A simple harmonic progressive wave is represented by \(y=A \sin (100 \pi t+3 x)\). The distance between two points on the wave at a phase difference of \(\frac{\pi}{3}\) radian is

Equation of the given harmonic progressive wave
\(y=A \sin (100 \pi t+3)\)
General equation of a harmonic wave
\(\mathrm{y}=\mathrm{A} \sin (\mathrm{wt}+\mathrm{kx})\)
From equations (i) and (ii),
\(\omega=100 \pi, \mathrm{k}=3\)
But, \(\mathrm{k}=\frac{2 \pi}{\lambda} \Rightarrow 3=\frac{2 \pi}{\lambda}\)
\(\therefore \quad \lambda=\frac{2 \pi}{3}\)
We also know,
Path difference \(\Delta \mathrm{x}=\frac{\lambda}{2 \pi} \times\) Phase difference \(\Delta \phi\)
\(\therefore \quad \Delta x=\frac{2 \pi}{3 \times 2 \pi} \times \frac{\pi}{3} \quad \ldots .\left(\right.\) Given \(\left.\Delta \phi=\frac{\pi}{3}\right)\)
\(=\frac{\pi}{9} \mathrm{~m}\)