Group velocity of waves wave package

By definition the monochromatic wave is boundless in space. The real wave is always limited in space and is emitted during a limited interval of time, which is why it cannot be monochromatic in full measure. However, any real wave can be considered as a result of the superposition of a large number of strictly monochromatic flat waves. As a result of interference, in one part of space these waves strengthen each other, and in other parts extingnish. Such waves have some features that can be discovered using a simple model of superposition of two plane monochromatic waves. 

Let two plane cross-sectional polarized monochromatic waves with equal amplitudes be distributed along an axis x. Such waves are described by equations = A cos(ft)iZ - kiX) and 1 2 = A cos(cOiZ k2X). Because of the superposition principle a combined wave can be represented as ( = if 1 + ( 2 cos(coj - k x) + A cos(cOit - k-pc), or 

We can see that the superposition of two monochromatic waves with equal amplitudes with slightly different frequencies and wavenumbers produces a new wave with variable in space and time amplitude 

Since Aco and Ak are small in comparison with coj and k a change of amplitude will take place comparatively slowly. Such a wave is called a wave with modulated amphtudes. 

Let us determine the rate of crest displacement of such a composed wave. In order to solve this problem we can repeat the method used in Section 2.8.2 when we evaluate the phase velocity rate. The crest corresponds to the constant phase in eq. (2.9.11), i.e.. 

---