Waves Class 11 Physics Notes

• Transverse waves: particles vibrate perpendicular to the direction of propagation (e.g. waves on a string, light). Can travel through solids and the surface of liquids.
• Longitudinal waves: particles vibrate parallel to the direction of propagation, forming compressions and rarefactions (e.g. sound). Can travel through solids, liquids and gases.
• Mechanical waves need a medium; electromagnetic waves (light, radio) do not.
• Categorised by wavelength: water ripples (tiny λ), sound (~15 m), light (~10⁻⁷ m).

MCQ Questions

1 Mark Questions

1. 1
   Define a wave. Distinguish between mechanical and electromagnetic waves.
2. 2
   Give one example each of a transverse and a longitudinal wave.
3. 3
   In what direction do particles vibrate in a longitudinal wave?

2 Mark Questions

1. 1
   Distinguish between transverse and longitudinal waves with respect to particle displacement.
2. 2
   Explain why sound cannot travel in vacuum but light can.
3. 3
   Why can transverse waves travel in solids but generally not in gases?

3 Mark Questions

1. 1
   A wave travels the length of a 20 m string in 5 s. If consecutive crests are 0.4 m apart, find the frequency and period of the wave.
2. 2
   Describe how a particle at the surface of water vibrates as a ripple passes, and how that differs from the motion of the ripple itself.

All particles of a wave oscillate with the same frequency and amplitude but with a continuously varying phase. The frequency is fixed by the source; the speed is fixed by the medium; the wavelength adjusts so that v = fλ. The unit of frequency is the hertz (Hz = s⁻¹).

• Tuning a radio dial selects the frequency of the transmitted signal.
• A 100 Hz sound in air at 20 °C (v ≈ 343 m/s) has wavelength about 3.43 m.
• Speed does not change with frequency of the wave in a given medium (no dispersion for sound in air).

MCQ Questions

1 Mark Questions

1. 1
   Define wavelength and frequency of a wave.
2. 2
   Write the relation connecting wave speed, frequency and wavelength.
3. 3
   Define the hertz.

2 Mark Questions

1. 1
   Derive the relation v = fλ starting from the definition of wave speed.
2. 2
   A tuning fork of frequency 512 Hz produces a wave in air with speed 345 m/s. Find the wavelength.
3. 3
   Why is the frequency of a wave fixed by its source and not by the medium?

3 Mark Questions

1. 1
   A source emits a wave of frequency 500 Hz. If the wave speed in the medium is 1500 m/s, find the wavelength and the time period, and state how far the wavefront travels in 2 s.
2. 2
   Explain the meaning of amplitude, frequency, wavelength and speed of a wave, and give their SI units.

Sound travels through a medium as a longitudinal wave, with a speed set by the elastic and inertial properties of the medium: the faster the medium returns to equilibrium and the lighter it is, the faster sound moves.

• Speed order: solids > liquids > gases (bulk modulus divided by density).
• In a gas, v ∝ √T — the speed rises with temperature (≈ 0.6 m/s per degree Celsius).
• At constant temperature, speed is independent of pressure of the gas.
• Moist air is denser-lighter: humidity increases the speed slightly because water-vapour molecules are light.

MCQ Questions

1 Mark Questions

1. 1
   State Newton's formula for the speed of sound in a gas.
2. 2
   What correction did Laplace make and why?
3. 3
   In which medium is the speed of sound maximum — solid, liquid or gas?

2 Mark Questions

1. 1
   Explain the Laplace correction and derive v = √(γP/ρ).
2. 2
   The speed of sound in air at 0 °C is 331 m/s. Estimate its speed at 20 °C.
3. 3
   Why does the speed of sound increase with humidity?

3 Mark Questions

1. 1
   Derive an expression for the speed of sound in a gas using Newton's formula and apply the Laplace correction to obtain the standard result.
2. 2
   Compare the speed of sound in solids, liquids and gases, explaining the physical reasons for the order.
3. 3
   Show that the speed of sound in a gas is proportional to the square root of its absolute temperature.

The sign between kx and ωt fixes the direction: (kx − ωt) is a wave moving in the +x direction, (kx + ωt) in the −x direction. The phase (kx − ωt) is what travels; its constancy, kx − ωt = constant, gives dx/dt = ω/k = v.

• Phase difference Δφ and path difference Δx are related by Δφ = (2π/λ)Δx.
• A path difference of λ gives a phase difference of 2π (one full cycle).
• Phase and time difference: Δφ = ωΔt.

MCQ Questions

1 Mark Questions

1. 1
   Define a progressive wave.
2. 2
   State the superposition principle for waves.
3. 3
   What is the wave number? Write its relation with wavelength.

2 Mark Questions

1. 1
   Write the general equation of a one-dimensional progressive wave and identify each symbol.
2. 2
   Show that the wave speed v = ω/k follows from the constancy of the phase.
3. 3
   Derive the relation between phase difference and path difference.

3 Mark Questions

1. 1
   A progressive wave is described by y = 0.05 sin π(200t − 2x) m. Find its (a) frequency, (b) wavelength and (c) speed of propagation.
2. 2
   State and explain the principle of superposition of waves, and give three phenomena that it explains.

When two identical harmonic waves travel in opposite directions along a string, their superposition gives a stationary (standing) wave. The pattern does not travel: energy is stored, not transported, and the medium divides into nodes (zero displacement) and antinodes (maximum displacement).

• Nodes occur where cos(kx) = 0; antinodes where cos(kx) = ±1.
• Consecutive nodes (or antinodes) are λ/2 apart.
• The standing wave does not transfer energy along the string.
• A string fixed at both ends forms a standing wave with nodes at both ends.

MCQ Questions

1 Mark Questions

1. 1
   Define a stationary or standing wave.
2. 2
   What is a node? What is an antinode?
3. 3
   What phase change does a wave suffer on reflection at a fixed boundary?

2 Mark Questions

1. 1
   Show that consecutive nodes (or antinodes) of a stationary wave are λ/2 apart.
2. 2
   Obtain the equation of a standing wave from two progressive waves of equal amplitude and frequency travelling in opposite directions.
3. 3
   Explain why a stationary wave does not transport energy along the medium.

3 Mark Questions

1. 1
   Describe the reflection of a travelling wave at a rigid boundary and at a free boundary, stating the phase change in each case with the reasoning.
2. 2
   Two waves y₁ = A sin(kx − ωt) and y₂ = A sin(kx + ωt) superpose on a string. Obtain the resultant wave and identify the positions of nodes and antinodes.

A stretched string fixed at both ends supports standing waves only at certain frequencies, called normal modes. The ends are nodes, so an integral number of half wavelengths must fit into the length: L = nλₙ/2, giving fₙ = nv/2L for n = 1, 2, 3, …

• Fundamental (first harmonic): f₁ = v/2L.
• All harmonics are present: f₂ = 2f₁, f₃ = 3f₁, … — the string can vibrate in any of these modes simultaneously.
• The wave speed on a string is v = √(T/μ), where T is the tension and μ the mass per unit length.

MCQ Questions

1 Mark Questions

1. 1
   Write the expression for the fundamental frequency of a stretched string.
2. 2
   State the frequency formula for an open organ pipe and a closed organ pipe.
3. 3
   Why does a closed organ pipe produce only odd harmonics?

2 Mark Questions

1. 1
   Derive the expression for the frequencies of the normal modes of a string fixed at both ends.
2. 2
   A string of length 1 m has a fundamental frequency of 110 Hz. Find its wavelength and the speed of waves on the string.
3. 3
   Show that the fundamental frequency of a closed pipe is v/4L and that only odd harmonics are possible.

3 Mark Questions

1. 1
   Compare the harmonics produced by an open organ pipe and a closed organ pipe of the same length.
2. 2
   A stretched wire of length 0.90 m and mass 3.6 × 10⁻³ kg is under a tension of 90 N. Calculate its fundamental frequency and the frequencies of its first three harmonics.
3. 3
   A closed organ pipe and an open organ pipe of the same length produce their fundamentals. Find the ratio of their frequencies.

Physically, the two waves fall alternately in phase (constructive — loud, sound reinforced) and out of phase (destructive — faint). The resultant amplitude varies slowly at the beat frequency while the wave oscillates rapidly at the average frequency. Beats are heard clearly only when the two frequencies are close (difference ≲ 10 Hz).

• Beats are a demonstration of the superposition principle.
• They are used to tune musical instruments: adjust a string until the beeps against a standard tuning fork vanish.
• A slightly loaded tuning fork shifts its frequency; a fork of known frequency can be calibrated by counting beats with another.

MCQ Questions

1 Mark Questions

1. 1
   Define beats.
2. 2
   Write the formula for the beat frequency when two waves of frequencies f₁ and f₂ superpose.
3. 3
   Why are beats heard only when the two frequencies are close together?

2 Mark Questions

1. 1
   Explain how beats arise from the principle of superposition.
2. 2
   Two tuning forks A (300 Hz) and B produce 4 beats per second. What are the possible frequencies of B?
3. 3
   How are beats used to tune musical instruments?

3 Mark Questions

1. 1
   Derive an expression for the resultant intensity of two waves of slightly different frequencies and show that the intensity oscillates at the beat frequency.
2. 2
   A fork of unknown frequency produces 5 beats/s with a 256 Hz fork. When the unknown fork is slightly loaded with wax, the beats drop to 3 beats/s. Find the unknown frequency and justify your answer.

• Observer moving towards a stationary source: f′ = f(v + v₀)/v.
• Source moving towards a stationary observer: f′ = fv/(v − vₛ).
• Both moving towards each other: f′ = f(v + v₀)/(v − vₛ).
• No shift when the relative velocity is zero, or when the motion is perpendicular to the line of sight (no radial component).

Applications: radar and speed guns, Doppler echocardiography, tracking weather and storms, and measuring the recession of stars and galaxies in astronomy.

MCQ Questions

1 Mark Questions

1. 1
   State the Doppler effect for sound.
2. 2
   Write the expression for apparent frequency when the observer moves towards a stationary source.
3. 3
   What happens to the apparent frequency as a source recedes from a stationary observer?

2 Mark Questions

1. 1
   Derive the expression f′ = fv/(v − vₛ) for a source moving towards a stationary observer.
2. 2
   A whistle of frequency 400 Hz is sounded by a train moving at 30 m/s. For a stationary observer towards whom the train moves, find the apparent frequency. (Speed of sound = 330 m/s.)
3. 3
   Give two applications of the Doppler effect in technology.

3 Mark Questions

1. 1
   Derive a general expression for the apparent frequency of sound when both the source and the observer are moving, and discuss the sign conventions.
2. 2
   A source of sound of frequency 500 Hz moves towards a stationary observer at 20 m/s while the observer walks towards the source at 5 m/s. Find the apparent frequency heard. (Speed of sound = 340 m/s.)

Exercises (NCERT Q14.1–Q14.19)

• 14.1) A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?
• 14.2) A stone dropped from the top of a tower of height 300 m splashes into the water of a pond near the base of the tower. When is the splash heard at the top given that the speed of sound in air is 340 m s⁻¹? (g = 9.8 m s⁻²)
• 14.3) A steel wire has a length of 12.0 m and a mass of 2.10 kg. What should be the tension in the wire so that speed of a transverse wave on the wire equals the speed of sound in dry air at 20 °C = 343 m s⁻¹?
• 14.4) Use the formula v = √(γP/ρ) to explain why the speed of sound in air (a) is independent of pressure, (b) increases with temperature, (c) increases with humidity.
• 14.5) You have learnt that a travelling wave in one dimension is represented by a function y = f(x, t) where x and t must appear in the combination x − vt or x + vt, i.e. y = f(x ∓ vt). Is the converse true? Examine if the following functions for y can possibly represent a travelling wave: (a) (x − vt)² (b) log [(x + vt)/x0] (c) 1/(x + vt)
• 14.6) A bat emits ultrasonic sound of frequency 1000 kHz in air. If the sound meets a water surface, what is the wavelength of (a) the reflected sound, (b) the transmitted sound? Speed of sound in air is 340 m s⁻¹ and in water 1486 m s⁻¹.
• 14.7) A hospital uses an ultrasonic scanner to locate tumours in a tissue. What is the wavelength of sound in the tissue in which the speed of sound is 1.7 km s⁻¹? The operating frequency of the scanner is 4.2 MHz.
• 14.8) A transverse harmonic wave on a string is described by y(x, t) = 3.0 sin (36t + 0.018x + π/4) where x and y are in cm and t in s. The positive direction of x is from left to right. (a) Is this a travelling wave or a stationary wave? If it is travelling, what are the speed and direction of its propagation? (b) What are its amplitude and frequency? (c) What is the initial phase at the origin? (d) What is the least distance between two successive crests in the wave?
• 14.9) For the wave described in Exercise 14.8, plot the displacement (y) versus (t) graphs for x = 0, 2 and 4 cm. What are the shapes of these graphs? In which aspects does the oscillatory motion in travelling wave differ from one point to another: amplitude, frequency or phase?
• 14.10) For the travelling harmonic wave y(x, t) = 2.0 cos 2π (10t − 0.0080 x + 0.35), where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of (a) 4 m, (b) 0.5 m, (c) λ/2, (d) 3λ/4.
• 14.11) The transverse displacement of a string (clamped at its both ends) is given by y(x, t) = 0.06 sin (2πx/3) cos (120πt), where x and y are in m and t in s. The length of the string is 1.5 m and its mass is 3.0 × 10⁻² kg. Answer the following: (a) Does the function represent a travelling wave or a stationary wave? (b) Interpret the wave as a superposition of two waves travelling in opposite directions. What is the wavelength, frequency, and speed of each wave? (c) Determine the tension in the string.
• 14.12) (i) For the wave on a string described in Exercise 15.11, do all the points on the string oscillate with the same (a) frequency, (b) phase, (c) amplitude? Explain your answers. (ii) What is the amplitude of a point 0.375 m away from one end?
• 14.13) Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all: (a) y = 2 cos (3x) sin (10t) (b) y = 2√(x − vt) (c) y = 3 sin (5x − 0.5t) + 4 cos (5x − 0.5t) (d) y = cos x sin t + cos 2x sin 2t
• 14.14) A wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of 45 Hz. The mass of the wire is 3.5 × 10⁻² kg and its linear mass density is 4.0 × 10⁻² kg m⁻¹. What is (a) the speed of a transverse wave on the string, and (b) the tension in the string?
• 14.15) A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.
• 14.16) A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?
• 14.17) A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a 430 Hz source? Will the same source be in resonance with the pipe if both ends are open? (speed of sound in air is 340 m s⁻¹).
• 14.18) Two sitar strings A and B playing the note 'Ga' are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?
• 14.19) Explain why (or how): (a) in a sound wave, a displacement node is a pressure antinode and vice versa, (b) bats can ascertain distances, directions, nature, and sizes of the obstacles without any 'eyes', (c) a violin note and sitar note may have the same frequency, yet we can distinguish between the two notes, (d) solids can support both longitudinal and transverse waves, but only longitudinal waves can propagate in gases, and (e) the shape of a pulse gets distorted during propagation in a dispersive medium.

Answers at a glance

1. 1
   Q14.1 → The disturbance reaches the other end in 0.5 s.
2. 2
   Q14.2 → The splash is heard about 8.7 s after the stone is dropped.
3. 3
   Q14.3 → Required tension ≈ 2.06 × 10⁴ N.
4. 4
   Q14.4 → (a) P cancels with ρ via ρ = PM/RT; (b) v ∝ √T; (c) humid air is lighter, so v increases.
5. 5
   Q14.5 → Functions (b) and (c) can represent travelling waves (along −x); (a) cannot.
6. 6
   Q14.6 → (a) 3.4 × 10⁻⁴ m (b) 1.5 × 10⁻³ m.
7. 7
   Q14.7 → Wavelength in the tissue ≈ 4.0 × 10⁻⁴ m (0.40 mm).
8. 8
   Q14.8 → (a) Travelling, 20 m s⁻¹ towards the left; (b) 3.0 cm, 5.73 Hz; (c) π/4; (d) ≈ 3.5 m.
9. 9
   Q14.9 → All three graphs are sine curves of identical shape; the oscillatory motion differs only in phase, not in amplitude or frequency.
10. 10
    Q14.10 → (a) 6.4π rad (b) 0.8π rad (c) π rad (d) 3π/2 rad.
11. 11
    Q14.11 → (a) Stationary wave; (b) λ = 3 m, f = 60 Hz, v = 180 m s⁻¹ each way; (c) T = 648 N.
12. 12
    Q14.12 → (i) Same frequency and phase for all points, but different amplitudes; (ii) ≈ 4.2 cm.
13. 13
    Q14.13 → (b), (c): travelling waves; (a): stationary wave; (d): none at all.
14. 14
    Q14.14 → (a) 78.75 m s⁻¹ (b) ≈ 248 N.
15. 15
    Q14.15 → Speed of sound ≈ 366 m s⁻¹.
16. 16
    Q14.16 → Speed of sound in steel ≈ 5.1 × 10³ m s⁻¹.
17. 17
    Q14.17 → Fundamental (first harmonic) mode of the closed pipe, since 430 Hz ≈ 425 Hz; no resonance with both ends open (nearest harmonic is 850 Hz).
18. 18
    Q14.18 → fB = 318 Hz.
19. 19
    Q14.19 → (a) Node ⇔ pressure antinode; (b) echolocation; (c) timbre from different overtones; (d) rigidity lets solids support transverse waves; (e) components travel at different speeds in a dispersive medium.

MCQ Questions

1 Mark Questions

1. 1
   Define wavelength and frequency of a progressive wave.
2. 2
   Define beats and write the beat frequency in terms of the combined wave frequencies.
3. 3
   What are nodes and antinodes in a stationary wave?
4. 4
   State the Doppler effect for sound.

2 Mark Questions

1. 1
   A wave travels with speed 330 m/s and frequency 660 Hz. Find its wavelength. (Ans: λ = v/f = 0.5 m)
2. 2
   A string of length 2 m and mass 50 g is under a tension of 200 N. Find the speed of a transverse wave on it. (Ans: v = √(T/μ) = 89.4 m/s)
3. 3
   Two tuning forks of frequencies 256 Hz and 260 Hz are sounded together. Find the beat frequency. (Ans: 4 Hz)

3 Mark Questions

1. 1
   Derive the expression for the velocity of a transverse wave on a stretched string. (Ans: v = √(T/μ))
2. 2
   Write and explain the equation of a plane progressive wave and derive relations among particle velocity, wave velocity and slope. (Ans: vp = −v × slope)
3. 3
   Derive the conditions for a stationary wave on a string fixed at both ends and find the frequencies of the first three harmonics.

5 Mark Questions

1. 1
   Describe the formation of stationary waves on a stretched string and derive the expressions for the allowed frequencies. Hence find the fundamental frequency and its overtones for a string of length L with tension T and linear density μ. (Ans: fn = [n/(2L)]√(T/μ))
2. 2
   State and derive the Doppler effect for sound — obtain the observed frequency when a source moves towards a stationary observer and when the observer moves towards a stationary source.