Concept

Doppler — where it appears

The change in observed frequency and amplitude when a source moves relative to a listener. It is exactly the derivative of the mapping between emission time and arrival time, so it becomes singular where a source approaches at the speed of sound.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

compressible · Aeroacoustics

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.

Aeroacoustics Speed of sound Acoustic analogy Multipole Noise Retarded time Boundary condition Convolution Correlation Dissipation Efficiency Far field

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