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.

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.