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Adding Harmonic Motions

Module by: Paul Padley. E-mail the author

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 with adding 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 ) ] 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 vibrating with the frequency of the sound you are listening two. In a noise canceling headphone there is also a microphone "listening" to the noice coming from outside the headphone. This oscillation is inverted and then added to membrane producing the sound you listen to. The net result is a signal that contains the 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 different frequencies: 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 ] 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 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 ] 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 cos θ = e i θ + e i θ 2 So in an acoustical system we will get a dominant sound that has the average of the 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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Definition of a lens

Lenses

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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