Acoustic Field Modes Around a Rigid Sphere

Each panel shows the spatial pattern of one mode: j_n(kr) · P_n(cos θ). These are the building blocks of the scattered field in Rudgers (1969), Eq. 9–11. The z-axis (vertical) is the polar axis; the rigid sphere is shown in gray at the center.

ka = 2.5 (wavenumber × sphere radius)

n = 0 — monopolar

Spherical symmetry

n = 1 — dipolar

2 lobes (+ / −)

n = 2 — quadrupolar

4 lobes (±/∓)

negative

positive

Field: j_n(kr) · P_n(cos θ) | x-axis horizontal, z-axis vertical (polar) | domain: −5a to +5a

n = 0: P₀(cos θ) = 1 — no angular dependence. Rings are perfect circles. Dominant at very low ka.

n = 1: P₁(cos θ) = cos θ — field is zero at the equator (θ = 90°) and opposite in sign between hemispheres.

n = 2: P₂(cos θ) = ½(3cos²θ − 1) — two angular nodes at θ ≈ 54.7°, producing four lobes.

ka slider: increasing ka tightens the Bessel function oscillations radially. In Rudgers' Eq. 11, the full scattered field is a weighted sum of all these modes.