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|=(n+\tfrac12)\lambda,\quad n=0,1,2,\ldots

problems.proof.analysis

Destructive interference requires an odd multiple of $\pi$ , hence $|\Delta r|=(n+1/2)\lambda$ . Taking $n=0,1,\ldots$ and the magnitude avoids duplicate negative orders.