If I have a sphere with radius r, a distance d away from a sound source of intensity I_{0}. What will the intensity I of the sound wave be on the point of the sphere directly opposite the source? Preferably I would like to find the intensity of the wave at any point on the sphere.
Any point would be difficult, because the sound source is "directly visible" from half the sphere at most. Other points can be reached only indirectly, and the overall analysis becomes very complex. For the sound that is directly incident, the intensity is inversely proportional to the square of the distance from the source.
I'm only concerned with the indirect half of the sphere. I want to know how to figure out the intensity of sound after it has diffracted around an object. I'm also assuming that there are no reflected sound waves. Is there any formula to calculate the intensity of a diffracted sound wave around a sphere?
Rayleigh's Theory of Sound had a section on sound propagation in the presence of a spherical obstacle. But that was anything but simple. I do not know whether there is a more recent and simpler exposure. I would guess there could also be studies on the diffraction of light by spherical bodies, I think their results could be adapted.
You might get some information by looking up the math of "head related transfer functions" - i.e the methods used in video games etc to convert sound coming from an arbitrary point in space, into the two inputs at your ears. Or, see if there is any theory relating to "dummy head" binaural sound recording techniques. More recent than Rayleigh - yes. Simpler - not necessarily!