Physics - Sound Wave Question with Solution | TestHub

The equations of the two waves are $y_1=a\sin (\omega t+{\varphi }_1)$ and $y_2=a\sin (\omega t+{\varphi }_2)$.

The amplitude of the resultant wave is given by $A_R=\sqrt{a_1^2+a_2^2+2a_1{a}_2\cos ({\varphi }_1-{\varphi }_2)}$.

Given that $a_1=a$, $a_2=a$, and $A_R=a$.

Substituting these values, we get $a=\sqrt{a^2+a^2+2a^2\cos ({\varphi }_1-{\varphi }_2)}$.

Squaring both sides, $a^2=2a^2+2a^2\cos ({\varphi }_1-{\varphi }_2)$.

Dividing by $a^2$, $1=2+2\cos ({\varphi }_1-{\varphi }_2)$.

$2\cos ({\varphi }_1-{\varphi }_2)=-1$

$\cos ({\varphi }_1-{\varphi }_2)=\frac{-1}{2}$

${\varphi }_1-{\varphi }_2={\cos }^{-1}\left(\frac{-1}{2}\right)$

Answer: ${\cos }^{-1}\left(\frac{-1}{2}\right)$