A unit-amplitude, monochromatic plane wave of a scalar field (r) scatters from an origin-centered obstacle of finite

A unit-amplitude, monochromatic plane wave of a scalar field ψ(r) scatters from an origin-centered obstacle of finite size. Apart from a factor of exp(−iωt ), the field takes the asymptotic form

Focus on the almost-forward direction where θ > z/k collects the energy of the wave at a distance z >> R from the origin. Show that

where ƒ(0) is the scattering amplitude evaluated on the z-axis. Use this result to argue that

Transcribed Image Text:

(r, 0, 0)~ exp(ikz) + exp(ikr) r -f(0, $).