Superposition of 2 Waves: Equation

In summary, the equation for the superposition of two waves is 2Acos[(k1-k2)x/2 - (ω1 - ω2)t/2]*sin[(k1 + k2)x/2 - (ω1 + ω2)t/2], which can be simplified by reducing the values of k1 and k2. However, if the waves have different angular frequencies, k cannot be left out and the equation may need to be rewritten in terms of w1 and w2.
  • #1
jumbogala
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Homework Statement


What is the equation for the superposition of 2 waves?

Homework Equations


The Attempt at a Solution


Displacement of the first wave is D1 = Asin(k1x - ω1t)

Displacement of second wave is D2= Asin(k2x - ω2t)

Add together: D1 + D2 = 2Acos[(k1-k2)x/2 - (ω1 - ω2)t/2]*sin[(k1 + k2)x/2 - (ω1 + ω2)t/2]

But this isn't in terms of what the problem is asking! Help, can the form be changed somehow?

Is there a way to reduce the k1 and k2 so that it is in that form?
 
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  • #2
But it IS in terms of w1-w2 and w1+w2! Maybe the question meant that you could use y=Asin(w*t) instead of y=Asin(kx-wt), but otherwise, your work is correct.
 
  • #3
Good point. But if the waves have different angular frequencies, wouldn't k be different? I don't see how you can just leave it out.

When a question asks for something in terms of w1 and w2, doesn't that mean I shouldn't have anything else in there?
 

1. What is the equation for the superposition of two waves?

The equation for the superposition of two waves is given by y(x,t) = A1sin(k1x - ω1t) + A2sin(k2x - ω2t), where A1 and A2 are the amplitudes, k1 and k2 are the wave numbers, and ω1 and ω2 are the angular frequencies of the two waves.

2. How do the amplitudes of the two waves affect the resulting superposition?

The amplitudes of the two waves determine the height or intensity of the resulting superposition wave. If the amplitudes are equal, the resulting wave will have a larger amplitude. If the amplitudes are unequal, the resulting wave will have a smaller amplitude.

3. What happens if the two waves have the same frequency and are in phase?

If the two waves have the same frequency and are in phase, their superposition will result in a wave with a larger amplitude. This is known as constructive interference.

4. Can the superposition of two waves result in a wave with a smaller amplitude?

Yes, if the two waves have different frequencies and are out of phase, their superposition will result in a wave with a smaller amplitude. This is known as destructive interference.

5. How does the phase difference between the two waves affect the resulting superposition?

The phase difference between the two waves affects the resulting superposition by determining whether it will be constructive or destructive interference. If the phase difference is 0 or a multiple of 2π, the superposition will be constructive. If the phase difference is π or a multiple of π, the superposition will be destructive.

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