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/m^{3},
rho_{0} 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/m^{3} for rho-0, this sets a limit on the product
Ma^{3}:

M(kg/m^{3})*a(m)^{3} > 3.039 x
10^{-11}