Problem 3

Two coherent sources of equal amplitude emit sound in phase of wavelength

\lambda

. At point

P

their path difference is

\Delta r

. Find the conditions for constructive and destructive interference at

P

, including the corresponding integer orders. (a) Derive the phase difference at

P

from signed path difference

\Delta r

. (b) Obtain constructive and destructive conditions with integer orders, distinguishing signed difference from its magnitude. (c) Decide whether reversing the path difference or reversing one source’s phase changes the fringe type; check zero and half-wavelength path differences.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\Delta r\mapsto-\Delta r:\text{same extrema}

problems.proof.analysis

Replacing $\Delta r$ by $-\Delta r$ reverses only the phase sign; the maximum and minimum conditions remain true. It exchanges the paths without changing the interference type at $P$ .

problems.proof.pitfall. Pitfall: forgetting the initial phase when sources are in antiphase. It swaps maxima and minima; it is not merely a sign change in path difference.