Let,
$\zeta(x,t) = A_0sin(k_0x)cos(\omega t) + \frac{2k_0A_0}{\pi} \{\int_{0}^{\infty}\frac{cos(kx)cos(\beta t)-cos(k_0x)cos(\omega t)}{k^2-k_0^2}dk\}$
Here $\beta ^2 = gktanh(kh)\ and\ \omega^2 = gk_0tanh(k_0h)$ where $g,h$ are given constants as well as $k_0$. $x$ can be treated as a spatial coordinate while $t$ as time coordinate. As such $x$ & $t$ can be viewed as parameters both greater than $0$.
I need to prove that $\lim_{t\rightarrow\infty} \zeta(x,t) = A_0cos(k_0x-\omega t)$. However, I think the answer must come out as $A_0sin(k_0x-wt)$
As such, one knows that the solution is dominated by $k_0$ ,thus one can expand the integrand about $k_0$. However, the problem is the denominator i.e. $(k^2-k_0^2)$ makes the integrand become singular at $k_0$.
The problem is one of the transient analysis of a wavemaker given here.
Hint: In the integral add and subtract $\sin(kx)\sin(\beta t)$ and $\sin(k_0 x)\sin(\omega t)$