Deep water waves
The behavior of the waves is quite different if the water is deep or if the water is shallow. 'Deep' is generally measured with respect to wavelength. A body of water is deep if the depth h is much greater than the wavelength L. Since L=2p/k we say that the water is deep if kh>>1.
Recall that the motion of our waves is governed by the formulas
If we assume that a body of water is deep and that we are only intested in water close to the surface (-y<<h) then we may make the following approximations.
sinhkh » (1/2)ekh
sinh(k(h+y)) » cosh(k(h+y)) » (1/2)ek(h+y)