Points of constructive and destructive interference

[AndreAo]

Homework Statement

Two point sources of sound waves of identical wavelength lambda and amplitude are separated by distance D = 2.0lambda. The sources are in phase. (a) How many points of maximum signal (constructive interference) lie along a large circle around the sources? (b) How many points of minimum signal (destructive interference)?

Homework Equations

$$\phi$$ = $$\Delta$$d/$$\lambda$$*2$$\pi$$
where $$\Delta$$d is the difference of distance between the two sources and the receptor.
$$\lambda$$ is the wavelength
$$\phi$$ is the phase difference

The Attempt at a Solution

To occur constructive interference $$\phi$$ have to be a 2$$\pi$$m, where m$$\in$$N. Using this fact:
2$$\pi$$m = $$\Delta$$d/$$\lambda$$*2$$\pi$$
m = $$\Delta$$d/$$\lambda$$
$$\Delta$$d = m$$\lambda$$
For m = 0, we have a straight passing between the two sources, so it will hit this circle in two points,if m = 1..n , we have points in the side of the two fonts that will hit the circle, again just two points. But the answer is 8 points. Can I say that it would have the same quantity of points for destructive interference too?

 

[Delphi51]

If you haven't already done it in high school, I suggest you dig out a compass and draw a series of concentric circles with radii 1 cm, 2 cm, 3 cm, and 4 cm to represent the crests from one source with wavelength 1 cm. Start again 2 cm away from the first source and draw the concentric circles again. Points where the crests intersect are points on the constructive interference lines. Draw the lines of constructive interference - you will see there are 8 of them all right. If you don't have a compass, use this animation:
http://id.mind.net/~zona/mstm/physics/waves/interference/twoSource/TwoSourceInterference1.html
For the destructive interference pattern you must draw circles with radii of .5, 1.5, 2.5, etc. in a different colour to represent the troughs of the waves. Mark the spots where a trough meets a crest to see the pattern of destructive interference.

 

[AndreAo]

I didn't draw it before, but I think this isn't the way to solve if a can find the locus of that points. Thanks for answering.

 

[Delphi51]

Even more fun to draw it in a computer Draw program so you can group the crests and troughs from each source together and move them relative to each other so you can see the patterns for separations of 1λ, 1.5λ, etc. Once you have done a few, you will see the pattern and can write down a simple formula for the number of constructive interference lines given the separation of the sources.

I'll be interested in seeing your more sophisticated solution!

 

[AndreAo]

I've found the solution for it, but I'll not reproduce in the answer: http://www.fisica.ufs.br/CorpoDocente/egsantana/ondas/interferencia/Interferencia.html , it's in portuguese.

Thanks