Principle of Superposition of Waves

The principle of superposition of waves states that when two or more waves travel through the same region of a medium simultaneously, the resultant (net) displacement of any particle of the medium, at any instant, is simply the algebraic sum of the displacements that each wave, acting alone, would have produced at that same point and instant. A direct and important consequence is that the waves themselves pass through one another completely unaffected -- after they have crossed, each continues on with its own original amplitude, frequency, and direction, exactly as if the other had never been there; only while they overlap does their combined effect appear. Where two waves arrive at a point in the same phase (crest meeting crest, or more generally a path or phase difference equal to a whole number of wavelengths, $n\lambda$, or a whole multiple of $2\pi$), their displacements add directly, an effect called constructive interference, producing a resultant of larger amplitude than either wave alone. Where two waves arrive exactly out of phase (crest meeting trough, a path difference of $\left(n+\tfrac12\right)\lambda$ or a phase difference of $\pi$, $3\pi,\dots$), their displacements subtract, an effect called destructive interference, which can cancel the disturbance completely if the two waves have equal amplitude. As a worked case, consider two waves of the same frequency and the same amplitude $A$, arriving at one point with a constant phase difference $\phi$ between them: $y_1=A\sin(\omega t)$ and $y_2=A\sin(\omega t+\phi)$. Superposing, $y=y_1+y_2=2A\cos\!\left(\frac{\phi}{2}\right)\sin\!\left(\omega t+\frac{\phi}{2}\right)$ so the resultant is itself simple harmonic, of the same frequency, with amplitude $A_R=2A\cos(\phi/2)$: maximum, $A_R=2A$, when $\phi=0$ (fully constructive), and zero when $\phi=\pi$ (fu …