# Superposition of two waves with different frequencies

**1. The problem statement, all variables and given/known data**

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)

**2. Relevant 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

**3. 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. 🙂

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

http://ift.tt/1j7WWtk

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