Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Retarded time — where it appears

The time at which a sound now arriving at a listener left its source, earlier than the present by the distance over the speed of sound. A moving source is heard where it was at that earlier time, which compresses or stretches the signal and decides how far its parts cancel.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

When the sound arriving now was made. The arrival time at a fixed observer against the emission time, for a source passing at Mach 0.8. The curve is monotone and its slope is not one: while the source approaches, a long stretch of emission arrives in a short stretch of time, and while it recedes the reverse.

The sound now is the source then

Every acoustic calculation is an exercise in bookkeeping about when. The pressure arriving at a listener was emitted at an earlier time, at a place the source has since left, and the Doppler shift is not a separate effect at all — it is the slope of the curve relating the two.

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

Named alongside it

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

AeroacousticsDopplerNoiseSpeed of soundAcoustic analogyConvolutionFar fieldMeasurementMemory kernelModel limitModel validityMultipole

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