Far-field scattered pressure (Rudgers 1969, Eq. 11): f(ka, θ) = −(2/ka) Σn Pn(cosθ)(2n+1) sinηn eiηn, where ηn is the rigid-body phase shift from Eq. 10. The left panel shows the full angular scattered pattern |f(ka,θ)|. The right panel shows the real weight each mode contributes: (2n+1)|sinηn|. The dashed line marks n ≈ ka — modes beyond this contribute negligibly.
Polar plot: Each point on the curve is the scattered amplitude |f| in that direction. Incident wave arrives from the left along the z-axis. θ=0 (right) is forward scatter; θ=π (left) is monostatic backscatter (the geometry of Rudgers' Fig. 6–7).
Bar chart: The highlighted bar is the dominant mode. Note how the peak roughly tracks n ≈ ka. At low ka (Rayleigh regime), n=0 dominates. At high ka, many modes contribute, creating directional lobes.
Nmax slider: Reduce it to watch the sum converge mode by mode. Significant convergence requires Nmax ≳ ka + a few extra terms — exactly the truncation Rudgers uses when integrating Eq. 8.
ηn (phase shifts): These encode the Neumann boundary condition (∂p/∂r=0 at r=a for a rigid sphere). They are the information the sphere "writes" onto each spherical harmonic of the scattered field.