The path difference between two waves, represented by \(\mathrm{y}_1=\mathrm{a}_1 \sin \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}\right)\) and \(\mathrm{y}_2=\mathrm{a}_2 \cos \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}+\phi\right)\) is

\(\begin{aligned}
& \mathrm{y}_1=\mathrm{a}_1 \sin \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}\right) \\
& \mathrm{y}_2=\mathrm{a}_2 \cos \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}+\phi\right)
\end{aligned}\)
\(\mathrm{y}_2\) can also be written as
\(\begin{aligned}
\mathrm{y}_2 & =\mathrm{a}_2 \sin \left[\frac{\pi}{2}+\left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}+\phi\right)\right] \\
& =\mathrm{a}_2 \sin \left(\omega \mathrm{t}-\frac{2 \pi \mathrm{x}}{\lambda}+\phi+\frac{\pi}{2}\right)
\end{aligned}\)
The phase difference between the two waves is
Path difference \(=\frac{\lambda}{2 \pi} \times\) Phase difference
Path difference \(=\frac{\lambda}{2 \pi} \times\left(\phi+\frac{\pi}{2}\right)\)