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

$\begin{array}{l}{\mathrm{y}}_1={\mathrm{a}}_1\sin \left(\mathrm{\omega t}-\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}\right)\\ {\mathrm{y}}_2={\mathrm{a}}_2\sin \left(\mathrm{\omega t}-\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}+\mathrm{\varphi }+\frac{\mathrm{\pi }}{2}\right)\end{array}$

Phase difference

                       $=\left(\mathrm{\omega t}-\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}+\mathrm{\varphi }+\frac{\mathrm{\pi }}{2}\right)-\left(\mathrm{\omega t}-\frac{2\mathrm{\pi x}}{\mathrm{\lambda }}\right)=\left(\mathrm{\varphi }+\frac{\mathrm{\pi }}{2}\right)$

$\text{ Path difference }=\frac{\mathrm{\lambda }}{2\mathrm{\pi }}\times \text{ Phase difference }=\frac{\mathrm{\lambda }}{2\mathrm{\pi }}\left(\mathrm{\varphi }+\frac{\mathrm{\pi }}{2}\right)$