Q.Find the amplitude of the resultant wave produced due to interference of two waves given as $y_1=A_1\sin\omega t$ and $y_2=A_2\sin(\omega t+\phi)$.

[!TLDR] Expanding $y=y_1+y_2$ and collecting $\sin\omega t$/$\cos\omega t$ terms gives resultant amplitude $A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi …

Given $y_1=A_1\sin\omega t$ and $y_2=A_2\sin(\omega t+\phi)$, the principle of superposition gives the resultant $y=y_1+y_2=A_1\sin\omega t+A_2\sin(\omega t+\phi)$. Expanding the second term using $\sin(\omega t+\phi)=\sin\omega t\cos\phi+\cos\omega t\sin\phi$ gives $y=A_1\sin\omega t+A_2\sin\omega t\cos\phi+A_2\cos\omega t\sin\phi=(A_1+A_2\cos\phi)\sin\omega t+(A_2\sin\phi)\cos\omega t$. Writing $A\cos\theta=A_1+A_2\cos\phi$ and $A\sin\theta=A_2\sin\phi$ for a convenient auxiliary angle $\theta$, the resultant collapses to the single simple-harmonic form $y=A\sin(\omega t+\theta)$ -- so the resultant is itself oscillating at the SAME angular frequency $\omega$, with a new amplitude A and phase $\theta$. Squaring and adding the two boxed relati …

Expand the second wave's sine term using the compound-angle formula, collect coefficients of $\sin\omega t$ and $\cos\omega t$, write them as $A\cos\theta$ and $ …

Forgetting the cross term $2A_1A_2\cos\phi$ entirely and simply writing $A=\sqrt{A_1^2+A_2^2}$ (as if the two waves were perpendicular vectors rather than sharing the same direction wi …