Оь амплитудах преломленных волн

Summary

A study was made of the amplitudes of head waves P₁₂₁, produced on the incidence of a harmonic longitudinal wave on a plane bourndary. Theoretically the paper is based on reference [30]. It is shown how the amplitudes vary as a function of the distance Δ from the source, the depthH of the interface, the frequencyf, the ratio of the velocities of the longitudinal waves$\frac{{a_1 }}{{a_2 }} = n_2$, the ratio of the densities$\frac{{\rho _1 }}{{\rho _2 }} = \rho$ and partly also of the Poisson constants σ₁ and σ₂ (where the suffix “1” denotes the medium with a source and the suffix “2” the medium without a source).

The dependence on the distance is dealt with in paragraph 2. It is shown that the greater the frequency and depth of the interface and that the smaller the distance and that the closern ₂ approaches unity, the more quickly the amplitude decreases in a medium without absorption with growing distance. In an absorbing medium the amplitude decreases with the distance still more rapidly than in a medium without absorption and thus more quickly the greater the coefficient of absorption. It is shown that the measurement of the coefficients of absorption from the amplitudes of head waves is hampered by a systematic error which is greater for larger frequency, smaller distance andn ₂ closer to unity. At the end of the paragraph it is then shown that on incidence of a shock wave the amplitude decreases with the distance more slowly than in the case of an incident harmonic wave.

The dependence of the amplitude on the frequency is dealt with in para. 3. In a medium without absorption the amplitude of the head waves always decreases with growing frequency more rapidly the greater the distance Δ and the frequency and the smallern ₂ and the depth of the interface. In the case of an shock source the prevailing frequency (mainly in the range of high frequencies) in the neighbourhood of the point of contact with the reflected wave will substantially decrease with growing distance. In a medium with absorption the higher components of the frequency will be absorbed still more rapidly, particularly at large distances and forn ₂ approaching zero. It will no longer generally hold, however, as in the case without absorption, that high frequency waves will be more damped in the case of smaller depth.

Paragraph 4 deals with the dependence of the amplitude on the depth of interface. In a medium without absorption the amplitude in most practical cases grows with the depth the more quickly the greater the frequency and depth of the interface, the smaller the distance Δ and the nearern ₂ approaches unity. In a medium with absorption the situation is more complicated. With growing depth the amplitude decreases at great distances Δ, for large coefficient of absorption in the first medium and small coefficient in the second medium, for very small and very large frequencies, forn ₂ close to zero and small depths. In other cases the amplitude grows with incerasing depth.

The dependence of the amplitude on the parameters of the medium is dealt with in para. 5. It is shown that for a given distance, depth and frequency the amplitude will be greater forn ₂ approaching nearer to unity. It is further shown that the amplitude increases with decreasing Poisson constants σ₁ and σ₂ and with increasing velocity of the longitudinal waves in the first medium forn ₂ constant. Forn ₂ close to zero the amplitude is smaller for larger ρ and forn ₂ close to unity the amplitude is maximum for ρ=1.