Coherent and Incoherent Addition of Waves :

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.

Coherent and Incoherent Addition of Waves :

$y = y_1 + y_2 \quad \dots (i)$

$But \quad y_1 = y_2 = A \sin \omega t$

$\therefore \quad y = 2A \sin \omega t$

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.



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