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The superposition of waves gives rise to the phenomenum of beats


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 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 ) ] ) . 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 ) ] ) . Now we will use sin ( θ + φ ) + sin ( θ φ ) = 2 sin θ cos φ and set θ + φ = k 1 x ω 1 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 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 ] ) . Now set Δ k = k 1 k 2 Δ ω = ω 1 ω 2 k = k 1 + k 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 ) .

The above equation shows beats. For example you can set t = 0 and see that you get y = 2 A cos ( x Δ k 2 ) sin ( k x ) . Likewise you could pick 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 ) or y = 2 A cos ( t Δ ω 2 ) sin ( ω t ) . You get a traveling wave that has an oscillating amplitude.

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

Phase and group velocities

When we look at 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 wavecrests: 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 Energy and momentum normally move with the group velocity.

Questions & Answers

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Source:  OpenStax, Waves and optics. OpenStax CNX. Nov 17, 2005 Download for free at http://cnx.org/content/col10279/1.33
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