Spherical coordinates and standing waves

May 2010
Hello, for some reason I can't seem to figure this out. Hopefully someone can point me in the right direction.

I am trying to solve for the following function f(r, theta, phi), with boundary conditions: f(0, theta, phi) = 1. The function must have a defined wavelength, and is symmetrical in terms of theta and phi. The volume from r = 0 to half the wavelength is equal to the (negative) volume from r = half of the wavelength to the wavelength.

I've simplified the problem down to:
integral( p(r) dr, 0, wavelength / 2) = - (12/(wavelength)^2)*integral( p(r) dr, wavelength / 2, wavelength)

where p(r) is equivalent to f(r, theta, phi) at any angle.