Superposition of two waves with different frequencies

In summary, the conversation discussed a question about two waves traveling through a dispersive medium with different frequencies and wave numbers. The goal was to find the sum of the two waves, and the conversation included using identities and equations to manipulate the waves and eventually reach a solution. The final solution involved using the average and difference of the wave and frequency numbers. The conversation ended with the suggestion that the solution may require some adjustments to the signs.
  • #1
SinSinger
2
0

Homework Statement


Hi all! It's a superposition question: Two waves travel through dispersive medium, with different frequencies and wave number.
P1(t)=Acos(k1x-w1t)
P2(t)=Acos(k2x-w2t)
Obtain the P(t)=P1(t)+P2(t)

Homework Equations


Well I used identity:
cosα+cosβ=2 cos 1/2(α+β)cos1/2(α-β)
and the following:
w(av)=(w1+w2)/2 Δw=w1-w2
k(av)= (k1+k2)/2 Δk=K1-k2

The Attempt at a Solution


So, this is what I tried to do:
P(t)=A0(2cos(1/2)((k1x-w1t)+(k2x+w2t))cos(1/2)((k1x-w1t)-(k2x-w2t)

=2A0(cos(((k+k)/2)x)-((w-w)/2)t))cos(((k-k)/2)x)-((w+w)/2)t))

=2A0(cos(k(av)-(1/2)Δwt)cos((1/2)Δkx-w(av)t))

And, from here on I'm stuck: Is this all that needed? Help would be very appreciated. :)
 
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  • #2
I think this is all that's needed. But you have to be a bit sharper with the signs, to get k(av) and w(av) in the first and Δk and Δw in the second cosine - I think...
 
  • #3
Hmm, what do you mean?
Thank for the help by the way :)
 

1. What is the concept of superposition of two waves with different frequencies?

The superposition of two waves with different frequencies refers to the phenomenon where two waves with different frequencies overlap or combine to form a new wave. This new wave is a result of the additive combination of the individual waves, and its properties are determined by the characteristics of the two original waves.

2. How does the amplitude of the resulting wave change when two waves with different frequencies superpose?

The amplitude of the resulting wave depends on the amplitudes of the two original waves. If the two waves have the same amplitude, the resulting wave will have a higher amplitude. However, if the two waves have different amplitudes, the resulting wave's amplitude will lie somewhere in between the two original waves' amplitudes.

3. What happens to the frequency of the resulting wave when two waves with different frequencies superpose?

The frequency of the resulting wave is equal to the average of the two original waves' frequencies. This means that if one wave has a higher frequency and the other has a lower frequency, the resulting wave's frequency will be somewhere in between the two frequencies.

4. Can the superposition of two waves with different frequencies result in a wave with a frequency of 0?

No, the superposition of two waves with different frequencies can never result in a wave with a frequency of 0. This is because the frequency of the resulting wave is always the average of the two original waves' frequencies, and as long as the two original waves have non-zero frequencies, the resulting wave will also have a non-zero frequency.

5. How is the phase difference between two waves with different frequencies related to the phase of the resulting wave?

The phase difference between two waves with different frequencies does not affect the phase of the resulting wave. The resulting wave's phase is determined by the phase of the two original waves at the point where they superpose. However, the phase difference between the two waves can affect the resulting wave's amplitude and frequency.

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