the depth slowly vary for water wave
water wave page 93
show:
\[\begin{align} &\phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0,\\ &\phi_z=\delta^2 \eta_t \text { and } \phi_t+\eta=0 \text { on } z=1 \text {, }\\ &\phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right) \text { on } z=b(x, y) \end{align} \]
\(\widetilde{u} \rightarrow\) orginal \(\quad u \rightarrow\) current
\[\widetilde{u}=c u \quad \widetilde{x}=\lambda x \]we start from:
\[\tilde{u}=\frac{\partial \phi}{\partial \tilde{x}}, \quad u=\frac{\tilde{u}}{c}=\frac{\partial \phi}{c \partial x} \cdot \frac{\partial x}{\partial \tilde{x}}=\frac{1}{c \lambda} \phi_x \]Since:
\[\small\tilde{u}_{\tilde{x}}+\tilde{w}_{\tilde{z}}=0, \tilde{u}_{\tilde{x}}=\frac{\partial \tilde{u}}{\partial \tilde{x}}=\frac{\partial \tilde{u}}{\partial x} \frac{\partial x}{\partial \tilde{x}}=\frac{1}{\lambda} \frac{\partial \tilde{u}}{\partial x}=\frac{c}{\lambda} u_x \]then:
\[\widetilde{u}_{\tilde{x}}=\frac{c}{\lambda} \frac{\partial}{\partial x}\left(\frac{1}{c \lambda} \phi_x\right)=\frac{1}{\lambda^2} \phi_{x x} \]we have: $$\widetilde{w}_{\tilde{z}}=\frac{\partial \tilde{w}}{\partial \tilde{z}}=\frac{\partial \tilde{w}}{\partial z} \cdot \frac{\partial z}{\partial \tilde{z}}=\frac{1}{h_0} \widetilde{w}_z,\left(\tilde{z}=h_0 z\right)$$
from:
therefore:
\[\tilde{w}_z=\frac{1}{h_0}\left(\frac{1}{h_0} \phi_z\right)=\frac{1}{h_0^2} \phi_{z z} \]for incompressible flow:
\[\widetilde{u}_{\tilde{x}}+\tilde{w}_{\tilde{z}}=0 \]consequently: $$\frac{1}{h_0^2} \phi_{z z}+\frac{1}{\lambda^2} \phi_{x x}=0, \quad\left(\delta=\frac{h_0}{\lambda}\right)$$
extend it to 3-D: $$\phi_{z z}+\delta^2\left(\phi_{x x}+\phi_{y y}\right)=0. \quad \text{(equ 2.66)} $$
Show: $$\phi_z=\delta^2 \eta_t$$
we have: $$w=\eta_t \quad\text{(equ.2.1)}$$
note that: \(w\) here is dimensionless (not \(\tilde{w}\) )
we have: \(\tilde{w}=w h_0 \sqrt{gh_0}/\lambda\) and from previous derivation:
but probably wrong (only valid when \(\sqrt{g h_0}=1/\lambda\) )
Show: $$\phi_t+\eta=0,\quad\text{on} \quad z=1$$
we start from: \(p=\eta \quad\) ( \(W_e=0\) for gravity wave),for inviscid fluid: \(u_t=-p_x\)
where \(u\) has been described:
then
\[u_t=\frac{1}{c \lambda} \phi_{t x}=-p_x \]\[\Rightarrow \frac{1}{\sqrt{g h_0} \cdot \lambda} \phi_t+P=0 \]from previous equations only when \(\lambda=1/\sqrt{g h_0}\)
\[\phi_t+p=0 \]Show: $$\phi_z=\delta^2\left(\phi_x b_x+\phi_y b_y\right)\quad \text{on} \quad z=b(x, y)$$
we have: $$w=u b_x+v b_y, \quad z=b$$
Also: $$\quad \delta^2 w=\phi_z \Rightarrow \phi_z=\delta^2\left(u b_x+v b_y\right)$$
note that: $$w=\frac{D z}{D t}=\frac{D}{D t}(1+\varepsilon\eta)=\varepsilon\left[\eta_t+\left(u_{\perp} \cdot \nabla\right) \eta\right]$$
After scaling: $$(u, v) \rightarrow \varepsilon(u, v) \quad w \rightarrow \varepsilon \ w$$
then: $$\varepsilon w=\varepsilon\left[\eta_t+\varepsilon\left(u_{\perp} \cdot \nabla\right)\eta\right]$$
[1]: Johnson, R. (1997). A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge Texts in Applied Mathematics). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511624056
标签：phi,frac,theory,quad,water,tilde,partial,lambda,Ray From： https://www.cnblogs.com/cicada-math/p/16989964.html