I have those two equations below:$$\begin{cases}\dfrac{\partial u}{\partial t}+g\dfrac{\partial h}{\partial x}=0\\\dfrac{\partial h}{\partial t}+H\dfrac{\partial u}{\partial x}=0\end{cases}$$And I have$$u(x, t)=u_0e^{i(k x+\omega t)},\quad h(x,t)=h_0e^{i(k x+\omega t)}.$$ I want to establish the relationship that connects the $u_0$ and $h_0$ in one expression.
I know like we have two equations with two variables but I can't figure out how to plug all this together to obtain only one expression?