A spherical radiator undergoing arbitrarily vibration, its normal surface velocity can be expressed as a linear combination of spherical modes in the form of infinite series.

For a pulsating spherical radiator(n=0), the modal coefficients  can be calculated by applying the orthogonality property of the Legendre functions.

And we found the modal coefficients  only has value with the n=0 order, and for other orders ( n=1,2,3,…..) the corresponding values are all zero, which agree with the physical meaning of pulsating sphere.

My questions are following:

(1)   For a spherical piston radiator with its cap angle  , how can I calculate its modal coefficents  ? And if the  for a pulsating piston radiator has the same rule with the pulsating whole sphere radiator ?

(2)   For a oscillating sphere radiator(n=1), or other higher order e.g. n=2, 3,4,…., how can I calculate the corresponding modal coefficients ?

The details of my  questions are in the attached file, would you please help me ?