4.11 Sphere in a Sound Wave

Let us now return to the translational motions at small Reynolds numbers. In Section 4.8, we studied the motion of a sphere in a fluid executing translational oscillations and obtained the particle s velocity using steady and unsteady force results, (4.4.1) and (4.8.2), respectively These results, derived on the assumption that the fluid is incompressible, are sometimes used to study the motion of small spheres in sound waves. This might appear to be inconsistent, because sound waves owe their existence to the compressibility of the fluid. However, the wavelike motion of the fluid is not evident unless the spatial variations of the pressure and velocity are noticeable. This occurs when the length scales of interest are comparable with the acoustic wavelength ?. The length scale relevant to the force on a particle is the size of the particle. Hence, when ?, is very large, compared with the size of the particles, the particle sees only a fluid motion that, everywhere around it, oscillates in time with the same pressure and velocity. In this situation, the forces on the sphere are well described by the incompressible results.

It follows from the above discussion that the ratio of size to wavelength gives an indication of the importance of compressibility effects. In acoustic work, it is more convenient to use the ratio of particle circumference to the wavelength, b=2 ?a/ ?. This can also be expressed in terms of the wavenumber,