A note on the similarity between acoustic streaming and gravity wave drift in irrotational fluid motion

Abstract For inviscid irrotational fluid motion, the nonlinear Lagrangian equations for periodic plane acoustic waves and long gravity waves are formally similar. It then follows that the Stokes drift is similar and can be calculated for the two problems. However, the lack of dissipative processes means that the Eulerian mean current cannot be determined, and hence the acoustic streaming velocity and the Lagrangian mean surface-wave drift remain unknown. To remedy this without altering the irrotational character of the fluid motion, we add a small frictional force which is linear in the velocity, or a so-called Rayleigh friction. Then, the Lagrangian mean drift (Stokes drift $$+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>+</mml:mo> </mml:math> Eulerian current) is uniquely determined. With this assumption, the acoustic streaming velocity is $$\left( \gamma +1 \right) /2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:mi>γ</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mfenced> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> times the Stokes drift in sound waves, where $$\gamma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>γ</mml:mi> </mml:math> is the adiabatic constant. For long gravity waves, the Lagrangian mean drift is 3/2 times the Stokes drift in surface waves. These results are valid whatever small the Rayleigh friction coefficient is, as long as it is not zero.