Name: Speed of Sound Wave in Gases
Uploaded: 2019-07-19
Duration: 8 min 7 s
Description: This animation enables students to study Newton’s formula for the sound wave in a gas. Also, the Laplace correction to Newton's formula is more accurate by s...

Question

Figure shows four isotropic point sources of sound that are uniformly spaced on the $x$ -axis. The sources emits sound at the same wavelength $λ$ and same amplitude $S_{m}$ , and they emit in phase. A point $P$ is shown on the $x$ -axis. Assume that as the sound waves travel to $P$ , the decrease in their amplitude is negligible. What multiple of $S_{m}$ is the amplitude of the net wave at $P$ , if the distance $d$ in the figure is (a) $\frac{λ}{4}$ , (b) $\frac{λ}{2}$ and (c) $λ$ ?

Accepted Answer

Given:

Four isotropic point sources of sound are uniformly spaced on $x$ -axis.
Wavelength of sound $= λ$

Amplitude of sound $x$

Phase difference, $ϕ = 0$

Let the distance between $S_{4}$ and $P$ be $x$ .

Then, distance between $S_{3}$ and $P$ is $d + x$ .

The distance between $S_{2}$ and $P$ is $2 d + x$

The distance between $S_{1}$ and $P$ is $3 d + x$

(a) When $d = \frac{λ}{4}$ , path difference between $(1)$ and $(3)$ is $2 d = \frac{λ}{2}$ ,

$ϕ = \frac{2 π}{λ} \times \frac{λ}{2} = π$

Therefore, the phase difference between $(1)$ and $(3)$ is $x$ .

Similarly, the path difference between $(2)$ and $P$ is $P$ ,

$ϕ = \frac{2 π}{λ} \times \frac{λ}{2} = π$

Sources $(1)$ and $(3)$ show destructive interference and $(2)$ and $(4)$ show destructive interference. Hence, amplitude at a point $P$ is $0$ .

(b) When $d = \frac{λ}{2}$ , path difference between sources $(1)$ and $(2)$ ,

Path difference $= d = \frac{λ}{2}$

Phase difference $= \frac{2 π}{λ} \times \frac{λ}{2} = \frac{2 π}{2} = π$

Similarly, for sources $(3)$ and $(4)$ ,

Phase difference $= π$

Thus, there is destructive interference between the sources $(1)$ and $(2)$ and sources $(3)$ and $(4)$ .

Therefore, at point $P$ , amplitude is $0$ .

(c) When $d = λ$ , path difference between sources $(1)$ and $(2)$ ,

Path difference $= λ$

Phase difference $= \frac{2 π}{λ} \times λ = 2 π$

Similarly, path difference between the sources $(3) and (4)$ ,

Path difference $= λ$

Phase difference $= \frac{2 π}{λ} \times λ = 2 π$

Thus, constructive interference at point $P$ , therefore the amplitude at point $P$ is $4 S_{m}$ .

Important Questions on Waves-II

Learn and Prepare for any exam you want !