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The superposition of waves gives rise to the phenomenum of beats

Superposition

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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Crow Reply
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RAW Reply
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Damian
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Rafiq
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Damian
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scanning tunneling microscope
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Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Adin Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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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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