Vertical Distribution of Amplitudes of the Rayleigh Type Dispersive waves (2nd report)

Abstract

In the previous paper, the writer reported results of systematic calculations of the vertical distribution of amplitudes and particle orbits to clearify the general properties of M₁₁ waves which are propagated along the surface of a superficial layer overlying a semi-infinite elastic medium.
It is commonly believed that the particle orbit for M₁₁ waves at the surface is of a retrograde rotation to the direction of wave propagation. By the writer's investigation, nevertheless, it is found that the particle orbit for waves of which the wavelengths are longer than about 4.6H (H: thickness of the superficial layer), changes its rotational direction, if the substratum is rigid.
Though M₁₁ waves are of one type of branch waves, they are classified into two groups of different types of particle orbit depending upon the structure and the wavelength as above mentioned. This shows that the particle orbit does not indicate the general properties of M₁₁ waves. In other words, it is unreasonable to decide the type of M₁₁ waves only by the rotational direction of the particle orbit, on the basis above mentioned.
In order to see whether this complexity also occurs for M₂₁ waves or not, the writer studied vertical distribution of amplitudes and particle orbits for M₂₁ waves which are of another type of dispersive Rayleigh wave.
In general, the rotational direction of the particle orbit is an anti-retrograde one to the direction of wave propagation, but if the wavelengths are longer than about 4.6H, the particle orbit changes its rotational direction with the increase of the phase velocity, even for the same wavelength. Consequently, a line which divides the existence domain in two parts regarding the phase velocity can be obtained. In the lower velocity part, the rotational direction of the particle orbit is an anti-retrograde one, while on the other hand in the higher velocity part, it is a retrograde one. This complexity for M₂₁ waves is comparable with that for M₁₁ waves.
In accordance with the fact that the particle orbit for M₁₁ waves changes its rotational direction in the existence domain. There must be a line which devides the domain into two parts with respect to the rotational direction of the particle orbit for M₁₁ waves. In the previous paper, the writer could not get such a line. The reason seems to be that the line for M₁₁ waves is so close to the μ=∞ dispersion curve that it could not be distinguished.
These complexities for both M₁₁ and M₂₁ waves occur only at and near the surface, and the rotational direction of the particle orbit at a point of some depth in the superficial layer is similar to what is usually believed.