# Superposition of 1-d Waves

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

A string with linear mass density μ = 0.0250 kg/m under a tension of T = 250. N is oriented in the x-direction. Two transverse waves of equal amplitude and with a phase angle of zero (at t = 0) but with different frequencies (ω = 3000. rad/s and ω/3 = 1000. rad/s) are created in the string by an oscillator located at x = 0. The resulting waves, which travel in the positive x-direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative x-direction. Find the values of x at which the first two nodes in the standing wave are produced by these four waves.

**3. The attempt at a solution**

So to begin I described the phases of the two waves travelling in the positive x direction.

[itex]\phi_{1} = (ω_{1}/v)x – ω_{1}t[/itex], where [itex]ω_{1} = 3000 rad/s[/itex] and [itex]\phi_{2} = (ω_{1}/3v)x – (ω_{1}/3)t[/itex].

And the wave speed, v = [itex]\sqrt{T/\mu}[/itex].

Now the waves get reflected, and we get phases:

[itex]\phi^{‘}_{1} = (ω_{1}/v)x + ω_{1}t[/itex] and

[itex]\phi^{‘}_{2} = (ω_{1}/3v)x + (ω_{1}/3)t[/itex].

So I’m not really sure what to do know. Should I just take the superposition of all the waves functions, and determine when their sum is equal to 0 (signifying no amplitude, hence a nodal point)?

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