ATTENUATION DES ONDES SISMIQUES DANS LES SOLIDES*

For the propagation of acoustic waves in formations, laboratory experiments yield an expression of the form:

in which

A is the amplitude (strain or stress)

A0 some constant coefficient

x the coordinate according to which propagation takes place

α the attenuation coefficient

C velocity

ω the angular frequency of sinusoidal oscillation.

No analogy between this law and that of electromagnetic waves can be made, since:

• 1)   coefficient α is not independent of frequency: its expression is α=Kω (K constant);
• 2)   however, with present recording technique, no noticeable dispersion can be found.

The present paper shows how results are compatible if losses of energy are accounted for by a hysteresis phenomenon that is analyzed (see Figure). Stress T is in abscissae, and strain δu/δx, multiplied by a coefficient E′ characteristic of the elastic properties of the solid, is in ordinates. The horizontal part of the curve is θ. It is supposed that the absorption properties of the ground are given by some dimensionless coefficient

λ=θ/T−

Then one gets

α=ωλ/4C

If λ≪ 1, we get the propagation law

An attenuation takes place without phase shift, and consequently without dispersion.

The author reverts to the inapplicability of the superposition principle, and foresees theoretically, for instance, that a strain T′ cos ω′t, where T′ > T and ω′ > ω, can completely cancel coefficient α.