Interference and wave nature of light

[lovelyrwwr]

Two in-phase sources of waves are separated by a distance of 4.00 m. These sources produce identical waves that have a wavelength of 5.00 m. On the line between them, there are two places at which the same type of interference occurs. Where are the places located?



I already know the answers from this website http://lev.ccny.cuny.edu/~hmakse/TEACHING/204soln27.pdf problem 3. However I cannot for the life of me understand the solution. I especially don't understand why d1 + d2 has to be less than 4. I have spent an hour on this problem and re-reading the explanation and my textbook. But it just doesn't click.

 

[haruspex]

I followed that link, but unfortunately some of the pages appear blank for me, including the one with the relevant solution. Are d1 and d2 the distances from a particular source? If so, let F1, f2 be the distances from the other source. You know d1 + d2 + f1 +f2 = 8m, so either d1 + d2 <= 4m or f1+f2 <= 4m. So choose as reference the source which is nearer on average.

 

[lovelyrwwr]

Here is the solution. I think I am having trouble envisioning the scenario :/

 

[haruspex]

I don't like the wording of the question. Interference will occur everywhere along the line. It ought to specify totally constructive or totally destructive.
I agree the d1+d2=4m argument is a bit glib. By symmetry, if one point is d1 from one source then there must be another such point d1 from the other source. Assuming there are exactly two such points then you can conclude d1+d2=4m.

 

[Nickie]

I can understand your frustration with this concept. Interference and the wave nature of light can be complex and difficult to grasp at first. Let me try to explain the solution to this problem in a different way.

When two waves meet, they can either interfere constructively (adding up to a larger amplitude) or destructively (canceling each other out). In this problem, we have two in-phase sources of waves, meaning their crests and troughs align perfectly. This results in constructive interference at certain points along the line between the two sources.

To determine where these points are located, we need to consider the path difference, which is the difference in distance traveled by the waves from each source to a particular point. In constructive interference, the path difference must be a whole number multiple of the wavelength (5.00 m in this case).

For the first point of interference, the path difference must be 0 (since the waves are coming from the same source and have traveled the same distance). This means that the distance between the two sources (4.00 m) must be a whole number multiple of the wavelength, which is why d1 + d2 must be less than 4.00 m.

For the second point of interference, the path difference must be equal to one wavelength. So, we can set up the equation d1 + d2 = 5.00 m and solve for d1. This gives us one possible location for the second point of interference.

I hope this explanation helps you to better understand this problem. It may also be helpful to visualize the interference patterns using diagrams or animations. Keep practicing and don't hesitate to ask for clarification if needed. Understanding these concepts takes time and effort, but it will eventually click for you.