8. Acoustic Radiator (FW‑H)

The raw pressure trace at the tailpipe exit does not directly sound like a bang; it must be converted to radiated sound. The Ffowcs Williams–Hawkings (FW‑H) equation [1] provides the exact far‑field pressure generated by a surface with mass and momentum injection.

FW‑H Equation (Stationary Source)

For a stationary exit plane, neglecting the quadrupole term (valid for exit Mach < 0.9), the radiated pressure at listener distance $r$ is:

$$ p'(t) = \frac{\rho_0}{4\pi r}\,\dot{Q}\!\left(t - \frac{r}{c_0}\right) \;+\; \frac{1}{4\pi r c_0}\,\dot{F}\!\left(t - \frac{r}{c_0}\right), $$

where

$Q(t) = \dot{m}_{\text{exit}} / \rho_0$ – volume flow rate.
$F(t) = (p_{\text{exit}} - p_0)A_{\text{exit}}$ – net force on the exit plane.
Derivatives $\dot{Q}, \dot{F}$ convert step changes into strong acoustic pulses, exactly the crack of a gunshot or exhaust pop.
The retarded time $r/c_0$ models the propagation delay.

Implementation

The code below computes the source signal for one audio sample. The delay line first writes the current source, then reads the delayed value corresponding to the propagation time.

// Per sample (C# float arithmetic for audio)
float Q  = massFlow / rho0;
float F  = (pressure - p_atm) * exitArea;
float dQ = (Q - lastQ) * sampleRate;
float dF = (F - lastF) * sampleRate;
float monopole = (rho0 / (4.0f * MathF.PI * r)) * dQ;
float dipole   = (1.0f  / (4.0f * MathF.PI * r * c0)) * dF;
float source   = monopole + dipole;

int delaySamples = (int)MathF.Round((r / c0) * sampleRate);
delayLine.Write(source);
float output = delayLine.Read(delaySamples);

lastQ = Q;
lastF = F;

For moving sources or listeners the distance $r$ changes continuously; using integer‑sample delays introduces zipper noise. In that case a fractional‑delay interpolation (linear, Lagrange, or Thiran) should replace the simple integer‑read delay line.

Quadrupole (Jet Noise) Option

The quadrupole term $\frac{1}{4\pi r c_0^2}\ddot{T}_{ij}$ is small for subsonic exits but can be added as a filtered noise source whose amplitude scales with $U^8$ (Lighthill’s law [2]) to produce the characteristic jet roar. This is mixed in only when the exit Mach number exceeds ~0.5.

References

J. E. Ffowcs Williams, D. L. Hawkings, “Sound generation by turbulence and surfaces in arbitrary motion,” Phil. Trans. R. Soc. Lond. A, 1969.
M. J. Lighthill, “On sound generated aerodynamically. II. Turbulence as a source of sound,” Proc. R. Soc. Lond. A, 1954.

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