[water wave] Ray theory-2

标签：phi right partial theory water theta bar left Ray

the depth slowly vary for water wave

p94-1 show:

\[\phi_z=a \delta^2\left(\phi_x B_{\bar{x}}+\phi_y B_{\bar{y}}\right) \text { on } z=B(\bar{x}, \bar{y}) \]

Start from:

\[\begin{aligned} & \left\{\begin{array}{l} \phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right) \\ b(x, y)=B(a x, a y), \quad \bar{x}=a x, \bar{y}=a y \end{array}\right. \\ \end{aligned} \]

Also,we have

\[b_x=\frac{\partial B}{\partial x}=\frac{\partial B}{\partial \bar{x}} \cdot \frac{\partial \bar{x}}{\partial x}=a B_{\bar{x}} \\ \]

Similarly: $ b y=a B \bar{y}$
hence:

\[\phi_z=a \delta^2\left(\phi_x B_{\bar{x}}+\phi_y B_{\bar{y}}\right) \]

p95-1 : show:

\[\small \begin{aligned} \phi_{z z}+\delta^2\left[\left(k^2+l^2\right) \phi_{\theta \theta}\right. & + 2a\left(k \phi_{\theta \bar{x}}+l \phi_{\theta \bar{y}}\right) \\ & \left.+a(k \bar{x}+l \bar{y})\phi_\theta+\small a^2\left(\phi_{\bar{x} \bar{x}}+\phi_{\bar{y} \bar{y}}\right)\right]=0 \end{aligned} \]

consider that:

\[\phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0 \]

\[\phi_{x x}=\frac{\partial}{\partial x}\left(\frac{\partial}{\partial x} \phi\right)=\frac{\partial}{\partial x}\left(\frac{\partial \bar{x}}{\partial x} \phi_{\bar{x}}+\frac{\partial \theta}{\partial x} \phi_\theta\right) \]

Since :

\[\partial \bar{x} / \partial x=a, \partial \theta / \partial x=k(\bar{x}, \bar{y}, \bar{t}) \]

\[\Rightarrow \phi_{x x}=\frac{\partial}{\partial x}\left(a \phi_{\bar{x}}+k \phi_\theta\right) \]

\[\begin{split} \phi_{x x}=\frac{\partial \bar{x}}{\partial x} \frac{\partial}{\partial \bar{x}}\left(a \phi_{\bar{x}}\right) &+ \frac{\partial \theta}{\partial \bar{x}} \frac{\partial}{\partial \theta}\left(a \phi_{\bar{x}}\right) \\ & +\frac{\partial \bar{x}}{\partial x} \frac{\partial}{\partial \bar{x}}\left(k \phi_\theta\right)+\frac{\partial \theta}{\partial \bar{x}} \frac{\partial}{\partial \theta}\left(k \phi_\theta\right) \end{split} \]

\[\begin{split} \ &=a^2 \phi_{\bar{x} \bar{x}}+k a \phi_{\bar{x} \theta}+a\left(k_{\bar{x}} \phi_\theta+k \phi_{\theta \bar{x}}\right)+k^2 \phi_{\theta \theta}\\ & =a^2 \phi_{\bar{x} \bar{x}}+k^2 \phi_{\theta \theta}+2 a k \phi_{\theta \bar{x}}+a k_{\bar{x}} \phi_\theta \end{split} \]

Similarly:

\[\phi_{y y}=a^2 \phi_{\bar{y} \bar{y}}+l^2 \phi_{\theta \theta}+2 a l \phi_{\theta \bar{y}}+a l_{\bar{y}} \phi_\theta \]

Q.E.D

p95-2 show:

\[\beta_{0 z z}-\delta^2\left(k^2+l^2\right) \beta_0=0 \]

We have:

\[\small \begin{aligned} \beta_{z z}+\delta^2[-\left(k^2+l^2\right) \beta & + 2 i a\left(k \beta_{\bar{x}}+l\beta_{\bar{y}}\right)\\ & + i a\left(k_{\bar{x}}+l_y\right) \beta+a^2\left(\beta_{\bar{x} \bar{x}}+\beta_{\bar{y} \bar{y}}\right)]=0 \\ \end{aligned} \]

we write:

\[\beta \backsim \sum_{n=0}^{\infty} a^n \beta_n(\bar{x}, \bar{y}, \bar{t}, z) \text { as } a \rightarrow 0 \]

then the term of \(\large a^0\)

\[\beta_{z z}=a^0 \beta_{0 z z} \]

\[\delta^2\left[-\left(k^2+l^2\right)\right] \beta=-\delta^2\left(k^2+l^2\right) a^0 \beta_0 \]

others are vanish for \(a^0\)
consequently:

\[\beta_{0 z z}-\delta^2\left(k^2+l^2\right) \beta_0=0 \quad \text { (equ. 2.74) } \]

Q.E.D

标签：phi,right,partial,theory,water,theta,bar,left,Ray
From： https://www.cnblogs.com/cicada-math/p/16991657.html