Standing Wave problem-In deep water limit $h\to\infty$ show that $\omega^2=gk$

Question

The equations I have are

$\phi=(-ag/\omega)\cos(kx)\sin(\omega t)e^{kz}$

and

$\eta=a\cos(kx)\cos(\omega t)$

I know that $d\phi/dz=d\eta/dt$

but when I partially differentiate and rearrange I get

$gk e^{kz}= \omega^2$ and I don't know how to get rid of the exponential function.