1. Consider the Helmholtz equation (*) vu+u=0, on the periodic cut disk domain Dex = {r > 0; 0

[answerhappygod]

1 Consider The Helmholtz Equation Vu U 0 On The Periodic Cut Disk Domain Dex R 0 0 O 0 Defined In Terms 1 (152.72 KiB) Viewed 22 times

1. Consider the Helmholtz equation (*) vu+u=0, on the periodic cut disk domain Dex = {r > 0; 0 <O< 0*}, defined in terms of polar coordinates (r,() and with 0) < 0* < 27. The domain is periodic in the O direction with period 0*. (a) Using a separation of variables technique, show that the most general *- periodic, bounded, separable solution to (*) is (1) u(r,0) = Ayeino Jun (r); Vn 2nt A* n=-00 for some coefficients An, where Jun is a Bessel function of the first kind of index Vn. (You may quote without proof the general solution of Bessel's equation with either integer or non-integer index.) (b) Now suppose (* = 27. Show that u = eiy, where y is the usual Cartesian u coordinate, satisfies (*) and can thus be written in the form of (t). In this case, show that the coefficients An satisfy 27 27A.Ju(r) = 5 * citrin 0-10) Ann de. (c) By evaluating this expression at a particular value of r, show that Ao = 1. By considering the limiting behaviour of the nth derivative of Jn(r) as r + 0, determine An for all n > 0. Explain clearly how you can also calculate the values of An for all n < 0, and hence deduce that "t2 er(e-t-1)/2 = 4"Jy("). tJnr). n=-00