Coherent and Incoherent Addition of Waves
When two identical vibrating needles are allowed to oscillate in phase when they just touch the free surface of water in the tank at points $S_1$ and $S_2$. These needles generate two waves, each of amplitude $A$ at any instant. Phase difference between their displacement does not change with time. Consider a point $P$ such that $S_1P = S_2P$. At this point, resultant displacement is the sum of the individual displacements $y_1$ and $y_2$ of the two waves respectively.
$y = y_1 + y_2 \quad \dots (i)$
$But \quad y_1 = y_2 = A \sin \omega t$
$\therefore \quad y = 2A \sin \omega t$
where $2A$ is the amplitude of the resultant wave.
Now, intensity, $I \propto (\text{amplitude})^2$
$\therefore$ resultant intensity, $\quad I = K(2A)^2 = 4KA^2 = 4I_0$
$I_0 = KA^2$ is the intensity of wave produced by each source.
Thus, the intensity of the resultant wave at any point on the perpendicular bisector of
$S_1S_2 = 4I_0$.
Now, consider another point $Q$
where $S_1Q \neq S_2Q$
Let \quad $S_2Q - S_1Q = \lambda$,
where $\lambda$ is wavelength of the wave.
$y_1 = A \sin \omega t$
$\because$ phase difference corresponding to $\lambda = 2\pi$ radian
$y_2 = A \sin(\omega t + 2\pi)= A \sin\omega t$
Substituting these values in (i) gives $y = 2A \sin \omega t$. Thus, the amplitude of the resultant wave is double the amplitude of the individual wave. Hence, intensity is increased (i.e., $I = 4I_0$) and the superposition is constructive.
Now consider one more point $R$ such that :
$S_2R - S_1R = \frac{\lambda}{2}$.
Then
$y_1 = A \sin \omega t$
$y_2 = A \sin (\omega t + \pi) = -A \sin \omega t$
Substituting the values of $y_1$ and $y_2$ in eqn. (i), we have
$y = 0$, giving zero amplitude (or intensity) or destructive superposition.
(1) Condition for constructive superposition (maximum intensity) can be generalised as,
$S_1Q - S_2Q = m\lambda, \quad \text{where } m = 0, 1, 2, \dots$
Thus, the path difference between two superimposing waves must be integral multiple of wavelength ($\lambda$) for constructive superposition.
(2) Condition for destructive superposition (minimum intensity) can be generalised as :
$S_1R - S_2R= \left( m + \frac{1}{2} \right) \lambda, \quad \text{where } m = 0, 1, 2, \dots$
Thus, the path difference between two superimposing waves must be odd multiple of wavelength ($\lambda$) for destructive superposition.