Two coherent point sources radiate sound pressure that falls off as 1/r (spherical spreading in 2D cross-section) and oscillates as a travelling wave:
p_i(r,t) = (A_i / max(r,r_min)) · sin(k_i·r − ω_i·t + φ_i)
k_i = 2π / λ_i, λ_i = v_sound / f_i, v_sound = 343 m/s
p_total(r,t) = p_1(r,t) + p_2(r,t)
The heatmap draws p_total across the whole field every frame (animation runs in slow motion so the fringe pattern is visible — the spatial wavelength is the real acoustic λ, only the frame-rate playback speed is scaled). Drag the probe (yellow ring) anywhere in the field to sample it directly.
- Path difference Δr = r₁ − r₂ at the probe, plus the phase-offset slider, sets the steady-state interference order n = Δr/λ + Δφ₀/360°. Integer n ⇒ constructive (crests align, +6 dB vs. one source alone); half-integer n ⇒ destructive (waves cancel).
- Relative level (dB) is 20·log₁₀(envelope amplitude / single-source reference), so 0 dB is one source at 1 m and +6 dB marks perfect constructive overlap of two equal sources — the classic result for coherent superposition.
- Beat frequency = |f₁ − f₂| appears only when the two sources are detuned; it is exact regardless of probe position, since it comes purely from the frequency difference, not from geometry.