Q.Give a mathematical analysis of the interference of sound and explain constructive and destructive interference.

The principle of superposition states that when two or more waves overlap in a medium, the net (resultant) displacement at any point and instant is simply the algebraic (or vector, in more than one dimension) sum of the individual displacements each wave would have produced there on its own, $y = y_1 + y_2$; this holds because the wave equation is linear, so any linear combination of its solutions is itself also a solution. Waves obeying this rule (of comparatively small amplitude relative to their wavelength) are called linear waves; larger-amplitude waves that violate strict superposition (such as some laser light) are called non-linear waves and are not treated further in this framework. Superposition explains three related phenomena: spatial interference (simply called interference), temporal interference (beats), and stationary (standing) waves. For two coherent waves of the same frequency and waveform but different amplitudes $A_1, A_2$ and a constant phase difference $\phi$, adding $y_1 = A_1\sin(kx-\omega t)$ and $y_2 = A_2\sin(kx-\omega t+\phi)$ using trigonometric identities gives a single resultant sinusoidal wave of the same frequency, $y = A\sin(kx-\omega t+\theta)$, with resultant amplitude found from $A^2 = A_1^2 + A_2^2 + 2A_1A_2\cos\phi$; since intensity is proportional to the square of amplitude ($I \propto A^2$), the resultant intensity is $I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\phi$, showing the outcome depends entirely on the phase difference $\phi$ at that point. Constructive interference occurs where crests overlap crests (and troughs overlap troughs), i.e. where $\phi = 2n\pi$ ($n = 0,1,2,\ldots$) or equivalently the path difference $\Delta r$ between the two waves' routes to that point is a whole number of wavelengths, $\Delta r = n\lambda$; there the amplitudes simply add, $A = A_1+A_2$, giving the maximum possible intensity. Destructive interfe …