Monopole · Harmonic
Interactive 2D visualization of monopole, dipole, and Green's function acoustic radiation.
This visualization shows the acoustic pressure field radiated by elementary point sources in 2D cross-section. These point-source solutions are fundamental building blocks for more complex acoustic radiation problems: any vibrating object can be represented (via boundary integral methods) as a distribution of point sources on its surface.
A pulsating volume source (imagine a tiny sphere whose radius oscillates). It radiates spherically symmetric waves whose amplitude decays as 1/r. The radiated pressure is proportional to the time derivative of the volume flow rate; a source with constant volume flux produces no sound. For a harmonic source q(t) = Q0 sin(ωt):
p(x, t) = ρ0 Q0 ω cos(ωt − kr) / (4πr)where k = ω/c is the wavenumber and r is distance from the source.
The limiting case of two equal-and-opposite monopoles at vanishing separation. It arises physically from an oscillating rigid body (no net volume change, just displacement of fluid). The radiation pattern has a cos(θ) directivity ("figure-8"): maximum along the oscillation axis, zero in the equatorial plane.
The dipole field has a near-field region (within a wavelength of the source) where the pressure decays as 1/r2 and oscillates nearly in phase with the source, and a far-field region (many wavelengths away) where it decays as 1/r and carries propagating wave energy. In the far field, the radiated pressure depends on the jerk (time derivative of acceleration) of the source.
p = −ρ0 cosθ / (4π) [ D''(τ)/(cr) + D'(τ)/r2 ]where τ = t − r/c is the retarded time.
The response to an impulsive point source: a delta function in space and time. The solution is a spherical shell expanding at the speed of sound, with 1/r amplitude decay. Any acoustic field can be constructed by superposing Green’s functions weighted by the source distribution—an idea central to boundary element methods for sound computation.
G(x, t) = δ(t − r/c) / (4πr)Visualized here as a narrow Gaussian pulse for clarity.
Two discrete monopoles of opposite sign, separated by a controllable distance h. As h decreases (becoming small relative to the wavelength), the interference pattern converges to the smooth dipole pattern, illustrating how the point-dipole solution arises as a limiting process.
The underlying theory follows Howe (2003), Chapters 1.4–1.7.