Question

Question: Figure shoes four isotropic point sources of sound that are uniformly spaced on an x axis. The sources emit sound at the same wavelength $\mathit{\lambda }$and same amplitude ${\mathit{s}}_{\mathbf{m}}$and they emit in phase. A point, is shown on the y axis. Assume that as the sound waves travel to, the decrease in their amplitude is negligible. What multiple of toP if distance din the figure is $\mathbf{\left(}\mathbf{a}\mathbf{\right)}\frac{\mathbf{\lambda }}{\mathbf{4}}\mathbf{,}\mathbf{\left(}\mathbf{b}\mathbf{\right)}\frac{\mathbf{\lambda }}{\mathbf{2}}\mathbf{,}\mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathit{\lambda }$

Short answer

Answer

1. Multiple of$s_m$ as the amplitude of the net wave at P if the distance$d=\lambda /4$ is zero.
2. Multiple of $s_m$as the amplitude of the net wave at P if the distancerole="math" localid="1661347340773" $d=\lambda /2$is zero.
3. Multiple of $s_m$as the amplitude of the net wave at P if the distance $d=\lambda$is .

Step-by-step solution

Step 1: Given data

• Point source uniformly spaced on the x axis.
• The wavelength of the source sound is $\lambda$.

The displacement amplitude of the source sound is s_m

Determining the concept

Use the principle of constructive and destructive interference to find the multiple of amplitude at point P.

Formulae are as follow:

1. $f=m2\pi$for constructive interference m = 0, 1, 2, ....
2. $f=\left(2m+1\right)\pi$for destructive interference m = 0, 1, 2, 3, ...

(a) Determining the multiple of  as the amplitude of the net wave at P if the distance 

From the problem, the sound wave has the same amplitude at the point P

But,$L=q\lambda$ , i.e., the path length difference is some constant times wavelength. The amplitude of wave passing through point P is the same, When,$d=\lambda /4$ , i.e., the net effect on amplitude must be zero.

There are four sources. First and third are $\lambda /2$distance apart; similarly, second and fourth are ,$\lambda /2,d=\lambda /2$ , distance apart, so destructive interference occurs at P.

Since, the interference is destructive and the amplitude of the resultant wave is zero.

Hence, the multiple of s_m as the amplitude of the net wave at P, if the distance $d=\lambda /4$is zero.

(b) Determining the multiple of sm  as the amplitude of the net wave at P if the distance  d=λ/2

Also, destructive interference forms between the first and second sources and third a $d=\lambda$and fourth sources, so there is no multiple of amplitude at point.

Since, the interference is destructive, if the amplitude of the resultant wave is zero.

Hence, Multiple of as the amplitude of the net wave at P, if the distance role="math" localid="1661348347215" $d=\lambda /2$ is zero.

(c) Determining the multiple of Sm as the amplitude of the net wave at P if the distance d=λ

The condition for constructive interference,

$$\begin{gathered} \frac{∆L}{\lambda }=0,1,2,3.. \\ f=m2\pi \end{gathered}$$

Difference between the two sources is$d=\lambda$, and that produces constructive interference. Four sound waves are in constructive interference, so that the amplitude increases by the factor of four, i.e.$4s_m$.

Hence, multiple of as the amplitude of the net wave at P, if the distance$d=\lambda$ is$4s_m$ .