Homework Help: Combination of two waves.

1. Jul 27, 2011

geoffreythelm

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

Two infinite waves Ψ1, Ψ2 have the same wavelength and polarisation and have amplitudes of E1 = 3 and E2 = 7 units. They are added together with a phase difference of 125 degrees.

1) What will be the relative intensity of the resultant wave? Assume initial phase of 0 degrees for Ψ1.

2) In the same conditions, what will be the phase of the resultant wave (in degrees)?

2. Relevant equations

N/A

3. The attempt at a solution

So far I've got Ψ1=3sin(kx-ωt) and Ψ2=7sin(kx-ωt+2.182)
And I know that I is proportional to A squared.

Now I'm pretty stuck!

2. Jul 27, 2011

ehild

3. Jul 27, 2011

geoffreythelm

Thanks for the help, ehild!

I've found a formula in my notes for resultant amplitude of 2 waves: (A3)^2 = (A1)^2 + (A2)^2 + 2(A1)(A2)cos(δ)

So using this, I get an (A3)^2 of 34, which I suppose is the relative Intensity. Not sure how I should answer this on the question though, I haven't really specified the relative intensities of A1 and A2.

As far as part 2 goes, when I add up Ψ1 and Ψ2, I get stuck with some terms I can't get rid of, and I certainly can't put it into a nice 'A sin(kx-wt +φ)' form.
I'm also getting a bit confused about usage of δ, φ, α, ε, phase, phase difference, phase constant, etc. In my notes I found the following for resultant phase difference (I think):

tanα=(A1sinα1+A2sinα2)/(A1cosα1+A2cosα2)

Not sure if this is of any use.

When I add together the 2 waveforms I get Ψ3=3+7cosδ(sin(kx-ωt))+7sinδ(cos(kx-ωt)).

Thanks again for the response!

4. Jul 27, 2011

ehild

You get the relative intensities of the original waves as the square of the amplitudes.
The sum of the two waves is equal with a third one, which is
A3 sin(kx-wt +α) with the terms in your notes. The whole argument of the sine function is the phase of the wave. You see, it changes both with time t and place x. The constant term , 2.182 radian in case of Ψ2 is the phase constant. It is zero in case of Ψ1. The phase constant is the phase at t=0 and x =0, the problem refers to it as initial phase.

The resultant amplitude is given for waves with phase difference δ which is the same as the difference of the phase constants:
δ=(wt-kx+2.182)-(wt-kx)=2.182.

The phase constant of the resultant wave is given in your notes for two waves: one with amplitude A1 and phase constant α1, the other with amplitude A2 and phase constant α2. A1=3 and the phase constant of the first wave α1=0 in the problem, and A2=7, α2 = 2.182 or 125°. From these, you get the phase constant α of the resultant wave.

ehild