Speed of sound waves - Equations, Practice problems, FAQs

What does a crying baby, a flute, and a teacher have in common? They all produce sound. For instance, when a teacher teaches, her vocal cords vibrate, thereby producing sound. The sound travels in the form of waves and reaches the listener's ears. For this to be possible, the sound wave has to transfer momentum and energy from one particle of the medium to another particle. Unlike non-mechanical waves, sound waves need a material medium to travel. However, the speed of the sound is not the same in all media. For instance, sound travels faster in solids–this is the reason why a person keeping their ear in a steel pipe hears sound faster compared to listening to sound in a gas. Furthermore,factors like humidity and temperature play a major role in influencing the speed of sound. In this article, we will explore the speed of sound waves in detail.

Table of contents

What is a sound wave?
Newton’s formula
Laplace’s correction
Effect of different factors on the speed of sound
Practice problems
FAQs

What is a sound wave?

A sound wave is a longitudinal wave which is produced due to vibration of molecules in a direction parallel to the direction of propagation of the wave. A compression is a region in which the molecules are closer to each other, the density of the air is high in this region, and the pressure is high as well. On the other hand, rarefaction refers to the region where the molecules of the wave are farther apart, the density and pressure are very low in this region.

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Based on their frequency range, they are further classified into

(i)Infrasonic waves-The frequency of these waves are less than 20 Hz

(ii)Audible waves-Their frequency lies between 20 Hz-20 kHz.

(iii)Ultrasonic waves-Their frequency is greater than 20 kHz.

Newton’s formula

Sound waves travel with different speeds in different media. Newton assumed that while sound travels in the form of compression, the temperature rises. The reverse happens in rarefaction due to expansion. Assuming isothermal behavior,(temperature of the surrounding gas medium-constant), applying the ideal gas equation,

P V= R T

P- Pressure R-gas constant

V-Volume T- temperature

- no.of moles

P V=constant

Differentiating, we get PdV+VdP=0

$P d V = - V d P \Rightarrow P = - \frac{d P}{\frac{d V}{V}}$ , where $B = - \frac{d P}{\frac{d V}{V}}$ is called the Bulk modulus.

The bulk modulus is a measure of how compressible a medium is. Higher the bulk modulus, the more difficult it is to compress the medium.

The velocity of sound in medium, $v = \sqrt{\frac{B}{ρ}} = \sqrt{\frac{E l a s t i c p r o p e r t y}{I n e r t i a l p r o p e r t y}}$

- density of the medium.

The speed of longitudinal waves in a solid bar is given by,

$v = \sqrt{\frac{Y}{ρ}}$ is the Young’s modulus.

The speed of longitudinal waves in a gaseous medium, is given by,

$v = \sqrt{\frac{P}{ρ}}$

P- pressure of the gas,

- density of the gas

At STP P=1.013 105 Pa (atmospheric pressure), =1.3 kg/m3

$v = \sqrt{\frac{1.013 \times 10^{5}}{1.3}} \approx 279 m / s$

Newton’s above theoretical value is not quite in agreement with the experimentally determined value of sound.

Laplace’s correction

Laplace assumed adiabatic behavior of air. He assumed that the compressions and rarefactions occur rapidly, which is quite contrary to what happens in an isothermal process. For an adiabatic process,

P V= constant

Differentiating, we get

$P V^{γ - 1} γ d V + V^{γ} d P = 0$

$P γ = - \frac{V^{γ} d P}{V^{γ - 1} d V}, P γ = - \frac{d P}{\frac{d V}{V}}$

P=B

Hence, speed of sound, $v = \sqrt{\frac{γ P}{ρ}}$

=1.4(for air) is called the adiabatic constant.

Substituting the values,we get v=332 m/s as the speed of sound–which is quite close to the actual value.

Note:

Speed of sound is highest in solids and lowest in gases. This is because solids have high inertia and elasticity–making wave propagation from one atom to another atom easier.

Effect of different factors on the speed of sound

Factors like temperature and humidity affect the speed of sound in a medium.

$P V = μ R T; P V = \frac{m}{M} R T$

m- given mass, M- molar mass

$P = \frac{m}{M V} R T$

$P = \frac{ρ R T}{M}$

$\frac{P}{ρ} = \frac{R T}{M} = c o n s t a n t$

$v = \sqrt{\frac{γ R T}{M}} \Rightarrow v \propto \sqrt{T}$

Hence, the speed of sound is directly proportional to the square root of temperature. This is why sound travels faster during summers compared to winters.

Let vt indicate the velocity of sound at a temperature T and v0 be its velocity at 00C. Then,

$\frac{v_{t}}{v_{0}} = \sqrt{\frac{273 + T}{273}} = {(1 + \frac{T}{273})}^{1 / 2} = 1 + \frac{1 \times T}{2 (273)} = 1 + \frac{T}{546}$

(By binomial expansion)

$v_{t} = v_{0} (1 + \frac{T}{546})$

Effect of humidity

The density of humid air is lesser than the density of dry air. From $v = \sqrt{\frac{γ P}{ρ}}$ , we have $v \propto \frac{1}{\sqrt{ρ}} \Rightarrow$ As humidity increases, the speed of sound increases.

Practice problems

Q. Calculate at what temperature, will the speed of sound in hydrogen, be the same as that of its speed in oxygen at 1000C . Given that the molar masses of oxygen and hydrogen are in the ratio 16:1

We have, $v = \sqrt{\frac{γ R T}{M}}; \sqrt{\frac{γ_{H_{2}} R T_{H_{2}}}{M_{H_{2}}}} = \sqrt{\frac{γ_{O_{2}} R T_{O_{2}}}{M_{O_{2}}}}$

Being diatomic gases, H2=O2

$T_{H_{2}} = (\frac{M_{H_{2}}}{M_{O_{2}}}) (T_{o_{2}}) = (\frac{1}{16}) (100 + 273) = 23.31 K \approx - 249 . 7^{0} C$

Q. Calculate the speed of sound in hydrogen gas at 270 C. Given, H2=1.4 and universal gas constant R=8.31 J/mol/K

$T = 27 + 273 = 300 \ K, {\ g a m m a = 1.4}_$

$v = \sqrt{\frac{γ R T}{M}} = \sqrt{\frac{1.4 \times 8.31 \times 300}{2 \times 10^{- 3}}} = 1321 m / s$

Q. The density of air at NTP is around 1.29 kg/m3. If air is said to be diatomic, what will be the velocity of sound in air at 1270 C?

Given, T2=127+273=400 K. and

T1=27+273=300 K.

Applying Laplace correction,

$v = \sqrt{\frac{γ P}{ρ}} = \sqrt{\frac{1.4 \times 1.013 \times 10^{5}}{1.29}} = 331.6 m s^{- 1}$

Since,

Let v2 be the velocity at temperature T2 and v1 be the velocity at temperature T1.

$\frac{v_{2}}{v_{1}} = \sqrt{\frac{T_{2}}{T_{1}}}$

$v_{2} = v_{1} \sqrt{\frac{T_{2}}{T_{1}}} = 331.6 \sqrt{\frac{400}{300}} = 382.8 m / s .$

Q. In a gaseous medium, find out at what temperature will the velocity of sound will be double of its velocity at 270 C ?

Let v2 be the velocity at temperature T2 and v1 be the velocity at temperature T1. Now T1=270 C=27+273=300 K

T2=?

$\frac{v_{2}}{v_{1}} = \sqrt{\frac{T_{2}}{T_{1}}}; v_{2} = 2 v_{1}$

${2 v}_{1} = v_{1} \sqrt{\frac{T_{2}}{T_{1}}}$

$\frac{T_{2}}{T_{1}} = 4; T_{2} = 4 \times 300 = 1200 K;$

T2=9270C.

FAQs

Q. Which factors of the medium affect the speed of sound?
A. The factors of a medium which affect the speed of sound are: elasticity,inertia and density. Sound travels fastest through solids—since they are more elastic and the molecules are closely packed. As a result, transmission of momentum and energy of the sound waves is easier in the case of solids.

Q. What environmental factors affect the speed of sound?
A. The speed of sound is affected by pressure, humidity and temperature. If the temperature increases, the velocity of sound increases. In hot air, the speed of sound is higher, since it is less dense.

Q. Does speed of sound depend upon pressure?
A. No, the speed of sound is independent of pressure. As pressure is increased, the density also increases, maintaining the ratio of pressure to density constant. Hence, the speed of sound does not change with pressure.

Q. Is the speed of sound constant?
A. The speed of sound is constant while traveling in one medium. It is the product of wavelength and frequency of the wave. Since both do not change while traveling the medium, the speed does not vary. On the other hand, while changing mediums, the wavelength changes, hence the speed changes.