sin o in the two- dimensional Laplacian, and using the chain rule for derivatives. Changing the geometry of the wave changes the solutions, of course. Instead of sines and cosines, one instead finds solutions in terms of Bessel functions. You get to investigate these solutions, a little bit. 1. Show that Eq. (2) is solved by f(r,0, t) = Jm(kr) (aleimó + aze-im) (bzewt +bzeit), (3) where aı, az, bi, b2, m, and k are all constants, and Jm is the mth order Bessel function of the first kind. Determine k in terms of w, m, v, or whatever variables it depends on. Hint: look up Bessel functions!

Please Help! If possible show every step my biggest issue is understanding how to get rid of the Bessel Function. Thank you!

Answer 1

Answer: Given that, The walle equation takes the form of veno And, we have a circolar drowhead of radius a, the Vesti-O amplitude fired, t) will outiefies the wave equation flary) flrid) alf +1 at the 22f latt! heart to + Form (1) show that Eq③ is solved by forud,t) - Imlker) lacine tazeina) (bieth the eint) let, are function flridt is variable seperable thus, we brenlite the function as, flr,d,t) = Rem(T! Im (d) TH) with I (4) = A etimo be the solution for Imd) other solutions are: 1 2 3 4 1 aak?, gle + oh yo me R = ker Y2

solutich for t is: Titl=beint tbz etkut k² a we Solution for yis a bassed ope" in the Wariable kr with ye Solution ImRr) the that is nou Signular every where on drumhead. The function Iml Rv) must vanish at r=a. so the point Ra must be keros of Im Solution for p is: B & (6)= a, eine ťazenima Thus the wave equation may be reduced to flrid,t) = Im1R6) la, e impreine) (be ellot the enint) Where R²= we