Two cases of interest are (1) normal incidence, θI = θT = βT = 0, and (2) when βT = π4. In the

In the case of normal incidence, only compression waves are excited, whereas in the second case only shear waves are excited.

5.5 Attenuation

Attenuation refers to a weakening of the wave amplitude. For cylindrical or spherical waves, the amplitude will weaken with distance from the source. This weakening is referred to as geometric spread. Interaction with the medium can lead to other types of attenuation by the processes of absorption or scattering. Mathematically this is accounted for by a complex wavenumber, κ = k + iα. Fluid media can exhibit a variety of attenuation strengths. At the micro level, the cause of attenuation is due to the interaction of waves and the particles in the medium. At a macro level, the effects are rolled up into the attenuation coefficient, α. This coefficient can depend on the frequency of the sound as well as on environmental factors such as temperature, relative humidity, and pressure. Attenuation acts opposite the direction of propagation, like air resistance for a moving object. In 3-dim a plane wave with attenuation is expressed as

where s is the distance traveled along the propagation path or direction. The same result applies to a spherical wave, exp(−αR). The unit of the attenuation coefficient is m− 1. Models of the atmospheric attenuation of air and attenuation in water can be found in Refs. [6] and [4, 7], respectively.

5.6 Exercises

1.Sound in a fluid medium is incident on a thin layer of a solid medium of thickness L, with two Lamé parameters.

(a)Determine the boundary conditions at each interface of the solid medium.

(b)Solve for the transmission and reflection coefficients and the coefficients of the wave in the solid medium.

2.Calculate the 2-dim Fourier transform of the following spatial window functions.

(a)Step function, 1 for x < Lx2 and y < Ly2, zero otherwise

(b)Circular step function, 1 for ρ < a, zero otherwise

(c)Radial quadratic window, 1 − (ρ/a)2 for ρ < a, zero otherwise

(d)What effect does the change in window have on the spectrum?

3.Given the transmission and reflection coefficient for a plane wave incident at the boundary of two media,

(a)Develop a procedure for modeling the transmitted and reflected field from a point source a distance, D, above the plane separating the medium.

(b)If possible work out the results analytically; otherwise write a procedure in the language of your choice.

References

[1]Körner, T. W., Fourier Analysis, Cambridge University Press, Cambridge, 1990.

[2]Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes in C++, The Art of Scientific Computing, Second Edition, Cambridge University Press, Cambridge, 2005.

[3]Tolstoy, I. and Clay, C. S., Ocean Acoustics, Theory and Experiment in Underwater Sound, McGraw-Hill, New York, 1966.

[4]Ainslie, M. A. and McColm, J. G., A simplified formula for viscous and chemical absorption in sea water, J. Acoust. Soc. Am., Vol. 103, No. 3, pp. 1671–1672, 1998.

[5]Brekhovskikh, L. M., Waves in Layered Media, Academic Press, New York, 1960.

[6]Rickley, E. J., Fleming, G. G., and Roof, C. J., Simplified procedure for computing the absorption of sound by the atmosphere, Noise Control Eng. J., Vol. 55, No. 6, pp. 482–494, 2007.

[7]Fisher, F. H. and Simmons, V. P., Sound absorption in sea water, J. Acoust. Soc. Am., Vol. 62, No. 3, pp. 558–564, 1977.