Concept

Sound power — where it appears

The rate at which a source sends acoustic energy away to the far field, independent of where it is measured once outside the near field. For aerodynamic sources it grows as a high power of speed — the sixth for a dipole, the eighth for a quadrupole.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

The sound is what fails to cancel. A compact quadrupole of kd = 0.01, followed outwards. The upper curve is what one of its four sources produces on its own at each radius — every one of them is as loud as a monopole — and the lower curve is what all four produce together. Inside the source they barely cancel at all; by one radian of wavelength the sum is 9.3e-5 of what a single source is doing, and it is falling as 1/R from there on because that is what radiation does. Between the two the field falls as the cube of the distance, which is a near field rather than a sound.

The sound is what does not cancel

A quadrupole is not a weak source. Every one of the four monopoles in it is as loud as a monopole of the same strength, and what makes the assembly quiet is that they very nearly cancel — one part in ten thousand survives at a hundredth of a wavelength. The eighth-power law is a statement about how nearly, not about how little.

compressible · Aeroacoustics

The body's dipole outshouts the flow's quadrupoles by one over the Mach number squared. The power of a wire's lift tone over the generous bound on its wake's quadrupole noise, against the wind's Mach number, for rms lift coefficients 0.2, 0.45 and 0.6 — the spread of the measurements. At Mach 0.03, a 10 m/s wind, the ratio is 9.4·10⁴; it falls as one over the Mach number squared and is still 104 at Mach 0.9, where a real cylinder's shedding has long since changed.

A wire sings with the force on it, not the flow round it

A free flow can only radiate as a quadrupole, four sources nearly cancelling, which is why a jet's noise grows as the eighth power of its speed and so little of its power becomes sound. Put a body in the flow and the cancellation is broken: the fluctuating force the body exerts is a dipole, its sound grows as the sixth power, and at the speed of any wind it outshouts the turbulence that made it by a factor of tens of thousands. The whistle of a wire is the force of its own wake, radiated.

compressible · Aeroacoustics

A steady force carried round a circle sings, and each harmonic rises steeply with the tip's Mach number. The loudest level, 30 m away, of the first four blade-passing harmonics of a two-bladed 1.88 m propeller giving 2 kN of thrust from 127 kW, against its tip Mach number. At a tip Mach number of 0.4 the fundamental is 89.8 dB and the fourth harmonic 51; at 0.78, the climb, 98.2 and 88.6; at 0.95, 101 and 98.9.

More blades hush a propeller only while its tips are slow

A propeller's thrust and torque are steady in the blade's own frame, and a steady force held still makes no sound. Carried round a circle it sings at the blade-passing frequency, and Gutin's formula says how loudly: through a Bessel function whose order is the harmonic's and whose argument is that order times the tip's speed towards the listener over the speed of sound. Such a function is exponentially small until its argument nears its order. So spreading the same thrust over more blades, which raises the order of the fundamental, silences it by twenty-five decibels at a tip Mach number of 0.4 and by one at 0.95.

compressible · Aeroacoustics

A five per cent distortion gives six blades a tone that no longer falls with the tip's speed, and two blades hardly notice. The sound power of the blade-passing fundamental against tip Mach number, thrust and shaft power held, for two and six blades: from the steady load, and from the thrust's fluctuation as each blade cuts an inflow distortion of 5 per cent rms. At tip Mach 0.4 six blades give 100 dB from the steady load and 113 from the distortion; two blades 127 and 109. The six-bladed steady tone overtakes its distortion tone at tip Mach 0.593.

A distorted inflow gives back the tone more blades took away

A propeller's steady thrust sings through a Bessel function whose order is the number of blades times the harmonic, and that function is exponentially small at low tip speeds — which is why six blades are quieter than two. But a blade that cuts a non-uniform inflow carries a thrust that changes round the circle, and its s-th harmonic radiates through a Bessel function of order six minus s. The sixth needs no tip speed at all. A distortion of five per cent turns six blades' tone into one that hardly falls as the tips slow, fills the axis where the steady tone is silent, and cuts the six-blade advantage over two from twenty decibels to thirteen. Two blades never notice.

compressible · Aeroacoustics

Named alongside it

The objects these essays reach for when they reach for this one.

Aeroacoustics Far field Multipole Acoustic analogy Cancellation Compactness Measurement Model limit Noise Averaging Bessel function Correlation

All concepts