Mechanical Waves
Class 12 Physics Notes • NEB Syllabus 2083
Types and examples of mechanical waves, diagrams and formulas, 16 solved multiple choice questions, step-by-step numericals, handwritten and typed PDF notes and frequently asked questions on the speed of sound and Laplace’s correction.
1. Syllabus
Unit: Waves and Optics
Chapter 7: Mechanical Waves (4 hours)
7.1 Speed of wave motion; velocity of sound in solid and liquid
7.2 Velocity of sound in gas
7.3 Laplace’s correction
7.4 Effect of temperature, pressure and humidity on velocity of sound
These notes follow the NEB Class 12 Physics syllabus (+2 Physics, Nepal). Read the quick overview first, practise the multiple choice questions, then work through the numericals, and finish with the PDF notes for quick revision.
2. Mechanical Waves at a Glance
Mechanical waves are waves that need a material medium (solid, liquid or gas) to travel. The particles of the medium vibrate about their mean positions and pass the disturbance on, so energy moves but the medium as a whole does not. Waves that do not need a medium, such as light, are called non-mechanical waves.
Types of mechanical waves and diagram
Transverse waves: particles vibrate perpendicular to the direction of travel (waves on a string, S-waves of earthquakes). Longitudinal waves: particles vibrate parallel to the direction of travel, forming compressions and rarefactions (sound in air, P-waves). Surface waves: particles move in a combination of both, as on the surface of water.
Figure: Transverse wave. Wavelength is the distance between two successive crests; \(A\) is the amplitude.
Figure: Longitudinal wave. Wavelength is the distance between two successive compressions.
10 examples of mechanical waves
| No. | Example | Type |
|---|---|---|
| 1 | Sound waves in air | Longitudinal |
| 2 | Waves on a guitar string | Transverse |
| 3 | Ripples on the surface of water | Surface (combined) |
| 4 | Seismic P-waves of an earthquake | Longitudinal |
| 5 | Seismic S-waves of an earthquake | Transverse |
| 6 | Push-pull pulse on a slinky spring | Longitudinal |
| 7 | Side-to-side pulse on a slinky spring | Transverse |
| 8 | Ultrasound used in medical scanning | Longitudinal |
| 9 | Sonar waves in water | Longitudinal |
| 10 | Sound travelling through a railway track | Longitudinal (and transverse in solids) |
Mechanical waves vs electromagnetic waves (non-mechanical waves)
| Basis | Mechanical waves | Electromagnetic waves |
|---|---|---|
| Medium | Need a material medium; cannot travel in vacuum | Need no medium; travel through vacuum |
| Cause | Vibration of particles of the medium | Oscillating electric and magnetic fields |
| Type | Transverse or longitudinal | Always transverse |
| Speed | Depends on the medium (about 343 m/s for sound in air) | \(3\times10^8\ \mathrm{m\,s^{-1}}\) in vacuum |
| Examples | Sound, water waves, waves on a string | Light, radio waves, X-rays |
Mechanical wave formulas
| Situation | Formula |
|---|---|
| Any wave | \(v=f\lambda\) |
| Stretched string (tension \(T\), mass per unit length \(\mu\)) | \(v=\sqrt{T/\mu}\) |
| Longitudinal wave in a solid rod (Young’s modulus \(Y\)) | \(v=\sqrt{Y/\rho}\) |
| Liquid (bulk modulus \(B\)) | \(v=\sqrt{B/\rho}\) |
| Gas, Newton (isothermal) | \(v=\sqrt{P/\rho}\) |
| Gas, Laplace (adiabatic) | \(v=\sqrt{\gamma P/\rho}=\sqrt{\gamma RT/M}\) |
| Effect of temperature | \(v_2/v_1=\sqrt{T_2/T_1}\), or \(v\approx331+0.6\theta\ \mathrm{m\,s^{-1}}\) |
3. Multiple Choice Questions
Question 1. Which of the following is correct about the speed of wave motion in a medium?
- a) It depends on the frequency of the wave
- b) It depends on the wavelength of the wave
- c) It depends on the properties of the medium
- d) It is constant for all waves in any medium
Correct Answer: c) It depends on the properties of the medium
Reason: The speed of a mechanical wave depends on the inertial (density) and elastic properties of the medium, \(v=\sqrt{E/\rho}\). It does not depend on the frequency or wavelength: in a given medium, if \(f\) changes then \(\lambda\) adjusts so that \(v=f\lambda\) stays constant.
Question 2. Which of the following factors affect the velocity of sound in air?
- a) Temperature alone
- b) Humidity alone
- c) Pressure alone
- d) Both temperature and humidity
Correct Answer: d) Both temperature and humidity
Reason: For a gas
so \(v\) increases with \(\sqrt T\). Humidity adds lighter water vapour, lowering the average molar mass \(M\) and raising \(v\). Pressure alone has no effect at constant temperature, because \(P/\rho\) stays constant.
Question 3. In which medium does sound travel fastest?
- a) Air
- b) Water
- c) Iron
- d) Wood
Correct Answer: c) Iron
Reason: \(v=\sqrt{E/\rho}\). Solids such as iron have a very large elastic modulus (Young’s modulus). Although they are denser, the rise in elasticity far outweighs the rise in density, so sound travels fastest in solids like iron.
Question 4. Which of the following expressions represents the velocity of sound in a gas?
- a) \(v=\sqrt{\dfrac{\gamma RT}{M}}\)
- b) \(v=\sqrt{\dfrac{RT}{M}}\)
- c) \(v=\sqrt{\dfrac{PT}{M}}\)
- d) \(v=\sqrt{\dfrac{P}{M}}\)
Correct Answer: a) \(v=\sqrt{\dfrac{\gamma RT}{M}}\)
Reason: Newton assumed isothermal propagation, giving \(v=\sqrt{P/\rho}\). Laplace corrected this: sound propagation is adiabatic, so \(v=\sqrt{\gamma P/\rho}\). With \(PV=nRT\), \(\dfrac P\rho=\dfrac{RT}{M}\), hence
Question 5. What is the effect of rising temperature on the velocity of sound in air?
- a) It increases the velocity
- b) It decreases the velocity
- c) No effect on velocity
- d) It depends on the humidity
Correct Answer: a) It increases the velocity
Reason: From \(v=\sqrt{\gamma RT/M}\), \(v\propto\sqrt T\), so the velocity of sound increases as the temperature rises.
Question 6. What does Laplace’s correction to Newton’s formula for the speed of sound account for?
- a) Sound travels isothermally, so the bulk modulus is \(P\)
- b) Compressions and rarefactions in sound are adiabatic, so the bulk modulus is \(\gamma P\)
- c) The effect of humidity on the velocity of sound
- d) The effect of pressure on the velocity of sound
Correct Answer: b) Compressions and rarefactions in sound are adiabatic, so the bulk modulus is \(\gamma P\)
Reason: Newton assumed an isothermal process (\(B=P\)), which gives about 280 m/s in air at STP. Laplace showed that the compressions and rarefactions are so rapid that no heat is exchanged, so the process is adiabatic and \(B=\gamma P\). The speed is multiplied by \(\sqrt\gamma\), giving about 331 m/s, in agreement with experiment.
Question 7. If the temperature of air increases, how does it affect the speed of sound?
- a) The speed decreases
- b) The speed increases
- c) The speed remains constant
- d) It depends on the type of wave
Correct Answer: b) The speed increases
Reason: As in Question 5, \(v\propto\sqrt T\); a higher absolute temperature gives a higher speed.
Question 8. A sound wave travels through a liquid medium at a speed of 1500 m/s. If the same wave travels through a solid medium of density 8000 kg/m³ and Young’s modulus \(2\times10^{11}\ \mathrm{Pa}\), how does its speed compare?
- a) The speed will be higher in the solid medium
- b) The speed will be lower in the solid medium
- c) The speed will be the same in both mediums
- d) Not enough information to determine
Correct Answer: a) The speed will be higher in the solid medium
Since \(5000>1500\), the speed is higher in the solid.
Question 9. What is the effect of humidity on the velocity of sound in air?
- a) It increases the velocity
- b) It decreases the velocity
- c) It has no effect
- d) It depends on the temperature
Correct Answer: a) It increases the velocity
Reason: Water vapour (molar mass 18) is lighter than the \(\mathrm{N_2}\) (28) and \(\mathrm{O_2}\) (32) it displaces, so humid air has a lower density. Since \(v\propto\dfrac1{\sqrt\rho}\), the velocity of sound increases.
Question 10. If the frequency of a sound wave is doubled, what happens to the wavelength in a given medium?
- a) The wavelength is doubled
- b) The wavelength is halved
- c) The wavelength remains the same
- d) The wavelength becomes zero
Correct Answer: b) The wavelength is halved
Reason: In a given medium \(v\) is constant, so \(f_1\lambda_1=f_2\lambda_2\). With \(f_2=2f_1\):
Question 11. A student measures the speed of sound in air at \(20^\circ\mathrm{C}\) and finds it to be 343 m/s. What will the speed of sound be at \(0^\circ\mathrm{C}\)?
- a) Approximately 331 m/s
- b) Approximately 320 m/s
- c) Approximately 343 m/s
- d) Approximately 300 m/s
Correct Answer: a) Approximately 331 m/s
Reason: \(v\propto\sqrt T\) with \(T_1=293.15\ \mathrm{K}\) and \(T_2=273.15\ \mathrm{K}\):
Question 12. The frequency of a sound wave is doubled while keeping the medium constant. What happens to the speed and wavelength?
- a) Speed remains the same, wavelength is halved
- b) Speed is doubled, wavelength is the same
- c) Both speed and wavelength are doubled
- d) Speed remains the same, wavelength remains the same
Correct Answer: a) Speed remains the same, wavelength is halved
Reason: The speed is fixed by the medium, so it stays constant. From \(v=f\lambda\), doubling \(f\) halves \(\lambda\).
Question 13. A sonar signal is emitted from a submarine and returns after 4 seconds. If the speed of sound in seawater is 1500 m/s, how far away is the target?
- a) 3000 m
- b) 2500 m
- c) 2000 m
- d) 1500 m
Correct Answer: a) 3000 m
Reason: The 4 s is the round-trip time, so the total distance is \(vt=1500\times4=6000\ \mathrm{m}\) and the target is half of that:
Question 14. In a Kundt’s tube experiment, a metallic rod vibrates in resonance and the measured wavelength in air is 50 cm. If the speed of sound in air is 340 m/s and in the metal rod is 5100 m/s, find the frequency of vibration.
- a) 680 Hz
- b) 1020 Hz
- c) 1200 Hz
- d) 5100 Hz
Correct Answer: a) 680 Hz
Reason: At resonance the frequency is the same in the rod and in the air column. Using the data for air:
Question 15. The velocity of sound in hydrogen at \(0^\circ\mathrm{C}\) is 1250 m/s. What will be the velocity of sound in oxygen at \(0^\circ\mathrm{C}\)? (Molar masses: hydrogen 2 g/mol, oxygen 32 g/mol)
- a) 312.5 m/s
- b) 280.5 m/s
- c) 331.0 m/s
- d) 250.5 m/s
Correct Answer: a) 312.5 m/s
Reason: Both gases are diatomic (same \(\gamma\)) and at the same temperature, so \(v\propto\dfrac1{\sqrt M}\):
Question 16. A person shouts near a 150 m high vertical cliff and hears the echo after 0.88 seconds. What is the approximate speed of sound in air?
- a) 320 m/s
- b) 340 m/s
- c) 360 m/s
- d) 380 m/s
Correct Answer: b) 340 m/s
Reason: The sound travels to the cliff and back, a distance \(2\times150=300\ \mathrm{m}\):
4. Numerical and Subjective Questions
a) Discuss how the velocity of sound differs in solids, liquids and gases. Why does sound travel faster in solids than in gases?
The speed of sound in any medium depends on its elasticity (an elastic modulus \(E\)) and its inertia (density \(\rho\)):
Gases: the relevant modulus is the adiabatic bulk modulus \(B=\gamma P\). Gases are highly compressible (low modulus) and have low density; the low elasticity makes sound slow, about 343 m/s in air.
Liquids: the relevant modulus is the bulk modulus \(B\). Liquids are far less compressible than gases, so \(B\) rises much more than the density does, and sound travels faster, about 1500 m/s in water.
Solids: for a longitudinal wave along a rod the relevant modulus is Young’s modulus \(Y\). Solids are very rigid, with very high moduli, and this outweighs their higher density.
Conclusion: sound travels faster in solids than in gases because the ratio \(E/\rho\) is much larger in solids. The strong intermolecular bonds of a solid resist deformation and pass the disturbance on quickly.
b) The bulk modulus of a liquid is \(2.2\times10^{9}\ \mathrm{N\,m^{-2}}\) and its density is \(1000\ \mathrm{kg\,m^{-3}}\). Calculate the velocity of sound in the liquid.
a) Derive the formula for the velocity of sound in a gas using Newton’s assumption. Then explain why Laplace’s correction was necessary.
Newton assumed the compressions and rarefactions are slow enough for heat to flow, so the temperature stays constant (an isothermal process). By Boyle’s law \(PV=\text{constant}\); differentiating:
The right side is the isothermal bulk modulus, so \(B_{\text{iso}}=P\). Putting this in \(v=\sqrt{B/\rho}\):
Why Laplace’s correction was needed: at STP, \(P=1.013\times10^5\ \mathrm{Pa}\) and \(\rho=1.293\ \mathrm{kg\,m^{-3}}\), so
The measured speed is about 331 m/s, so Newton’s value is roughly 16% too low. Laplace realised that the compressions and rarefactions are so rapid that no heat can flow, so the process is adiabatic, not isothermal.
b) A gas has a density of \(1.5\ \mathrm{kg\,m^{-3}}\) and a bulk modulus of \(2.25\times10^{5}\ \mathrm{N\,m^{-2}}\). Find the speed of sound in the gas.
a) Describe Laplace’s correction to Newton’s formula for the velocity of sound in gases. How does considering adiabatic changes instead of isothermal ones correct the result?
Laplace pointed out that sound frequencies are high, so compressions (heating) and rarefactions (cooling) follow each other very rapidly, and gases are poor conductors of heat. Heat therefore has no time to flow between the compressed and rarefied regions, so the process is adiabatic.
For an adiabatic process \(PV^{\gamma}=\text{constant}\), with \(\gamma=C_p/C_v\). Differentiating:
The right side is the adiabatic bulk modulus, \(B_{\text{ad}}=\gamma P\). Hence Laplace’s formula:
Since \(\gamma>1\) (\(\gamma\approx1.4\) for air), Newton’s value is multiplied by \(\sqrt\gamma\): \(280\times\sqrt{1.4}\approx331\ \mathrm{m\,s^{-1}}\), which matches experiment.
b) A gas has an adiabatic index \(\gamma=1.4\) and a molar mass of \(28.97\ \mathrm{g\,mol^{-1}}\). Calculate the speed of sound in this gas at 300 K. (\(R=8.314\ \mathrm{J\,mol^{-1}K^{-1}}\))
a) Explain how temperature affects the velocity of sound in gases. Write the expression for velocity in terms of temperature and explain the physical meaning of each term.
When the temperature of a gas rises, its molecules move faster and collide more frequently, so they pass on the compressional disturbance more quickly. The velocity of sound increases with temperature.
From \(v=\sqrt{\gamma P/\rho}\) and \(\dfrac P\rho=\dfrac{RT}{M}\):
\(v\): the velocity of sound in the gas.
\(\gamma\): the adiabatic index \(C_p/C_v\), a thermodynamic property of the gas.
\(R\): the universal gas constant, which links the energy scale to the temperature scale.
\(T\): the absolute temperature in kelvin; \(v\propto\sqrt T\).
\(M\): the molar mass; heavier gases carry sound more slowly at the same temperature.
b) Calculate the speed of sound in air at \(27^\circ\mathrm{C}\). (The velocity of sound in air at \(0^\circ\mathrm{C}\) is 331 m/s.)
a) Explain the effect of pressure on the velocity of sound in a gas. Why is the velocity of sound independent of pressure under ideal conditions?
At constant temperature a change of pressure has no effect on the velocity of sound in a gas. By Laplace’s formula \(v=\sqrt{\gamma P/\rho}\). By Boyle’s law, for a fixed mass of gas at constant temperature \(P\propto\dfrac1V\), and the density \(\rho\propto\dfrac1V\), so \(P\propto\rho\). The ratio \(\dfrac P\rho\) therefore stays constant: if the pressure increases, the density increases in the same proportion, and \(v\) is unchanged.
b) Calculate the velocity of sound in a metal rod whose Young’s modulus is \(2.0\times10^{11}\ \mathrm{N\,m^{-2}}\) and density is \(7800\ \mathrm{kg\,m^{-3}}\).
c) The pressure of a gas is doubled while keeping the temperature constant. What will be the effect on the speed of sound? Justify your answer mathematically.
The speed of sound remains unchanged.
From the ideal gas equation, \(\dfrac P\rho=\dfrac{RT}{M}\), so Laplace’s formula becomes
This depends only on \(T\), \(M\) and \(\gamma\). With \(T\) constant, doubling \(P\) doubles \(\rho\) as well, so \(\dfrac P\rho\) and hence \(v\) stay the same.
a) How does humidity affect the speed of sound in air? Why does moist air allow sound to travel faster than dry air?
An increase in humidity increases the speed of sound in air. Since \(v\propto\dfrac1{\sqrt\rho}\), a lower density gives a higher speed. Dry air is mainly nitrogen (molar mass about 28 g/mol) and oxygen (about 32 g/mol), while water vapour has a lower molar mass of about 18 g/mol. At the same pressure and temperature, equal volumes hold equal numbers of molecules (Avogadro’s law), so water vapour displaces heavier molecules. The average molar mass, and hence the density, of moist air is therefore lower than that of dry air, and sound travels faster in it.
b) The velocity of sound in dry air at \(0^\circ\mathrm{C}\) is 331 m/s. Calculate the velocity in air at \(30^\circ\mathrm{C}\) with 80% humidity. Assume the velocity increases by 0.1% for every 1% increase in relative humidity.
Step 1: velocity in dry air at \(30^\circ\mathrm{C}\)
Step 2: humidity correction (using the rule given in the question)
a) How does molar mass affect the speed of sound in air?
From \(v=\sqrt{\gamma RT/M}\), at constant temperature the speed of sound is inversely proportional to the square root of the molar mass, \(v\propto\dfrac1{\sqrt M}\). Sound therefore travels faster in lighter gases and slower in heavier gases, because lighter molecules have a higher root-mean-square speed at a given temperature and pass on the disturbance more quickly.
b) If the speed of sound in helium is 1005 m/s and in air is 343 m/s, compare the molar masses of helium and air using the velocity formula.
Take the same temperature for both gases. Helium is monatomic, so \(\gamma_{He}\approx1.66\); air is mainly diatomic, so \(\gamma_{air}\approx1.4\). The ratio of speeds is
Taking \(M_{air}\approx28.97\ \mathrm{g\,mol^{-1}}\):
So air is about 7.2 times heavier per mole than helium.
a) How do the bulk modulus and density of a medium affect the speed of sound? Explain this for different states of matter.
For a fluid, \(v=\sqrt{B/\rho}\), where \(B\) is the bulk modulus (elasticity) and \(\rho\) the density (inertia).
Bulk modulus: \(v\propto\sqrt B\). A larger \(B\) means a stiffer medium with stronger restoring forces, so the disturbance travels faster.
Density: \(v\propto\dfrac1{\sqrt\rho}\). A larger \(\rho\) means more massive particles that are harder to accelerate, so the disturbance travels slower.
Gases: very low density but extremely low bulk modulus; the low elasticity dominates, so sound is slowest.
Liquids: denser than gases but vastly harder to compress; the large rise in \(B\) outweighs the rise in \(\rho\), so sound is faster than in gases.
Solids: the highest densities but also very high elastic moduli (\(Y\), or \(B+\tfrac43S\)); the elasticity dominates, so sound is fastest.
b) At what temperature will the speed of sound in air be 350 m/s? Use \(v=331+0.6T\).
a) A student says that an increase in atmospheric pressure increases the speed of sound in air. Do you agree or disagree? Justify with reasoning and equations.
I disagree. By Laplace’s formula \(v=\sqrt{\gamma P/\rho}\). At constant temperature, Boyle’s law gives \(P\propto\dfrac1V\) and the density \(\rho=\dfrac mV\), so \(\rho\propto P\): raising the pressure raises the density by the same factor. The ratio \(\dfrac P\rho=\dfrac{RT}{M}\) is unchanged, and \(\gamma\) is constant, so
is unaffected by a change in pressure alone.
b) A tuning fork produces a sound wave with a frequency of 512 Hz in air. If the speed of sound is 344 m/s, calculate its wavelength.
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6. Frequently Asked Questions
What are mechanical waves?
What are the types of mechanical waves?
What are 10 examples of mechanical waves?
What does a mechanical wave diagram show?
What are non-mechanical waves?
What is the difference between mechanical waves and electromagnetic waves?
What is the mechanical wave formula?
What is the speed of sound in solids, liquids and gases?
What is Laplace’s correction to Newton’s formula for the speed of sound?
How does temperature affect the speed of sound?
How does pressure affect the speed of sound?
How does humidity affect the speed of sound?
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