Vanier 203-SN3-RE Waves and Modern Physics tutoring

From path difference to interference

Two coherent sound sources emit sinusoidal waves of the same frequency, in phase, in a uniform medium where linear superposition applies. At an observation point, the path lengths are 2.10 m and 2.40 m, the arriving sound-pressure amplitudes are equal, and reflections are neglected. The wavelength is 0.60 m. Is the interference constructive or destructive?

First distinguish wavelength from period on a wave graph. Wavelength measures spatial repetition; period measures repetition in time. Read wavelength and period from graphs.

The path difference is $\Delta r=2.40-2.10=0.30$ m. Convert it into phase difference:

$$\begin{aligned}\Delta\phi&=\frac{2\pi\Delta r}{\lambda}\\&=2\pi\frac{0.30}{0.60}\\&=\pi\ \mathrm{rad}.\end{aligned}$$

The waves arrive half a cycle apart, so they interfere destructively. The sound-pressure variations cancel at that observation point in this ideal model.