Note: j is used in this set of notes instead of i. ECE 6382 Fall 2016 David R. Jackson Notes 25 Scattering by a Sphere Note: j is used in this set of notes instead of i.

Scalar Wave Transformation y x z Incident wave: (scalar function, e.g. pressure of a sound wave) The general form of the solution in spherical coordinates is

Scalar Wave Transformation y x z Proposed form of incident wave: Notes: No  variation  m = 0 Must be finite at the origin (only jn) Must be finite on the entire z axis (no Q, only P with  = n)

Scalar Wave Transformation Use so that Multiply both sides by and integrate. Orthogonality:

Scalar Wave Transformation (cont.) We then have We can now relabel m  n.

Scalar Wave Transformation (cont.) Let We then have

Scalar Wave Transformation (cont.) The coefficients are therefore determined from To find the coefficients, take the limit as x  0.

Scalar Wave Transformation (cont.) Recall that Note: Therefore, as x  0, we have

Scalar Wave Transformation (cont.) As x  0 we therefore have

Scalar Wave Transformation (cont.) Problem: If we now let x  0 we get zero on both sides (unless n = 0). Solution: Take the derivative with respect to x (n times) before setting x = 0.

Scalar Wave Transformation (cont.) Hence Define:

Scalar Wave Transformation (cont.) Hence Next, we try to evaluate In: (Rodriguez’s formula)

Scalar Wave Transformation (cont.) Therefore Integrate by parts n times: Notes:

Scalar Wave Transformation (cont.) Schaum’s outline Mathematical Handbook Eq. (15.24):

Scalar Wave Transformation (cont.) Hence

Scalar Wave Transformation (cont.) We then have Note: Hence,

Scalar Wave Transformation (cont.) Now use so, Hence

Acoustic Scattering z Rigid sphere y x Acoustic PW so,

Acoustic Scattering (cont.)

Acoustic Scattering (cont.) We have where Choose Hence,

Acoustic Scattering (cont.) We then have

Acoustic Scattering (cont.) Incident wave Real part of total pressure near a sphere for ka = 1.0 http://www.paraffinalia.co.uk/Software/examples.shtml