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Same frequency, different phase

One of the most important concepts we encounter in vibrations and waves is the principle of superposition. Lets look at a couple of cases starting withadding two motions with the same frequency but different phases. It is easiest to calculate this if you use complex notation x 1 = A 1 e i ( ω t + α 1 ) x 2 = A 2 e i ( ω t + α 2 ) x = x 1 + x 2 = A 1 e i ( ω t + α 1 ) + A 2 e i ( ω t + α 2 ) x = e i ( ω t + α 1 ) [ A 1 + A 2 e i ( α 2 α 1 ) ] This comes up all the time in real life: For example noise canceling headphones use this technique. In headphones there is a membrane vibratingwith the frequency of the sound you are listening two. In a noise canceling headphone there is also a microphone "listening" to the noice coming fromoutside the headphone. This oscillation is inverted and then added to membrane producing the sound you listen to. The net result is a signal that containsthe desired sound and subtracts the noise resulting in quieter operation.

Different frequency

One can also consider the case of two oscillations with the same phase but differentfrequencies: x 1 = A 1 e i ( ω 1 t ) x 2 = A 2 e i ( ω 2 t ) x = x 1 + x 2 = A 1 e i ( ω 1 t ) + A 2 e i ( ω 2 t ) x = e i ( ω 1 t ) [ A 1 + A 2 e i ( ω 2 ω 1 ) t ] In an acoustical system, this gives beats, which is more easily seen if we take the case where A 1 = A 2 A , then: x = x 1 + x 2 = A e i ( ω 1 t ) + A e i ( ω 2 t ) = A e i ( ω 1 + ω 2 2 + ω 1 ω 2 2 ) t + A e i ( ω 1 + ω 2 2 ω 1 ω 2 2 ) t = A e i ( ω 1 + ω 2 2 ) t [ e i ( ω 1 ω 2 2 ) t + e i ( ω 1 ω 2 2 ) t ] = 2 A e i ( ω 1 + ω 2 2 ) t cos [ ( ω 1 ω 2 2 ) t ] Where the last step used cos θ = e i θ + e i θ 2 So in an acoustical system we will get a dominant sound that has the average ofthe two frequencies and and envelope of amplitude that slowly oscillates. This will be looked at more closes in the context of mechanical waves.

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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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