NCERT Exemplar Solutions Class 11 Physics Waves — Multiple Choice Questions Q1–17

NCERT Exemplar Problems, Class 11 Physics — Waves, MCQ I and MCQ II. All 17 questions solved.

The book numbers these questions 15.1 to 15.17 (the exemplar counts Waves as its fifteenth unit; it is Chapter 14 of the rationalised textbook); here they are Questions 1 to 17. Questions 1 to 10 are MCQ I (exactly one option is correct) and Questions 11 to 17 are MCQ II (one or more options may be correct). Four ideas carry the whole page:

• Wave speed belongs to the medium. On a string $v = \sqrt{T/\mu}$; in a fluid $v = \sqrt{B/\rho}$. Frequency is fixed by the source, so when the speed changes it is the wavelength that changes.
• Reading a wave equation. In $y = A\sin(\omega t \pm kx)$ the sign tells the direction ($+kx$ travels towards $-x$), and $v = \omega/k$, $\lambda = 2\pi/k$, $f = \omega/2\pi$.
• Reflection. From a denser (rigid) boundary the wave comes back reversed in direction with a phase change of $\pi$.
• Standing waves. $y = 2A\sin kx\cos\omega t$: every point oscillates at the same frequency, with an amplitude fixed by its position, and no energy crosses a node.

Key insight. Separate what the source fixes (frequency) from what the medium fixes (speed). Questions 2, 4, 9 and 16 are all decided by asking which of the two has changed.

Choose the correct answer from the given options. (MCQ I: one correct. MCQ II: one or more correct.)

Question 1

Water waves produced by a motor boat sailing in water are

Solution. A water particle at the surface of a wave moves in a roughly circular (or elliptical) path: it goes up and down, across the direction of travel, and also forwards and backwards, along it. The up-and-down part is transverse and the to-and-fro part is longitudinal. Surface water waves are therefore a combination of the two.

Question 2

Sound waves of wavelength $\lambda$ travelling in a medium with a speed of $v$ m/s enter into another medium where its speed is $2v$ m/s. Wavelength of sound waves in the second medium is

Solution. The frequency is set by the source and does not change when the wave crosses into a new medium; each crest arriving at the boundary sends exactly one crest onward. With $v = f\lambda$ and $f$ fixed, the wavelength is proportional to the speed:

$$\lambda’ = \frac{2v}{f} = 2\cdot\frac{v}{f} = 2\lambda.$$

Question 3

Speed of sound wave in air

Solution. By Laplace’s formula $v = \sqrt{\gamma P/\rho} = \sqrt{\gamma RT/M}$. At a fixed temperature, $P/\rho$ is constant, so changing the pressure changes nothing, and (b) is false; the $\sqrt{T}$ dependence rules out (a). Water vapour (molar mass $18$ g/mol) is lighter than dry air (about $29$ g/mol), so humid air has a smaller average molar mass and a smaller density at the same pressure. A smaller $\rho$ gives a larger speed.

Question 4

Change in temperature of the medium changes

Solution. Temperature changes the speed of sound, $v \propto \sqrt{T}$. The frequency is fixed by the source, and amplitude and loudness are fixed by how hard the source is driven, not by the medium’s temperature. Since $\lambda = v/f$ with $f$ unchanged, the quantity that changes is the wavelength.

Question 5

With propagation of longitudinal waves through a medium, the quantity transmitted is

Solution. In any mechanical wave each particle only oscillates about its own mean position; no particle travels along with the wave, so matter is not carried. What moves through the medium is the disturbance, and with it energy. That makes (b) the answer; (c) and (d) both claim that matter is transported.

Question 6

Which of the following statements are true for wave motion?

Solution. A transverse mechanical wave needs a restoring force against a sideways (shear) displacement, and only a solid has a shear modulus. Fluids resist compression but not shear, so in their bulk they carry longitudinal waves only; that also rules out (b), because sound travels in air and water. No mechanical wave travels through a vacuum, so (d) is false. (Surface waves on water are a special case driven by gravity and surface tension at the boundary, not a bulk transverse wave.)

Question 7

A sound wave is passing through air column in the form of compression and rarefaction. In consecutive compressions and rarefactions,

Solution. Compressions and rarefactions follow each other so quickly, and air conducts heat so poorly, that no heat has time to flow from a compressed (warmer) region to a rarefied (cooler) one. The changes are adiabatic. That is exactly Laplace’s correction to Newton’s formula: the modulus is $\gamma P$, not $P$, and Boyle’s law (an isothermal law) is not obeyed, so (b) is false. Density is precisely what changes in a compression, so (a) is false. The bulk modulus in the wave equation is a property of the undisturbed air, $\gamma P_0$, and is treated as constant for small disturbances, so (c) is not the intended answer.

Question 8

Equation of a plane progressive wave is given by $y = 0.6\sin 2\pi\left(t – \dfrac{x}{2}\right)$. On reflection from a denser medium its amplitude becomes $2/3$ of the amplitude of the incident wave. The equation of the reflected wave is

Solution. Three things happen on reflection. The amplitude becomes $\tfrac{2}{3}\times 0.6 = 0.4$. The direction reverses, so $(t – x/2)$ becomes $(t + x/2)$. And reflection from a denser medium adds a phase of $\pi$, which turns $\sin\theta$ into $\sin(\theta + \pi) = -\sin\theta$. Together:

$$y_r = 0.4\sin\left[2\pi\left(t + \frac{x}{2}\right) + \pi\right] = -0.4\sin 2\pi\left(t + \frac{x}{2}\right).$$

Option (c) forgets the phase change and (d) forgets the change of direction.

Question 9

A string of mass $2.5$ 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, the disturbance will reach the other end in

Solution. The mass per unit length is $\mu = 2.5/20 = 0.125$ kg/m, so the pulse speed is

$$v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{200}{0.125}} = \sqrt{1600} = 40 \text{ m/s}.$$

The time to cover the $20$ m string is $t = 20/40 = 0.5$ s.

Question 10

A train whistling at constant frequency is moving towards a station at a constant speed $V$. The train goes past a stationary observer on the station. The frequency $n’$ of the sound as heard by the observer is plotted as a function of time $t$ (Fig. 15.1). Identify the expected curve.

Solution. While the train approaches at constant speed $V$, the observer hears a constant raised frequency $n’ = n\,\dfrac{v}{v – V}$. After it passes and recedes at the same speed, the frequency is a constant lower value $n” = n\,\dfrac{v}{v + V}$. Neither depends on time, so the graph is two horizontal lines with a drop at the moment of passing, which is graph (c). (For an observer standing a little to one side of the track the drop would be smoothed over a short interval, but it would still fall from one constant level to another; none of the rising curves (a), (b), (d) can occur, because the frequency never increases.)

Question 11

A transverse harmonic wave on a string is described by $y(x, t) = 3.0\sin\left(36t + 0.018x + \pi/4\right)$, where $x$ and $y$ are in cm and $t$ is in s. The positive direction of $x$ is from left to right.

Solution. Compare with $y = A\sin(\omega t + kx + \phi)$: $\omega = 36$ rad/s and $k = 0.018$ rad/cm.

• (a) The phase $\omega t + kx$ stays constant if $x$ decreases as $t$ increases, so the wave moves towards $-x$, right to left. True.
• (b) $v = \omega/k = 36/0.018 = 2000$ cm/s $= 20$ m/s. True.
• (c) $f = \omega/2\pi = 36/2\pi = 5.73$ Hz. True.
• (d) The distance between successive crests is the wavelength, $\lambda = 2\pi/k = 2\pi/0.018 \approx 349$ cm, not $2.5$ cm. False.

Question 12

The displacement of a string is given by $y(x, t) = 0.06\sin(2\pi x/3)\cos(120\pi t)$, where $x$ and $y$ are in m and $t$ in s. The length of the string is $1.5$ m and its mass is $3.0\times10^{-2}$ kg.

Solution. The displacement is a product of a function of $x$ alone and a function of $t$ alone, $y = 2A\sin kx\cos\omega t$, which is the form of a stationary wave; a progressive wave depends on $x$ and $t$ only through $\omega t \pm kx$. So (a) is false and (b) is the right description, with $f = 120\pi/2\pi = 60$ Hz.

Such a wave is the sum of two waves $A\sin(kx – \omega t)$ and $A\sin(kx + \omega t)$ travelling in opposite directions, with $\lambda = 2\pi/k = 2\pi/(2\pi/3) = 3$ m, $f = 60$ Hz and $v = f\lambda = 180$ m/s. So (c) is true. The amplitude at position $x$ is $0.06\,|\sin(2\pi x/3)|$, which varies from zero at the nodes to $0.06$ m at the antinodes, so (d) is false.

Question 13

Speed of sound waves in a fluid depends upon

Solution. In a fluid $v = \sqrt{B/\rho}$. The speed is proportional to $\sqrt{B}$ and to $1/\sqrt{\rho}$, which are (d) and (c). It neither grows with $\rho$ nor with $B^2$.

Question 14

During propagation of a plane progressive mechanical wave

Solution. In $y = A\sin(\omega t – kx)$ every particle performs simple harmonic motion with the same amplitude $A$ (a plane wave does not spread out, and no damping is assumed), so (b) and (c) are true. The phase $\omega t – kx$ depends on $x$, so particles at different positions are at different stages of their motion and (a) is false. The speed is set by the elastic and inertial properties of the medium, $\sqrt{T/\mu}$ or $\sqrt{B/\rho}$, so (d) is true.

Question 15

The transverse displacement of a string (clamped at its both ends) is given by $y(x, t) = 0.06\sin(2\pi x/3)\cos(120\pi t)$. All the points on the string between two consecutive nodes vibrate with

Solution. Every point oscillates as $\cos(120\pi t)$, so all have the same frequency, $60$ Hz. Between two consecutive nodes $\sin(2\pi x/3)$ keeps one sign, so all those points move up together and down together: the same phase. Their amplitudes $0.06\,|\sin(2\pi x/3)|$ differ from point to point, and since the energy of an oscillating element goes as the square of its amplitude, the energies differ too. So (a), (b), (d) are true and (c) is false.

Question 16

A train, standing in a station yard, blows a whistle of frequency $400$ Hz in still air. The wind starts blowing in the direction from the yard to the station with a speed of $10$ m/s. Given that the speed of sound in still air is $340$ m/s,

Solution. The wind carries the sound with it, so relative to the ground the sound moves towards the platform at $340 + 10 = 350$ m/s: (b) is true. But the source and the observer are both at rest relative to the ground, so the distance between them does not change. Every wave sent out arrives, one after another, at the same rate it was emitted. The wind changes the speed and the wavelength together ($\lambda = 350/400$ m), not the frequency. The observer hears $400$ Hz, so (a) is true and (c), (d) are false.

Question 17

Which of the following statements are true for a stationary wave?

Solution. For $y = 2A\sin kx\cos\omega t$:

• (a) The amplitude $2A|\sin kx|$ is fixed in time for each particle but changes with position, so neighbours have different amplitudes. True.
• (b) Every particle has $y = 0$ whenever $\cos\omega t = 0$, so they all pass through the mean position together. True.
• (c) Contradicts (a). False.
• (d) The two component waves carry equal energy in opposite directions, so the net flow across any plane is zero; energy only sloshes between the nodes. True.
• (e) The nodes, where $\sin kx = 0$, never move. True.

Common mistakes

• Changing the frequency when the medium changes in Questions 2 and 4. The source fixes $f$; the medium changes $v$, and $\lambda$ follows.
• Thinking pressure raises the speed of sound in Question 3. At fixed temperature $P/\rho$ is constant, so pressure has no effect; humidity lowers the density and does.
• Forgetting one of the three changes on reflection in Question 8. Amplitude, direction and the phase of $\pi$ from a denser boundary must all be applied.
• Using the total mass instead of mass per length in Question 9. $\mu = m/L$, and $v = \sqrt{T/\mu}$.
• Picking a smooth peak for the passing train in Question 10. At constant speed the heard frequency is constant on approach and constant (lower) on recession; it only drops.
• Reading $+kx$ as motion towards $+x$ in Question 11. $\omega t + kx$ is a wave travelling towards $-x$.
• Expecting the wind to change the pitch in Question 16. With source and observer both at rest, the frequency is unchanged; only the speed and wavelength alter.

Practise next

• A wire of mass $0.5$ kg and length $4$ m is held at a tension of $50$ N. How long does a transverse pulse take to travel from one end to the other?

• A wave on a string is $y = 0.05\sin(20\pi t – 0.5\pi x)$ in SI units. Find its wavelength, frequency and speed, and its direction of travel.

• Exemplar Waves: Very Short, Short and Long Answer Questions — where the same ideas are applied to sonometers, organ pipes, beats, the resonance tube and seismic waves.
• NCERT textbook Chapter 14 — Waves for the theory of travelling and standing waves, the speed of sound and beats.
• Exemplar Oscillations — the previous unit, whose simple harmonic motion is what every particle in a wave performs.