Analyzing wave interference patterns

[sillyquark]

Hello, this is my first visit to PF. I have a question that I can tell isn't very difficult, but I can't seem get my head around it, maybe i just need to review my text and make an attempt tomorrow.

Homework Statement

Consider the point P in the following diagram. Analyze this diagram to show that the equation
|P^n S^1-P^2 S^2| = (n-(1/2))(lambda) is valid for this particular location for the point P.

the ^ indicates a subscript

Homework Equations

|P^n S^1-P^2 S^2| = (n-(1/2))(lambda)

The Attempt at a Solution

I haven't made an attempt at a solution, but I am not looking for any answers. If someone could help me understand the problem better, I am sure that I can solve the question.

 

[Spinnor]

If you have two different sources with the same frequency then there will be places where there is destructive interference. Those places are the points that lie on the lines labeled N1 and N2. For N=1 the formula in the attached thumbnail says that for any point P_1 that lies on the line N=1 then the difference in the distances P_1S_1 and P_1S_2 is one half the wavelength and there will be destructive interference.

Hope that helps.

 

[sillyquark]

I'm still a little unsure what the question wants, should I attempt to prove the statement algebraically? Should I be reasoning, and rationalize the equation in a manner similar to what you have done? The question is worth four marks so I assume that they are looking for four points of proof.

 

[sillyquark]

All that I can come up with at the moment is:
Since the point P lies on the third nodal line, the absolute value of the path difference between (S1,P) and (S2,P), is equal to 2 1/2 times the wavelength. Since 2 1/2 is a multiple of 1/2, the point lies on a path of destructive interference between the two waves.

 

[sillyquark]

One problem I have encountered is if the point lies on the normal line (line down the center). Would n=0? If so wouldn't my answer state that there is a difference in distance between point P and the two frequency generating sources, when clearly the line bisects them. Is this equation useless when calculating on the normal line since there is no difference in distance between the two points? Or does n=1/2 when a point lies on the normal line, this would allow multiplication by 0 which would agree with rational thought. That being said n>or=1/2?

I'm sorry to run on but the text was not concise. Should I include the correct version from the statement above in my answer or stick to evidence directly concerning point P. Finally, should I draw on a ratio between S1, S2, and any point (P for example) since they form a triangle.