The problem of oblique water wave diffraction by small undulation of the bottom of a laterally unbounded ocean is considered using linear water wave theory. A perturbation analysis is employed to obtain the velocity potential, the reflection and the transmission coefficients up to the first order in terms of integrals involving the shape functions c(x) representing the bottom undulation. Finite cosine transform is used to find the first order potential, and this potential is utilised in obtaining the first order reflection and transmission coefficients. Some particular forms of the shape function representing an exponentially damped undulation, a single hump and a patch of sinusoidal ripples are considered and the integrals for the reflection and transmission coefficients are evaluated. For the exponentially damped undulation, it is observed that the reflection ceases much before transmission while for the single hump, reflection and transmission go hand in hand up to a certain value of the wavenumber, after which they vanish. For the patch of sinusoidal ripples having the same wavenumber, the reflection coefficient up to the first order is found to be an oscillatory function in the quotient of twice the component of the wavenumber along x-axis and the ripple wavenumber. When this quotient becomes one, the theory predicts a resonant interaction between the bed and free surface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Also, when a patch of ripples having different wavenumbers is considered the same result follows. Known results for the normal incidence are recovered as special cases for the patch of sinusoidal ripples. The theoretical observations are shown computationally. 相似文献
Currently, scant attention has been paid to the theoretical analysis on dynamic response mechanism of the "Dualistic" structure roek slope. The analysis presented here provides insight into the dynamic response of the "Dualistie" structure rock slope. By investigating the principle of energy distribution, it is shown that the effect of a joint plays a significant role in slope stability analysis. A dynamic reflection and transmission model (RTM) for the "Dualistic" structure rock slope and explicit dynamic equations are established to analyze the dynamic response of a slope, based on the theory of elastic mechanics and the principle of seismic wave propagation. The theoretical simulation solutions show that the dynamic response of the "Dualistic" structure rock slope (soft-hard) model is greater than that of the "Dualistic" strueture rock slope (hard-soft) model, especially in the slope crest. The magnifying effect of rigid foundation on the dynamic response is more obvious than that of soft foundation. With the amplitude increasing, the cracks could be found in the right slope (soft-hard) crest. The crest failure is firstly observed in the right slope (soft-hard) during the experimental process. The reliability of theoretical model is also investigated by experiment analysis. The conclusions derived in this paper could also be used in future evaluations of Multi-layer rock slopes. 相似文献