I read Lighthill's work and it doesn't seem very convincing in hindsight. Is there better data now or a new explanation?
Direct wave A(t,x)=A0*sin(2Pi*(f*t -x/L)); f*t-x/L=const Phase with speed c=L/f.
Reflected wave A'(t,x)=A0*sin(2Pi*(f*t +x/L)); c'=-L/f.
Summ of A+A' produses positive impulse on reflector and zero impulse on stable A+A' wave.
But neither wave carries any mass with it, hence the momentum is zero. One can certainly superimpose a net flow on a sound wave to get a mass flux but then there are serious problems with reflection at a boundary.
I have a hazy recollection of water surface traveling wave as summ of line and circular movements. Line movement corresponds to average wave sped with non zero impulse. Circular movement correponds to mass movement in direct (on the top) and opposite (on the bottom) to the wave propagation and has zero impulse.
The true Stokes (not Gerstner) solutions give a slight helical advance in the Lagrangian flow called Stokes drift.
My dump explanation :-)
The acoustic wave interacts with moving plate. The interaction time for additional pressure is larger relative to negative pressare in case of plates moving in direction of wave.
The problem solution in difference between stable and moving wave
I guess the answer is in solution of more simple classical task - The long spring with discrete masses and additional bigger mass.
Is average offset of bigger mass not equal to zero?
If you assume linear behavior, then you'll never see any convective term (the flow term) in the acoustic wave equation. On the other hand, if you retain nonlinear terms, convection reveals itself. I would suggest R.T. Beyer's book or papers as a more reader-friendly resource than Lighthill.
I have read many of Beyer's papers and Lighthill's book "Waves and Fluids" in grad school. The nonlinear term is obviously essential. The problem with the mass spring system is that if one fixes the springs to the walls then there is no net mass transport and there is a backwards "drift" that must cancel and stokesian advance for a propagating wave. I wrote up a details description here on RG once. These men, like many others before them, thought they had solved this only to find that the data didn't quite work out. McIntyre and others created elaborate schemes to try to resolve this problem. In the last ten years there has been a problem with agreement with data that has not gone away. I just want to know if there is anything new to resolve this.
These men recommends to consider infinite spring with finite in time sinusoidal impulse.
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