the depth slowly vary for water wave
P100-1 Show:
\[\theta_x^2+\theta_y^2=c_0 \tag{1} \]has a solution in this form:
\[\theta=f(x+\lambda y) \]method-1
We can regard equ (1) as:
\[\theta_x \theta_x+\theta_y \theta_y=c_0 \]then the characteristic equation:
\[\begin{gathered} x_\tau=\theta_x,\quad y_\tau=\theta_y, \quad z_\tau=c_0 \\ \rightarrow z=\theta=c_0 \tau+c_1, \ldots \text { unsolved } \end{gathered} \]method-2
consider fourier transform:
provided that:
then:
\[F\left[\theta_x\right]=\text { is } \bar{\theta}(s) \]According to Energy integral
\[\int_{-\infty}^{+\infty}[f(t)]^2 d t=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}|F(\omega)|^2 d \omega \]rewrite equ (1)
\[\int_{-\infty}^{+\infty}\left[\theta_x^2+\theta_y^2\right] d x=\int_{-\infty}^{+\infty} c_0 d x \tag{2} \]or
\[\int_{-\infty}^{+\infty}\left[\theta_x^2+\theta_y^2\right] d y=\int_{-\infty}^{+\infty} c_0 d y \tag{3} \]equ (2) can be writed as
\[\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left[s^2 \bar{\theta}^2(s)+\bar{\theta}_y^2(s)\right] d s=\int_{-\infty}^{+\infty} c_0 d x \]... unsolved
method-3
[solution from stack exchange](partial differential equations - How to solve \((f_x)^2+(f_y)^2=4(1-f(x,y))(f(x,y))^2\)? - Mathematics Stack Exchange)
We look for particular solution in the form:\(\theta(x+y)\)
note that:
Let \(u=x+y\)
rewrite eq (1)
\[\left(\theta^{\prime}\right)^2+\left(\theta^{\prime}\right)^2=c_0 \]\[\left(\theta^{\prime}\right)^2=\frac{c_0}{2} \quad \theta^{\prime}=A_0 \quad A_0=\sqrt{\frac{C_0}{2}} \]\[\theta=A_0 u+c_1, \quad c_1 \rightarrow \text{ constant} \]Q.E.D
[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：prime,infty,right,theory,int,water,theta,left,Ray From： https://www.cnblogs.com/cicada-math/p/16997711.html