This paper presents an intuitive method for deriving the asymptotic solution of all detectable waves reflected at the free surface due to the incidence of a line P source, i.e., P- and S-disturbances that propagate horizontally at their respective velocities with a decrease rate of (where denotes the horizontal distance). At first, the line P source is regarded as a combination of plane-wave components incident with real angles and those incident with imaginary angles. Based on the theory of plane-wave reflection with an extension to the range of imaginary incidences, the two transmissible plane-wave incidences of small real angle and of finite imaginary angle which directly lead to the actual P- and S-disturbances at large distances, respectively, are identified. Consequently, the geometric ray representation for near-field reflection associated with the S-disturbance is clarified after a modification upon the conventional theory. In analogy with the classical plane-wave reflection theory, the reflected waves are obtained by multiplying the fully reflected waves due to the incident cylindrical wave by the corresponding reflection coefficients, which are determined by the stress-free condition at the surface. Hence, the present solution can capture all the reflected waves with a decrease rate of , unlike the conventional asymptotic solution. This intuitive method not only simplifies the calculation of asymptotic solutions using the method of steepest descent or stationary phase, but is physically more meaningful via establishment of a direct link between the reflection of plane waves and that of cylindrical waves. Results show that the S-disturbance due to a shallow or slowly oscillating line P source becomes significant especially for the responses beneath the surface. In contrast, the S-disturbance was recognized as a minor response and ignored in many existing studies.
扫码关注我们
求助内容:
应助结果提醒方式:
