Exercise Solutions Available
Step-by-step answers for all textbook questions in this chapter
How to Use This Page
1. Solve first, check later — Attempt every exercise question on your own before reading the solutions below. 2. Follow the sequence — Work through sections I to VI in order; later sections build on earlier ones. 3. Understand the method — Focus on the reasoning and formula selection, not just the final answer. 4. Practise numericals — Re-work the worked examples with different values until you can set them up without hints. 5. Review mistakes — Read the "Common Mistakes to Avoid" section before every exam.
Key Concepts (Quick Recap)
Wave Basics
| Quantity | Symbol | Formula | SI Unit |
|---|---|---|---|
| Wave speed | v | v = fλ | m/s |
| Frequency | f | f = 1/T | Hz |
| Time period | T | T = 1/f | s |
| Wavelength | λ | λ = v/f = vT | m |
| Angular frequency | ω | ω = 2πf = 2π/T | rad/s |
| Wave number | k | k = 2π/λ | rad/m |
| Wave equation | y | y = A sin(kx − ωt) | m |
Types of Waves
| Type | Motion of particles | Examples | Can polarise? |
|---|---|---|---|
| Transverse | Perpendicular to propagation | Light, string waves, water surface waves | Yes |
| Longitudinal | Parallel to propagation | Sound waves, spring compression waves | No |
Speed of Sound
| Medium | Formula | Notes |
|---|---|---|
| General | v = √(E/ρ) | E = elastic modulus, ρ = density |
| Solid (longitudinal) | v = √(Y/ρ) | Y = Young's modulus |
| Solid (transverse) | v = √(η/ρ) | η = shear modulus |
| Liquid | v = √(B/ρ) | B = bulk modulus |
| Gas | v = √(γP/ρ) = √(γRT/M) | γ = Cₚ/Cᵥ |
| Newton's formula (air) | v = √(P/ρ) | Incorrect (gives ~280 m/s) |
| Laplace correction | v = √(γP/ρ) | Correct (gives ~331 m/s at 0°C) |
| Temperature dependence | v ∝ √T | v_t = v₀√(1 + t/273) ≈ v₀ + 0.61t |
Standing Waves
| Property | String (fixed both ends) | Pipe (open both ends) | Pipe (closed one end) |
|---|---|---|---|
| Fundamental | f₁ = v/(2L) | f₁ = v/(2L) | f₁ = v/(4L) |
| Overtones | fₙ = nv/(2L), n = 1,2,3… | fₙ = nv/(2L), n = 1,2,3… | fₙ = nv/(4L), n = 1,3,5… |
| Harmonics | All harmonics present | All harmonics present | Only odd harmonics |
| Wavelength | λₙ = 2L/n | λₙ = 2L/n | λₙ = 4L/n |
Doppler Effect
| Case | Formula |
|---|---|
| Observer moving toward source | f' = f(v + v₀)/v |
| Observer moving away from source | f' = f(v − v₀)/v |
| Source moving toward observer | f' = fv/(v − vₛ) |
| Source moving away from observer | f' = fv/(v + vₛ) |
| Both moving (general) | f' = f(v ± v₀)/(v ∓ vₛ) |
Beats and Intensity
| Quantity | Formula | ||
|---|---|---|---|
| Beat frequency | f_beat = \ | f₁ − f₂\ | |
| Intensity | I = P/A (power per unit area) | ||
| Intensity level | β = 10 log₁₀(I/I₀) dB | ||
| Reference intensity | I₀ = 10⁻¹² W/m² (threshold of hearing) |
I. Choose the Best Answer (MCQs)
1. The speed of sound in air at 0°C is approximately:
- (a) 280 m/s
- (b) 331 m/s
- (c) 340 m/s
- (d) 380 m/s
2. In a stationary wave, the distance between two consecutive nodes is:
- (a) λ
- (b) λ/2
- (c) λ/4
- (d) 2λ
3. The Doppler effect is observed for:
- (a) Sound waves only
- (b) Light waves only
- (c) Both sound and light waves
- (d) Neither sound nor light waves
4. A pipe closed at one end has a fundamental frequency f. If the pipe is opened at both ends, the fundamental frequency becomes:
- (a) f
- (b) 2f
- (c) f/2
- (d) 4f
5. Two sound waves of frequencies 256 Hz and 260 Hz produce beats. The beat frequency is:
- (a) 258 Hz
- (b) 4 Hz
- (c) 516 Hz
- (d) 2 Hz
| Explanation: Beat frequency = | f₁ − f₂ | = | 260 − 256 | = 4 Hz. The beat frequency is simply the difference between the two frequencies. Four loudness maxima are heard per second. |
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6. Newton's formula for the speed of sound in air gave an incorrect value because:
- (a) He used the wrong density of air
- (b) He assumed isothermal conditions instead of adiabatic
- (c) He did not account for humidity
- (d) He used the wrong value of atmospheric pressure
7. A transverse wave is described by y = 0.05 sin(2πx − 100πt). The wavelength of the wave is:
- (a) 0.05 m
- (b) 1 m
- (c) 2π m
- (d) 0.01 m
II. Fill in the Blanks
1. The wave equation relating speed, frequency, and wavelength is v = fλ. 2. In a transverse wave, particles vibrate perpendicular to the direction of wave propagation. 3. The speed of sound in air increases by approximately 0.61 m/s for every 1°C rise in temperature. 4. In a closed pipe, only odd harmonics are present in the vibration pattern.
6. Newton's formula for the speed of sound is v = √(P/ρ), which was corrected by Laplace to v = √(γP/ρ). 7. The distance between a node and the nearest antinode in a standing wave is λ/4. 8. When a sound source moves away from a stationary observer, the observed frequency decreases (Doppler effect). 9. The intensity level of sound is measured in decibels (dB). 10. Sound waves are longitudinal waves that require a material medium for propagation. 11. In an open pipe of length L, the fundamental wavelength is 2L. 12. The phenomenon of a sound persisting after the source has stopped due to multiple reflections is called reverberation.
| 5. The beat frequency equals the difference of the two frequencies: f_beat = | f₁ − f₂ | . |
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III. True or False (with corrections)
IV. Match the Following
| Column A | Column B |
|---|---|
| 1. Transverse wave | (d) Particles vibrate perpendicular to wave direction |
| 2. Longitudinal wave | (f) Particles vibrate parallel to wave direction |
| 3. Standing wave | (a) Superposition of two identical waves travelling in opposite directions |
| 4. Doppler effect | (e) Apparent change in frequency due to relative motion |
| 5. Beats | (b) Periodic loudness variation from two close frequencies |
| 6. Resonance | (c) Maximum amplitude when driving frequency equals natural frequency |
V. Answer Briefly (2-Mark Questions)
VI. Answer in Detail (5-Mark Questions)
1. Derive the expression for the speed of sound in a gas (Laplace correction) and solve numericals.
Newton assumed isothermal propagation of sound (slow compressions and rarefactions at constant temperature):
Laplace's correction: Sound propagation involves rapid compressions and rarefactions. There is no time for heat exchange, so the process is adiabatic (PVᵞ = constant).
Effect of temperature:
Worked Example 1: Calculate the speed of sound in air at 30°C. (Speed at 0°C = 331 m/s)
Worked Example 2: Find the speed of sound in helium gas at NTP. (γ = 5/3, M = 4 × 10⁻³ kg/mol, R = 8.314 J/(mol·K), T = 273 K)
2. Derive expressions for the fundamental frequency and overtones of (a) an open pipe and (b) a closed pipe. Solve numericals.
(a) Open pipe (open at both ends):
For a pipe open at both ends, both ends are antinodes.
(b) Closed pipe (closed at one end, open at the other):
For a pipe closed at one end, the closed end is a node and the open end is an antinode.
Worked Example 1: An open pipe of length 0.5 m resonates at its fundamental frequency. Find the fundamental frequency and the first two overtone frequencies. (Speed of sound = 340 m/s)
Worked Example 2: A closed pipe of length 0.85 m has a fundamental frequency of 100 Hz. Find the speed of sound and the frequency of the next overtone.
3. State and derive the Doppler effect formula for sound and solve numericals.
The Doppler effect is the apparent change in frequency of a wave when there is relative motion between the source and the observer.
Case 1: Source stationary, observer moving toward source:
Case 2: Observer stationary, source moving toward observer:
General formula (both moving):
Worked Example 1: A train whistle of frequency 600 Hz is heard by a stationary observer as the train approaches at 30 m/s. What frequency does the observer hear? (Speed of sound = 340 m/s)
Worked Example 2: An ambulance with a siren of frequency 800 Hz approaches and then passes a stationary observer at 25 m/s. Find the apparent frequencies (a) while approaching, and (b) while receding. (v = 340 m/s)
Worked Example 3: Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. Find the beat frequency. How many beats are heard in 5 seconds?
Common Mistakes to Avoid
1. Confusing transverse and longitudinal waves — In transverse waves, particles move perpendicular to wave direction (e.g., light, string vibrations). In longitudinal waves, particles move parallel to wave direction (e.g., sound in air). Sound cannot be a transverse wave in air because air cannot sustain shear forces.
2. Using v = fλ with inconsistent units — Ensure that v is in m/s, f in Hz, and λ in metres. A common error is mixing cm/s with m, or kHz with km. Always convert to SI units before substituting.
3. Mixing up open and closed pipe harmonics — Open pipe: all harmonics present (f₁, 2f₁, 3f₁, ...). Closed pipe: only odd harmonics present (f₁, 3f₁, 5f₁, ...). The fundamental of a closed pipe is v/(4L), half that of an open pipe of the same length.
4. Wrong Doppler formula sign — The Doppler formula f' = f(v ± v₀)/(v ∓ vₛ) uses upper signs for approach and lower signs for recede. A mnemonic: "approaching increases frequency" — so use the sign that makes f' > f when source and observer approach each other.
5. Forgetting Laplace correction — Newton's formula v = √(P/ρ) gives an incorrect speed of sound (~280 m/s). Always use Laplace's corrected formula v = √(γP/ρ) for adiabatic propagation. For air, γ = 1.4, which gives ~331 m/s at 0°C.
6. Confusing node/antinode positions — In a standing wave on a string fixed at both ends: both ends are nodes. In an open pipe: both ends are antinodes. In a closed pipe: closed end is a node, open end is an antinode. Getting these wrong leads to incorrect wavelength calculations.
7. Applying Doppler effect incorrectly for light — The formulas given above are for sound waves in a medium. For light, the relativistic Doppler formula must be used, and only relative velocity matters (the medium-based formulas do not apply).
| 8. Confusing beat frequency with resultant frequency — Beat frequency is | f₁ − f₂ | (how many times the loudness varies per second). The resultant sound has an average frequency of (f₁ + f₂)/2 with an amplitude modulated at the beat frequency. These are different quantities. |
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Exam Checklist
- [ ] I can write and use the wave equation v = fλ and its variations
- [ ] I can distinguish between transverse and longitudinal waves with examples
- [ ] I can derive and apply Laplace's corrected formula for the speed of sound: v = √(γP/ρ)
- [ ] I can calculate the speed of sound at different temperatures using v = v₀ + 0.61t
- [ ] I can find fundamental and overtone frequencies for open and closed pipes
- [ ] I can explain why only odd harmonics are present in a closed pipe
- [ ] I can apply the Doppler effect formula for various source/observer motion combinations
- [ ] I can calculate beat frequency and explain how beats are used for tuning
- [ ] I can define and identify nodes, antinodes, and wavelengths in standing waves
- [ ] I can solve numericals involving wave speed, frequency, wavelength, Doppler effect, and resonance