The equation vxx + vyy + k2 v = 0 is a generalization of Laplaces equation and is

Chapter 11, Problem 7

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The equation vxx + vyy + k2 v = 0 is a generalization of Laplaces equation and is sometimes called the Helmholtz12 equation. (a) In polar coordinates the Helmholtz equation is vrr + (1/r)vr + (1/r2 )v + k2 v = 0. If v(r, ) = R(r)(), show that R and satisfy the ordinary differential equations r2 R + rR + (k2 r2 2 )R = 0, + 2 = 0. (b) Consider the Helmholtz equation in the disk r < c. Find the solution that remains bounded at all points in the disk, that is periodic in with period 2, and that satisfies the boundary condition v(c, ) = f(), where f is a given function on 0 < 2. Hint: The equation for R is a Bessel equation. See of Section 11.4.