# Light Interference Wave Equation - Assumption

1. Apr 12, 2013

### elemis

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

2. Relevant equations

In question.

3. The attempt at a solution

To be clear it's part (vi) that's unclear to me.

In order to ignore the cosine term it has to reduce to 1. This can happen, only if k(x1+x2)/2 = ωt

Is this a correct assumption ?

Also, it is known that k = 2∏/λ and ω=2∏/T

However, I'm trying to think in what way these two components could be equal but I can't get it.

Could someone give me some help, please ?

2. Apr 12, 2013

### Staff: Mentor

For a long exposure, you average y^2 over time - the constant phase in the cosine does not matter for the average.

3. Apr 12, 2013

### elemis

I really dont understand what you mean. Could you please break it down for me ?

I really do appreciate you taking the time out to help me.

4. Apr 25, 2013

### Simon Bridge

A photograph collects light over a period of time.
The lightness in the photo depends on the intensity of the incoming light and the time of the exposure.

He's telling you to find y^2, then average it over time.
This gives you a function of position alone - which will be what the photograph shows.
When you do this - the terms you are worried about will cancel out.

Do you know what y=?
Do you know how to square a function?
Do you know how to average a function?

5. Apr 25, 2013

### elemis

I can do the first two but not the last one. How do I average a function over time ?

EDIT: Why do I need to find y^2 ? Is it because intensity is directly proportional to Amplitude^2 ?

6. Apr 25, 2013

### Simon Bridge

7. Apr 26, 2013

### elemis

Because the limits are 0 to infinity the cosine function goes zero and hence we can ignore it completely, correct ?

8. Apr 26, 2013

### Simon Bridge

Did you read the links? The second one tells how to handle the average of a periodic function - what does it say to do?

9. Apr 26, 2013

### elemis

I've carried out the steps for the time average for cos2x for limits 0 to T.

I've gotten : $\frac{1}{T}$[$\frac{1}{2}$T+$\frac{1}{4}$sin2T]

What do I choose to be T in this case ? Infinity ?

10. Apr 26, 2013

### Simon Bridge

But it is not exactly cos2x that you have to average is it? x(t)=?

The trick with time averaging square trig-functions is to choose your interval carefully - if you let T be any integer number of quarter cycles, the average will make sense easily. I usually pick a singe period for T.
http://hyperphysics.phy-astr.gsu.edu/hbase/math/defint.html (3rd panel down)