water wave laplace's equation

Question

I have tired differentiating with respect to x ,z , and t however I dont seem to be able to show it equals zero.

Answer 1

We have $\phi(x,y,z|t)$ where $t$ is a fixed parameter for purposes here. Then, we have

$$\begin{align} &\frac{\partial^2 \phi(x,y,z|t)}{\partial x^2}=-k^2\phi\\\\ &\frac{\partial^2 \phi(x,y,z|t)}{\partial y^2}=0\\\\ &\frac{\partial^2 \phi(x,y,z|t)}{\partial z^2}=+k^2\phi \end{align}$$

Then, it is easy to see that

$$\begin{align} \nabla \phi(x,y,z|t) &=\frac{\partial^2 \phi(x,y,z|t)}{\partial x^2}+\frac{\partial^2 \phi(x,y,z|t)}{\partial y^2}+\frac{\partial^2 \phi(x,y,z|t)}{\partial z^2}\\\\ &-k^2\phi(x,y,z|t)+0+k^2\phi(x,y,z|t)\\\\ &=0 \end{align}$$

which implies that $\phi$ satisfies Laplace's Equation.

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Note that if $z$ is viewed as a parameter, then $\phi(x,t|z)$ satisfies the one-dimensional wave equation

$$\frac{\partial^2 \phi(x,t|z)}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 \phi(x,t|z)}{\partial t^2}=0$$

with $\omega=kc$