Question

A point sound source is situated in a medium of the bulk modulus $1.6 \times 10^{5} N m^{- 2}$ . An observer standing at a distance $10 m$ from the source writes down the equation for the wave as $y = A \sin (15 π x - 6000 π t)$ . Here, $y$ and $x$ are in metres and $t$ is in second. The maximum pressure amplitude received to the observer's ear is $(24 π) Pa$ , then find.
(a) the density of the medium,
(b) the displacement amplitude $A$ of the waves received by the observer and
(c) the power of the sound source.

Accepted Answer

The equation of the given travelling wave is $y = A \sin (15 π x - 6000 π t)$ .

Comparing the general wave equation of the travelling wave $y = A \sin (k x - ω t)$
where the angular wave number $k (= \frac{2 π}{λ} = 15 π)$ , so the wavelength $λ = \frac{2}{15} m$ .

The angular frequency is $ω (= 2 π f = 6000 π)$ , so the frequency $f = 3000 Hz$ .

So, the wave velocity, $v = f λ = 3000 \times \frac{2}{15} = 400 m s^{- 1}$ .

$(a)$ The bulk modulus $(B = 1.6 \times 10^{5} N m^{- 2})$ is related to the density of the medium $(ρ)$ and wave velocity $(v)$ is related as

$B = ρ ν^{2}$

$\Rightarrow ρ = \frac{B}{v^{2}}$

$= \frac{1.6 \times 10^{5}}{{(400)}^{2}}$

$= 1 kg m^{- 3}$ .

$(b)$ The pressure amplitude ${(∆ p)}_{m} = 24 π$ (given), is related to displacement amplitude $(A)$ , the bulk modulus $(B)$ and the angular wave number $(k)$ as

${(∆ p)}_{m} = B A k$

$\Rightarrow A = \frac{{(∆ p)}_{m}}{B k}$

$= \frac{24 π}{1.6 \times 10^{5} \times 15 π}$

$= 10^{- 5} m$ .

$(c)$ The average intensity $(I)$ is given by

$I = \frac{1}{2} ρ v ω^{2} A^{2} = \frac{Power}{surface area of the wave front}$

Due to the point source, the spherical wave fronts are produced.

Then at a distance $r = 10 m$ , the intensity will be

$I = \frac{1}{2} ρ v ω^{2} A^{2} = \frac{Power}{4 π r^{2}}$

$\Rightarrow power = \frac{1}{2} ρ ν ω^{2} A^{2} \times 4 π r^{2}$

$= \frac{1}{2} \times 1 \times 400 \times {(2 π \times 3000)}^{2} \times {(10^{- 5})}^{2} \times 4 π \times {(10)}^{2}$

$= 288 π^{3} W$ .

B M Sharma Solutions for Chapter: Sound Waves and Doppler Effect, Exercise 1: ILLUSTRATIONS

B M Sharma Physics Solutions for Exercise - B M Sharma Solutions for Chapter: Sound Waves and Doppler Effect, Exercise 1: ILLUSTRATIONS

Questions from B M Sharma Solutions for Chapter: Sound Waves and Doppler Effect, Exercise 1: ILLUSTRATIONS with Hints & Solutions

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