A hydrodynamic model, with the incorporation of elasticity, is considered to study oblique incident waves propagating over a small undulation on an elastic bed in a two-layer fluid, where the upper layer fluid is bounded above by a rigid lid, which is an approximation to the free surface. Following the Euler–Bernoulli beam equation, the elastic bed is approximated as a thin elastic plate. The surface tension at the interface of the layers is completely ignored since its contribution will be minimal. The behavior and properties of the roots of the dispersion relation are analyzed using counting argument and contour plot. In this case, time-harmonic waves propagate with only one wavenumber. Considering an irrotational motion in an incompressible and inviscid fluid, and applying perturbation technique, the first-order corrections to the velocity potentials are evaluated by an appropriate application of Fourier transform and, subsequently, the corresponding reflection and transmission coefficients are computed through integrals containing a shape function which depicts the bottom undulation. To validate the theory developed, two particular undulating bottom topography are taken up as examples in order to evaluate the hydrodynamic coefficients which are represented through graphs. It is noted that the reflection coefficient shows an oscillatory pattern, when the wavenumber of the undulating bed takes a value almost double the wavenumber of the incident wave. When the ratio of the wavenumber of the undulating bed and the wavenumber of the incident wave approaches 2, the theory predicts existence of resonance between the undulating elastic bed and the interface of the layers. It is worthwhile to note that reflected wave energy of the incident wave is higher if the number of ripples of the bottom undulation is large. It is further noted that the reflected energy increases in response to an increase in the values of the elastic parameters of the ocean bed. The results for a patch of sinusoidal ripples having different wavenumbers are found to be similar. Further, the reflected energy decreases for an increase in the angle of incidence. It is noted that reasonable changes in elasticity, ripple wavenumber and the number of the ripples of the bed have a remarkable effect when the propagating wave passing through a channel encounters a small bottom undulation. When results are compared with the available results, good agreement is observed.

中文翻译：

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在流过通道的两层流体中，斜波在小起伏弹性底部地形上的传播

考虑了结合弹性的流体动力学模型来研究倾斜入射波在两层流体中弹性床上的小起伏传播，其中上层流体由刚性盖限定，这是一个近似值到自由表面。根据 Euler-Bernoulli 梁方程，弹性床近似为薄弹性板。层界面处的表面张力被完全忽略，因为它的贡献将是最小的。使用计数参数和等高线图分析了色散关系根的行为和性质。在这种情况下，时谐波仅以一个波数传播。考虑不可压缩和无粘性流体中的无旋运动，并应用微扰技术，通过适当地应用傅里叶变换来评估对速度势的一阶校正，随后，通过包含描绘底部起伏的形状函数的积分来计算相应的反射和透射系数。为了验证所发展的理论，以两个特定的起伏底部地形为例，以评估通过图表表示的水动力系数。应该注意的是，当起伏床的波数取几乎两倍于入射波波数的值时，反射系数显示出一种振荡模式。当起伏床的波数与入射波的波数之比接近2时，该理论预测在起伏的弹性床和层界面之间存在共振。值得注意的是，如果底部起伏的波纹数量较大，则入射波的反射波能量较高。还应注意，反射能量响应于海床弹性参数值的增加而增加。发现一块具有不同波数的正弦波纹的结果是相似的。此外，反射能量随着入射角的增加而降低。值得注意的是，当传播波通过河道遇到较小的底部起伏时，弹性、波纹波数和河床波纹数的合理变化具有显着的效果。当结果与可用结果进行比较时，