Long surface waves incident on a submerged horizontal plate
Long surface waves incident on a submerged horizontal plate
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DOI：
10.1017/s0022112077001098
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发表时间：
1977-11
影响因子：
3.7
通讯作者：
P. Siew;D. Hurley
中科院分区：
文献类型：
--
作者：
P. Siew;D. Hurley
在深度为H的通道中，波长为λ的表面重力波序列入射到深度为c的长度为l的水平板上。在λ和l都比H大的假设下，采用匹配渐近展开的方法表明，对于一阶，反射系数R和透射系数T由\[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \]和\[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \]给出，其中\begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ为角频率，g为重力加速度。
A train of surface gravity waves of wavelength λ in a channel of depth H is incident on a horizontal plate of length l that is submerged to a depth c. Under the assumption that both λ and l are large compared with H, the method of matched asymptotic expansions is used to show that, to first order, the reflexion coefficient R and the transmission coefficient T are given by \[ R = \chi \left\{\frac{\sigma l}{(gH)^{\frac{1}{2}}}\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}-2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\} \] and \[ T =\chi\left\{2i\left[\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\right]\right\} \] where \begin{eqnarray*} \chi &=& 1\left/ \left\{2\bigg(\frac{c}{H}\bigg)^{\frac{1}{2}}\bigg(1-\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)+\frac{\sigma l}{b}\bigg(\frac{H}{g}\bigg)^{\frac{1}{2}}\bigg(1+\frac{c}{H}\bigg)\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}\right.\right.\\ &&\left. +2i\bigg(\sin\frac{\sigma l}{(gc)^{\frac{1}{2}}}+\frac{\sigma l}{b}\bigg(\frac{c}{g}\bigg)^{\frac{1}{2}}\cos\frac{\sigma l}{(gc)^{\frac{1}{2}}}\bigg)\right\}, \end{eqnarray*} σ is the angular frequency and g the acceleration due to gravity.