The dispersion relation for waves in deep water is

$$\omega^{2}=g|k|$$

At time $t=0$ the water is at rest and the elevation of its free surface is $\zeta=\zeta_{0} \exp (-|x| / b)$ where $b$ is a positive constant. Use Fourier analysis to find an integral expression for $\zeta(x, t)$ when $t>0$.

Use the method of stationary phase to find $\zeta(V t, t)$ for fixed $V>0$ and $t \rightarrow \infty$.

$$\left[\int_{-\infty}^{\infty} \exp \left(i k x-\frac{|x|}{b}\right) d x=\frac{2 b}{1+k^{2} b^{2}} ; \quad \int_{-\infty}^{\infty} \exp \left(-a x^{2}\right) d x=\sqrt{\frac{\pi}{a}}(\operatorname{Re} a \geqslant 0) \cdot\right]$$