Light Interference Wave Equation - Assumption

[elemis]

Homework Statement

Homework Equations

In question.

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 ?

 

[mfb]

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

 

[elemis]

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

I really don't understand what you mean. Could you please break it down for me ?

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

 

[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?

 

[elemis]

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?

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 ?

 

[Simon Bridge]

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

http://www.insula.com.au/physics/1279/L14.html

How do I average a function over time ?

http://www2.ensc.sfu.ca/~glennc/e220/e220l18b.pdf

 

[elemis]

http://www.insula.com.au/physics/1279/L14.htmlhttp://www2.ensc.sfu.ca/~glennc/e220/e220l18b.pdf

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

 

[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?

 

[elemis]

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

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 ?

 

[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)