Answer Happy

1. Consider the Helmholtz equation (*) Vºu+ 1 = 0, on the periodic cut disk domain De* = {r > 0; 0 <0 <*}, defined in terms of polar coordinates (r, 0) and with 0 < 6* < 27. The domain is periodic in the 0 direction with period 0*. (a) Using a separation of variables technique, show that the most general 0*- periodic, bounded, separable solution to (*) is ю 2пп , Ꮎ (+) u(r, ) = Aneivino Jun (r); = Vn Vn = A* n=-0 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 0* 27. Show that u = eiy, where y is the usual Cartesian coordinate, satisfies (*) and can thus be written in the form of (†). In this case, show that the coefficients An satisfy 27 AnJn() = ** citr sin o-no) = 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 In(r) as p = 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 er(t +-1)/2 = t" Jn(r). n=-