Suppose we have two waves, with the same amplitude but different wavelengths and velocities and we add them y 1 = A sin [ 2 π λ 1 ( x − v 1 t ) ] y 1 = A sin [ 2 π λ 1 ( x − v 1 t ) ] y 2 = A sin [ 2 π λ 2 ( x − v 2 t ) ] . y 2 = A sin [ 2 π λ 2 ( x − v 2 t ) ] . Then y 1 + y 2 = A ( sin [ 2 π λ 1 ( x − v 1 t ) ] + sin [ 2 π λ 2 ( x − v 2 t ) ] ) . y 1 + y 2 = A ( sin [ 2 π λ 1 ( x − v 1 t ) ] + sin [ 2 π λ 2 ( x − v 2 t ) ] ) . Lets rewrite using wave number and angular frequency y 1 + y 2 = y = A ( sin [ ( k 1 x − ω 1 t ) ] + sin [ ( k 2 x − ω 2 t ) ] ) . y 1 + y 2 = y = A ( sin [ ( k 1 x − ω 1 t ) ] + sin [ ( k 2 x − ω 2 t ) ] ) . Now we will use sin ( θ + φ ) + sin ( θ − φ ) = 2 sin θ cos φ sin ( θ + φ ) + sin ( θ − φ ) = 2 sin θ cos φ and set θ + φ = k 1 x − ω 1 t θ + φ = k 1 x − ω 1 t θ − φ = k 2 x − ω 2 t . θ − φ = k 2 x − ω 2 t . We can rearrange to get 2 θ = ( k 1 + k 2 ) x − ( ω 1 + ω 2 ) t 2 θ = ( k 1 + k 2 ) x − ( ω 1 + ω 2 ) t 2 φ = ( k 1 − k 2 ) x − ( ω 1 − ω 2 ) t . 2 φ = ( k 1 − k 2 ) x − ( ω 1 − ω 2 ) t . By substituting we can then see that y = 2 A ( cos [ k 1 − k 2 2 x − ω 1 − ω 2 2 t ] × sin [ k 1 + k 2 2 x − ω 1 + ω 2 2 t ] ) . y = 2 A ( cos [ k 1 − k 2 2 x − ω 1 − ω 2 2 t ] × sin [ k 1 + k 2 2 x − ω 1 + ω 2 2 t ] ) . Now set Δ k = k 1 − k 2 Δ k = k 1 − k 2 Δ ω = ω 1 − ω 2 Δ ω = ω 1 − ω 2 k = k 1 + k 2 2 k = k 1 + k 2 2 ω = ω 1 + ω 2 2 ω = ω 1 + ω 2 2 and we can rewrite the wave as y = 2 A cos ( x Δ k 2 − t Δ ω 2 ) sin ( k x − ω t ) . y = 2 A cos ( x Δ k 2 − t Δ ω 2 ) sin ( k x − ω t ) .

The above equation shows beats. For example you can set t = 0 t = 0 and see that you get y = 2 A cos ( x Δ k 2 ) sin ( k x ) . y = 2 A cos ( x Δ k 2 ) sin ( k x ) . Likewise you could pick x = 0 x = 0 and get the same figure, but now the horizontal axis is time y = 2 A cos ( − t Δ ω 2 ) sin ( − ω t ) y = 2 A cos ( − t Δ ω 2 ) sin ( − ω t ) or y = 2 A cos ( t Δ ω 2 ) sin ( − ω t ) . y = 2 A cos ( t Δ ω 2 ) sin ( − ω t ) . You get a traveling wave that has an oscillating amplitude.

Figure 1: Adding two waves of similar frequency together, gives rise to beats.

When we look at y = 2 A cos ( x Δ k 2 − t Δ ω 2 ) sin ( k x − ω t ) y = 2 A cos ( x Δ k 2 − t Δ ω 2 ) sin ( k x − ω t ) we see that there are two velocities. One, referred to as the phase velocity, is the speed of the individual wave crests: v p = ω k = ν λ . v p = ω k = ν λ . The group velocity is the velocity of the envelope v g = Δ ω Δ k → d ω d k v g = Δ ω Δ k → d ω d k Energy and momentum normally move with the group velocity.