Example

A wave with period T=10sec and height H=2ft propagates in infinite depth water. A wave with the same height but with 5 larger wave length is also propagating in the same direction at the first one. At a given moment the crests of the two waves occur at the same location. Determine the expression for the combined wave profile. Plot this profile over one period of the resulting envelope. What is the speed with which this envelope propagates? Compare this speed with the group velocity of the original wave. In which direction the combined profile propagates relative to the direction of the two waves?

Solution:

x (1.05) = 538.1ft

As shown in class:

We define:

so combined profile:

NOTE: There are 41 waves of wave length inside wave length

Envelope:     (propagates in the same direction as original)

wave speed of envelope is:    

group velocity of wave (1) (for deep water)    

   Actually definition of    

since x  not quite ``0'';