Amplitude of the Resultant Wave Produced due to Superposition of Two Waves

The two pulse examples above (6.4.1, 6.4.2) show the two EXTREME cases -- fully constructive and fully destructive interference. More generally, consider two waves of the SAME frequency $\omega$ but possibly different amplitudes $A_1$, $A_2$, differing in phase by a constant $\phi$, whose displacements at some fixed point (say x = 0) are $y_1=A_1\sin\omega t$ and $y_2=A_2\sin(\omega t+\phi)$. By the principle of superposition, the resultant is $y=y_1+y_2$; expanding $\sin(\omega t+\phi)$ and collecting the $\sin\omega t$ and $\cos\omega t$ terms gives $y=(A_1+A_2\cos\phi)\sin\omega t+(A_2\sin\phi)\cos\omega t$. Writing $A\cos\theta=A_1+A_2\cos\phi$ (Eq. 6.4) and $A\sin\theta=A_2\sin\phi$ (Eq. 6.5) for some convenient angle $\theta$, this collapses neatly to $y=A\sin(\omega t+\theta)$ (Eq. 6.6) -- so the RESULTANT is itself a simple harmonic oscillation of the SAME frequency $\omega$, just with a new amplitude A and phase $\theta$. …