Question

In figure, sound wavesand, both of wavelength$\lambda$, are initially in phase and traveling rightwards , as indicated by the two rays. Ways Ais reflected from four surfaces but ends up traveling in its original direction. Waveends in that direction after reflection from two surfaces. Let distance Lin the figure expressed as a multiple$\mathbf{q} \mathbf{\lambda }\mathbf{:}\mathbf{L}\mathbf{=}\mathbf{q\lambda }\mathbf{.}$What are the(a)Smallest (b)Second smallest value of qthat put Aand Bexactly out of phase with each other after the reflection ?

Short answer

Answer

1. The smallest value of q that put A and B exactly out of the phase with each other after the reflections is 0.5.
2. The second smallest value of P that put Aand B exactly out of the phase with each other after the reflections is1.5.

Step-by-step solution

Step 1: Given

$L=q\lambda$

The waves are out of phase after the reflections

Determining the concept

From the condition of constructive and destructive interference, find the smallest and second smallest value of q for which the two waves are out of phase.

Formulae are as follow:

1. $\varphi =m\left(2\pi \right)$for constructive interference m = 0,1,2,..
2. $\varphi =\left(2m+1\right)\pi$for destructive interference $m=0,1,23..$

(a) Determining the smallest value of  q that put A and B exactly out of the phase with each other after the reflections

The given condition states that the two waves are out of phase after the reflection. So, the interference should be destructive.

If the interference is destructive, then the path difference must be equal to .

So, from the figure, their path difference is , hence, from the condition, must be,

$L=\frac{\lambda }{2}$…… (i)

But, it is given that,

$L=q\lambda$

Substitute for into the equation (i)

$$\begin{gathered} q\lambda =\frac{\lambda }{2} \\ q=\frac{1}{2} \\ =0.5 \end{gathered}$$

Hence, the smallest value of q that put A and B exactly out of the phase with each other after the reflections is 0.5

(b) Determine the second smallest value of q that put A and B  exactly out of the phase with each other after the reflections

For the second smallest value of,

$L=3\lambda /2$…… (ii)

But,

$L=q\lambda$

Substitute role="math" localid="1661349019513" $q\lambda forL$into the equation (ii)

$$\begin{gathered} q\lambda =3\lambda /2 \\ q=1.5 \end{gathered}$$

Hence, the second smallest value of q that put and exactly out of the phase with each other after the reflections is 1.5 .