How Does Phase Difference Affect Wave Interference?

[geoffreythelm]

I'm getting so confused about this question, any help would be great. :)

Homework Statement

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)?

Homework Equations

N/A

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!

 

[ehild]

Add up Ψ1 and Ψ2. Decompose 7sin(kx-wt + δ) in terms of sines and cosines of (kx-wt) and δ. (Use the addition laws:

http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities

the sum has to be equal to a single wave with amplitude A and phase constant φ, A sin(kx-wt +φ) .

ehild

 

[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!

 

[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

 

[rwisz]

I can understand your confusion with this question. Let's break it down step by step and try to make sense of it.

First, we are given two waves, Ψ1 and Ψ2, that have the same wavelength (meaning they have the same distance between each peak or trough) and the same polarization (meaning the direction of the electric field is the same). We are also given their amplitudes, which represent the maximum displacement of the wave from its equilibrium position. In this case, E1 = 3 and E2 = 7 units.

Next, we are told that these two waves are being "added together with a phase difference of 125 degrees". This means that the two waves are being superimposed on each other at the same point in space and time, but one of the waves is shifted in phase by 125 degrees compared to the other. This phase difference is represented by the term "2.182" in the equation you wrote for Ψ2.

Now, let's tackle the first question: What will be the relative intensity of the resultant wave? Intensity is proportional to the square of the amplitude, so we need to find the amplitude of the resultant wave. To do this, we can use the principle of superposition, which states that when two waves are added together, the resulting amplitude is the sum of the individual amplitudes. In this case, the resultant amplitude will be E1 + E2 = 3 + 7 = 10 units.

So, the relative intensity of the resultant wave will be the square of the amplitude, which is (10)^2 = 100 units.

For the second question, we need to find the phase of the resultant wave. To do this, we can use the formula for the phase difference between two waves, which is given by the equation φ = 2πΔt/T, where Δt is the time difference between the two waves and T is the period (or wavelength) of the waves. In this case, we know that the two waves have the same wavelength, so T is the same for both. We also know that the phase difference between the two waves is 125 degrees, which is equivalent to 2.182 radians. So, we can plug these values into the equation and solve for Δt:

2.182 = 2πΔt/λ

Δt = 2.182λ/2