Theory of the Attenuation of Ultrasound by Suspended Sediment
( references |agu97 paper )
The scattering of ultrasonic waves in water by suspended sand grains has been discussed considerably in the literature [Sheng and Hay 1988, Lee and Hanes 1995, Hay 1983]. At the frequency used here, 200 kHz, the wavelength in water (7.5 mm) is much larger than the particle size. This means that the scattering cross section is relatively small, and strongly dependent on particle size.
The diagram shows plane waves incident on a sand grain suspended in water. The curved line represents the beam pattern of the scattered sound, which peaks backwards. The expression for the cross section is valid in the Rayleigh regime, in which ka << 1. The two gamma parameters depend on the compressibility and density of the particle and the medium; I have used values from [Hay 1983] for sand in water.

It is easy to calculate the exponential attenuation coefficient alpha-sub-s for the pressure amplitude; the result is

where M is the volume concentration of suspended sediment in kg/m3, rho0 is the density of the material of which the grains are composed, and a is the radius of the sand grain (assumed spherical).

For the experiment at 3-Mile Slough, a detectable attenuation would correspond roughly to an attenuation length of 1 km. (This would attenuate the pressure amplitude by about 20% over the pathlength, 220 m, of our measurement.) For a sound frequency of 200,000 Hz and a value of 2650 kg/m3 for rho-0, this sets a limit on the product Ma3:

M(kg/m3)*a(m)3 > 3.039 x 10-11