NEET Physics: Oscillations and Waves · Chapter 2

Waves

What to remember

A travelling wave y = a sin(kx − ωt) carries energy and not matter. Its speed is v = fλ = ω/k, where k = 2π/λ and ω = 2πf.
The speed of a transverse wave on a string is v = √(T/μ). The speed of sound in a gas is v = √(γP/ρ) (Laplace); Newton's formula √(P/ρ) gives a value too low. Sound speed in air rises with temperature (v ∝ √T) and with humidity, and does not depend on pressure at constant temperature.
A string fixed at both ends and an open pipe have frequencies f = nv/2L (n = 1, 2, 3, …). A closed pipe has only odd harmonics, f = (2n − 1)v/4L.
Two waves of nearly equal frequencies f₁ and f₂ give beats; the beat frequency is |f₁ − f₂|.

Wave motion, transverse and longitudinal waves

A wave is a disturbance that travels through a medium and carries energy from one place to another. The particles of the medium vibrate about their mean positions; they do not move along with the wave. Mechanical waves need a material medium that has elasticity and inertia.

Property	Transverse wave	Longitudinal wave
Particle motion	perpendicular to the direction of travel	parallel to the direction of travel
Pattern	crests and troughs	compressions and rarefactions
Medium needed	one that has shear (rigidity): solids, surface of liquids, strings	any medium with bulk elasticity: solids, liquids, gases
Examples	wave on a stretched string	sound in air, wave in a spring along its length

Transverse waves cannot travel through the interior of a gas or a liquid, because these cannot sustain shear. Sound in a gas is always longitudinal.

Displacement relation and wave parameters

A harmonic wave travelling in the positive x-direction is

y(x, t) = a sin(kx − ωt + φ)

For a wave travelling in the negative x-direction, use (kx + ωt).

Symbol	Name	Relation	Unit
a	amplitude	maximum displacement	m
λ	wavelength	distance between two consecutive points in the same phase	m
k	angular wavenumber	k = 2π/λ	rad m⁻¹
T	period	time of one full vibration	s
f	frequency	f = 1/T	Hz
ω	angular frequency	ω = 2πf = 2π/T	rad s⁻¹
v	wave speed	v = fλ = ω/k	m s⁻¹
(kx − ωt + φ)	phase	state of vibration at x and t	rad

Two points whose separation along the wave is Δx have a phase difference Δφ = kΔx = 2πΔx/λ. Points one wavelength apart are in the same phase; points half a wavelength apart are in opposite phase (Δφ = π). When a wave passes from one medium to another, its frequency does not change, but its speed and wavelength do.

Speed of a travelling wave

The speed depends on the properties of the medium, not on the frequency or amplitude.

Wave	Speed	Symbols
Transverse wave on a stretched string	v = √(T/μ)	T = tension (N), μ = mass per unit length (kg m⁻¹)
Longitudinal wave in a solid rod	v = √(Y/ρ)	Y = Young's modulus, ρ = density
Longitudinal wave in a liquid	v = √(B/ρ)	B = bulk modulus
Sound in a gas, Newton's formula	v = √(P/ρ)	P = pressure, ρ = density
Sound in a gas, Laplace's correction	v = √(γP/ρ)	γ = Cp/Cv

Newton's formula and Laplace's correction. Newton assumed that the compressions and rarefactions in a sound wave take place at constant temperature (an isothermal process), so the pressure of the gas is P. For air at STP, P = 1.013 × 10⁵ Pa and ρ = 1.29 kg m⁻³, which gives v ≈ 280 m s⁻¹. This is about 15% lower than the measured value, about 331 m s⁻¹ at 0 °C. Laplace pointed out that the changes are so rapid that heat has no time to flow, so the process is adiabatic and the effective elastic constant is γP. For air γ = 1.4, and the formula gives v ≈ 280 × √1.4 ≈ 331 m s⁻¹, which agrees with experiment.

Effect of temperature, pressure and humidity on the speed of sound in air

Temperature: v ∝ √T (T in kelvin). The speed rises by about 0.6 m s⁻¹ for each 1 °C rise.
Pressure: at constant temperature, P/ρ is constant (ρ is proportional to P), so the speed does not change when only the pressure changes.
Humidity: water vapour is lighter than dry air, so moist air has a lower density than dry air at the same temperature and pressure. The speed of sound is therefore greater in moist air.
The speed does not depend on the frequency or the amplitude of the sound.

Principle of superposition

When two or more waves meet in a medium, the resultant displacement at any point is the algebraic sum of the displacements that the individual waves would produce there:

y = y₁ + y₂ + …

After the waves cross, each continues as if the other were absent. Where the waves are in phase, the amplitudes add (constructive interference); where they are in opposite phase, they subtract (destructive interference). Standing waves and beats are results of this principle.

Reflection of waves

When a wave pulse reaches the boundary of the medium, it is reflected.

Boundary	Example	Displacement at the boundary	Reflected pulse
Rigid (fixed) end	string tied to a wall	zero (a node)	inverted; phase change of π
Open (free) end	string end free to move, or open end of a pipe	maximum (an antinode)	not inverted; no phase change

For a sound wave in a pipe, the closed end acts like a rigid boundary (a displacement node) and the open end acts like a free boundary (a displacement antinode).

Standing waves and normal modes

A wave and its reflection travelling in opposite directions form a standing (stationary) wave:

y = 2a sin(kx) cos(ωt)

Nodes are points of zero amplitude. They are at x = 0, λ/2, λ, … The distance between two consecutive nodes is λ/2.
Antinodes are points of maximum amplitude 2a. They are midway between nodes, so the distance between a node and the next antinode is λ/4.
All particles between two consecutive nodes vibrate in the same phase; particles on the two sides of a node vibrate in opposite phase. A standing wave does not transport energy along the medium.

The patterns of vibration that a system can sustain are its normal modes.

String fixed at both ends (length L). Both ends are nodes, so L = nλ/2.

f_n = nv/2L = n f₁, with n = 1, 2, 3, …, and f₁ = (1/2L)√(T/μ).
The fundamental is the first harmonic; 2f₁ is the second harmonic (first overtone), and so on. All harmonics are present.

Open organ pipe (open at both ends). Both ends are antinodes, so L = nλ/2.

f_n = nv/2L, with n = 1, 2, 3, … All harmonics are present; the frequencies are in the ratio 1 : 2 : 3 : …

Closed organ pipe (one end closed). The closed end is a node and the open end is an antinode, so L = (2n − 1)λ/4.

f_n = (2n − 1)v/4L = v/4L, 3v/4L, 5v/4L, … Only odd harmonics are present, in the ratio 1 : 3 : 5.
The first overtone is the third harmonic; the second overtone is the fifth harmonic.
The fundamental of a closed pipe is half that of an open pipe of the same length.
In a closed air column, two successive resonant lengths differ by λ/2.

End correction (in brief). The antinode at an open end is formed slightly outside the pipe. If the pipe radius is r, the end correction is e ≈ 0.6r for each open end. The effective length is L + e for a closed pipe and L + 2e for an open pipe.

System	Boundary condition	Allowed frequencies	Fundamental
String, both ends fixed	node – node	nv/2L (n = 1, 2, 3, …)	v/2L
Open pipe	antinode – antinode	nv/2L (n = 1, 2, 3, …)	v/2L
Closed pipe	node – antinode	(2n − 1)v/4L	v/4L

Beats

When two sound waves of slightly different frequencies f₁ and f₂ are heard together, the intensity rises and falls regularly. Each rise and fall is a beat. The beat frequency (number of beats per second) is

f_beat = |f₁ − f₂|

The time between two successive beats is 1/|f₁ − f₂|. The ear can separate beats only when the frequency difference is small (below about 10 Hz). Beats are used to tune musical instruments: if the beats vanish, the frequencies are equal. Loading a tuning fork with wax lowers its frequency; filing it raises its frequency.

Formula summary

Quantity	Formula	Unit
Wave equation	y = a sin(kx − ωt)	m
Wavenumber, angular frequency	k = 2π/λ, ω = 2πf	rad m⁻¹, rad s⁻¹
Wave speed	v = fλ = ω/k	m s⁻¹
Phase difference	Δφ = 2πΔx/λ	rad
String	v = √(T/μ)	m s⁻¹
Sound, Laplace	v = √(γP/ρ)	m s⁻¹
Temperature effect	v₁/v₂ = √(T₁/T₂)	–
String or open pipe	f_n = nv/2L	Hz
Closed pipe	f_n = (2n − 1)v/4L	Hz
End correction	e ≈ 0.6r	m
Beats	f₁ − f₂	Hz

Worked examples

Example 1. A wave is y = 0.02 sin(10πx − 200πt) in SI units. Find λ, f and v.

k = 10π, so λ = 2π/k = 0.2 m. ω = 200π, so f = ω/2π = 100 Hz. v = fλ = 100 × 0.2 = 20 m s⁻¹. Check: ω/k = 200π/10π = 20 m s⁻¹.

Example 2. A wire of linear density 0.01 kg m⁻¹ is stretched with a tension of 100 N between supports 0.5 m apart. Find the wave speed and the fundamental frequency.

v = √(T/μ) = √(100/0.01) = √10⁴ = 100 m s⁻¹. f₁ = v/2L = 100/(2 × 0.5) = 100 Hz.

Example 3. Take the speed of sound as 340 m s⁻¹. Find the fundamental frequency of a closed pipe and an open pipe, each 0.5 m long (ignore end correction).

Closed: f₁ = v/4L = 340/(4 × 0.5) = 170 Hz. Open: f₁ = v/2L = 340/(2 × 0.5) = 340 Hz.

Example 4. Two tuning forks of frequencies 256 Hz and 260 Hz sound together. Find the beat frequency and the number of beats in 5 s.

f_beat = 260 − 256 = 4 Hz, so there are 4 beats each second and 20 beats in 5 s.

Example 5. The speed of sound in air at 27 °C is 340 m s⁻¹. Find its value at 327 °C.

T₁ = 300 K, T₂ = 600 K. v₂ = v₁√(T₂/T₁) = 340 × √2 ≈ 481 m s⁻¹.

NEET traps

1. Newton's formula uses P (isothermal); Laplace's formula uses γP (adiabatic). The factor √γ links them.
2. The speed of sound in a gas depends on temperature (as √T) and not on pressure alone. Use kelvin temperature in the ratio, not Celsius.
3. Moist air is less dense than dry air, so sound travels faster in it, not slower.
4. A closed pipe has only odd harmonics; an open pipe has all harmonics. The first overtone of a closed pipe is the third harmonic, but that of an open pipe is the second harmonic.
5. The closed end of a pipe is a displacement node; the open end is a displacement antinode. The opposite is true for the pressure variation.
6. A rigid-end reflection inverts the pulse (phase change π); a free-end reflection does not.
7. The distance between a node and the next node (or antinode and antinode) is λ/2; between a node and the next antinode it is λ/4.
8. The frequency of a wave is fixed by the source and does not change when the wave enters another medium; the speed and wavelength change.
9. Beat frequency is the difference of the two frequencies, not the sum or the average.
10. The wave speed on a string depends on T and μ only. Changing the frequency changes the wavelength, not the speed.

One-liners

1. A wave transports energy and momentum, not matter.
2. In transverse waves the particles move perpendicular to the direction of the wave; in longitudinal waves they move parallel to it.
3. Wave speed v = fλ = ω/k, and k = 2π/λ.
4. Speed of a wave on a string is v = √(T/μ).
5. The Newton value for the speed of sound in air at STP is about 280 m s⁻¹; the measured value is about 331 m s⁻¹.
6. Laplace's formula: v = √(γP/ρ), with γ = 1.4 for air.
7. The speed of sound in air is proportional to √T and is independent of pressure at constant temperature.
8. Sound travels faster in humid air than in dry air.
9. Superposition: the resultant displacement is the sum of the individual displacements.
10. Reflection from a rigid boundary gives a phase change of π; from an open boundary, no change.
11. Adjacent nodes are λ/2 apart.
12. String fixed at both ends: f_n = nv/2L for n = 1, 2, 3, …
13. Open pipe: all harmonics, f_n = nv/2L. Closed pipe: odd harmonics only, f = (2n − 1)v/4L.
14. End correction is about 0.6r for each open end.
15. Beat frequency = |f₁ − f₂|.

Practice questions

A wave has a frequency of 500 Hz and a wavelength of 0.68 m. What is its speed?
1. 170 m s⁻¹
2. 340 m s⁻¹
3. 34 m s⁻¹
4. 735 m s⁻¹
Answer
B. 340 m s⁻¹
v = fλ = 500 × 0.68 = 340 m s⁻¹.
A travelling wave is y = 2 sin(6πx − 300πt) in SI units. What is its speed?
1. 50 m s⁻¹
2. 300 m s⁻¹
3. 1800π m s⁻¹
4. 150 m s⁻¹
Answer
A. 50 m s⁻¹
v = ω/k = 300π/6π = 50 m s⁻¹. Check: λ = 2π/6π = 1/3 m, f = 150 Hz, fλ = 50.
A stretched string has a tension of 64 N and a mass per unit length of 0.04 kg m⁻¹. What is the speed of a transverse wave on it?
1. 1600 m s⁻¹
2. 16 m s⁻¹
3. 40 m s⁻¹
4. 80 m s⁻¹
Answer
C. 40 m s⁻¹
v = √(T/μ) = √(64/0.04) = √1600 = 40 m s⁻¹.
The temperature of air is raised from 27 °C to 327 °C. The ratio of the speed of sound at the higher temperature to that at the lower temperature is
1. 2
2. 12.1
3. √12.1
4. √2
Answer
D. √2
v ∝ √T with T in kelvin: √(600/300) = √2. (Using Celsius values would wrongly give √12.)
Newton's formula v = √(P/ρ) gives the speed of sound in air at STP as about
1. 280 m s⁻¹
2. 331 m s⁻¹
3. 390 m s⁻¹
4. 340 m s⁻¹
Answer
A. 280 m s⁻¹
√(1.013 × 10⁵/1.29) ≈ 280 m s⁻¹, which is lower than the measured 331 m s⁻¹.
A string of length 0.8 m is fixed at both ends. The speed of transverse waves on it is 320 m s⁻¹. What is the frequency of its fundamental mode?
1. 100 Hz
2. 400 Hz
3. 256 Hz
4. 200 Hz
Answer
D. 200 Hz
f₁ = v/2L = 320/1.6 = 200 Hz. Check: λ = 2L = 1.6 m and f = 320/1.6.
A closed organ pipe has length 0.25 m. Take the speed of sound as 340 m s⁻¹ and ignore end correction. What is the frequency of its second overtone?
1. 340 Hz
2. 680 Hz
3. 1700 Hz
4. 1020 Hz
Answer
C. 1700 Hz
Fundamental = v/4L = 340 Hz. The second overtone is the fifth harmonic: 5 × 340 = 1700 Hz.
An open pipe has a fundamental frequency of 400 Hz. What is the fundamental frequency of a closed pipe of the same length (same speed of sound)?
1. 800 Hz
2. 200 Hz
3. 400 Hz
4. 100 Hz
Answer
B. 200 Hz
Open: v/2L = 400, so closed: v/4L = 200 Hz.
Two tuning forks of frequencies 480 Hz and 486 Hz are sounded together. How many beats are heard per second?
1. 3
2. 966
3. 483
4. 6
Answer
D. 6
f_beat = 486 − 480 = 6 per second.
When a transverse pulse on a string is reflected from a rigid support, the phase change of the reflected pulse is
1. 0
2. 2π
3. π
4. π/2
Answer
C. π
A rigid end is a node; the pulse returns inverted, a phase change of π.
The distance between two consecutive nodes of a standing wave of wavelength λ is
1. λ/2
2. λ
3. 2λ
4. λ/4
Answer
A. λ/2
Nodes are separated by λ/2.
Compared with dry air at the same temperature and pressure, the speed of sound in humid air is
1. the same
2. greater
3. smaller
4. zero
Answer
B. greater
Water vapour is lighter than dry air, so moist air has lower density and the speed is greater.
Sound waves travelling in air are
1. longitudinal, made of compressions and rarefactions
2. transverse, made of crests and troughs
3. longitudinal, made of crests and troughs
4. transverse, made of compressions and rarefactions
Answer
A. longitudinal, made of compressions and rarefactions
A gas cannot sustain shear, so sound in air is longitudinal.
The temperature of a gas is kept constant while its pressure is doubled. The speed of sound in the gas
1. doubles
2. remains the same
3. becomes half
4. becomes √2 times
Answer
B. remains the same
v = √(γP/ρ) and ρ ∝ P at constant temperature, so P/ρ is unchanged.
A closed organ pipe has length 24.4 cm and inner radius 1 cm. Use end correction e = 0.6r and take the speed of sound as 340 m s⁻¹. What is its fundamental frequency?
1. 170 Hz
2. 680 Hz
3. 340 Hz
4. 349 Hz
Answer
C. 340 Hz
Effective length = 24.4 + 0.6 = 25 cm = 0.25 m. f₁ = v/4L = 340/1 = 340 Hz.
By the principle of superposition, when two waves overlap in a medium, the resultant displacement at a point is
1. the product of the individual displacements
2. always the larger of the two displacements
3. always zero
4. the algebraic sum of the displacements due to the individual waves
Answer
D. the algebraic sum of the displacements due to the individual waves
The principle of superposition states that y = y₁ + y₂.
Given below are two statements: Statement I: Transverse waves can travel through solids. Statement II: Transverse waves can travel through the interior of gases.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
C. Statement I is correct but Statement II is incorrect
Solids have shear rigidity and carry transverse waves; gases do not.
Given below are two statements: Statement I: In a standing wave, all particles between two consecutive nodes vibrate in the same phase. Statement II: Particles on the two sides of a node vibrate in opposite phase.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
A. Both Statement I and Statement II are correct
Both are properties of a standing wave.
Which of the following statements are correct? 1. A closed pipe supports even harmonics as well as odd harmonics. 2. The closed end of a pipe is a displacement node. 3. The open end of a pipe is a displacement antinode.
1. 1 and 2 only
2. 1 and 3 only
3. 2 and 3 only
4. 1, 2 and 3
Answer
C. 2 and 3 only
A closed pipe supports only odd harmonics (1 false). Closed end: node (2 true). Open end: antinode (3 true).
Given below are two statements: Statement I: An open organ pipe contains both even and odd harmonics. Statement II: The first overtone of an open organ pipe has a frequency twice that of its fundamental.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
A. Both Statement I and Statement II are correct
f_n = nv/2L, so all n occur; the first overtone is n = 2, which is 2f₁.
Given below are two statements: Statement I: The speed of sound in air increases when the temperature increases. Statement II: The speed of sound in air is directly proportional to the temperature in degree Celsius.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
C. Statement I is correct but Statement II is incorrect
v ∝ √T (kelvin). It increases with temperature, but it is not proportional to the Celsius value.
Which of the following statements are correct for a string of length L fixed at both ends? 1. The fundamental wavelength is 2L. 2. All harmonics are present. 3. The frequency of the nth harmonic is n times the fundamental frequency.
1. 1 and 2 only
2. 1 and 3 only
3. 2 and 3 only
4. 1, 2 and 3
Answer
D. 1, 2 and 3
L = nλ/2 gives λ₁ = 2L; f_n = nv/2L = n f₁; every integer n is allowed.
Given below are two statements: Statement I: The beat frequency of two sound waves is the sum of their frequencies. Statement II: Beats are produced by the superposition of two waves of slightly different frequencies.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
D. Statement I is incorrect but Statement II is correct
The beat frequency is |f₁ − f₂|. Statement II is correct.
Given below are two statements: Statement I: Newton assumed that the pressure changes in a sound wave occur isothermally. Statement II: Laplace showed that these changes are adiabatic.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
A. Both Statement I and Statement II are correct
Newton used P (isothermal); Laplace used γP (adiabatic).
Which of the following statements are correct for reflection of waves at a boundary? 1. A pulse reflected from a rigid end is inverted. 2. A pulse reflected from a free end has no phase change. 3. A node is formed at a free end of a string.
1. 1 and 2 only
2. 1 and 3 only
3. 2 and 3 only
4. 1, 2 and 3
Answer
A. 1 and 2 only
A free end is an antinode, so 3 is false. 1 and 2 are correct.
Given below are two statements: Statement I: In a travelling wave, the particles of the medium move along with the wave at the wave speed. Statement II: A travelling wave transports energy from one place to another.
1. Both Statement I and Statement II are correct
2. Both Statement I and Statement II are incorrect
3. Statement I is correct but Statement II is incorrect
4. Statement I is incorrect but Statement II is correct
Answer
D. Statement I is incorrect but Statement II is correct
Particles only vibrate about their mean positions; the wave carries energy and not matter.
Assertion (A): The speed of sound in humid air is greater than in dry air at the same temperature and pressure. Reason (R): Water vapour is lighter than dry air, so humid air has lower density.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
A. Both A and R are true and R is the correct explanation of A
v = √(γP/ρ): lower ρ gives a higher speed.
Assertion (A): The speed of a transverse wave on a string increases when its tension increases. Reason (R): The speed of sound in air increases with temperature.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
B. Both A and R are true but R is not the correct explanation of A
v = √(T/μ) makes A true, and R is also true, but R says nothing about strings.
Assertion (A): Newton's formula gives the speed of sound in air at STP as about 280 m s⁻¹. Reason (R): Newton assumed that the pressure changes in a sound wave are adiabatic.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
C. A is true but R is false
A is true. Newton assumed isothermal changes, so R is false.
Assertion (A): The beat frequency of two waves is the sum of their frequencies. Reason (R): Beats are produced by the superposition of two waves of slightly different frequencies.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
D. A is false but R is true
A is false because f_beat = |f₁ − f₂|. R is true.
Assertion (A): The speed of sound in a gas at constant temperature does not change with pressure. Reason (R): At constant temperature the ratio P/ρ of a gas remains constant.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
A. Both A and R are true and R is the correct explanation of A
v = √(γP/ρ) depends on P/ρ, which is fixed when T is fixed.
Assertion (A): A node in a standing wave is a point where the displacement is always zero. Reason (R): At a node the amplitude of vibration is maximum.
1. Both A and R are true and R is the correct explanation of A
2. Both A and R are true but R is not the correct explanation of A
3. A is true but R is false
4. A is false but R is true
Answer
C. A is true but R is false
A is true; R is false because the amplitude at a node is zero (it is maximum at an antinode).
Match List I with List II. List I: a. String fixed at both ends, fundamental b. Closed pipe, fundamental c. Closed pipe, first overtone d. Open pipe, first overtone. List II: i. v/L ii. 3v/4L iii. v/2L iv. v/4L.
1. a-i, b-ii, c-iii, d-iv
2. a-iii, b-iv, c-ii, d-i
3. a-iv, b-iii, c-ii, d-i
4. a-iii, b-iv, c-i, d-ii
Answer
B. a-iii, b-iv, c-ii, d-i
String: v/2L. Closed fundamental: v/4L; closed first overtone: 3v/4L; open first overtone: 2 × v/2L = v/L.
Match List I with List II. List I: a. Speed of a wave on a string b. Newton's formula for sound in a gas c. Laplace's formula for sound in a gas d. Speed of a longitudinal wave in a solid rod. List II: i. √(P/ρ) ii. √(γP/ρ) iii. √(T/μ) iv. √(Y/ρ).
1. a-iii, b-ii, c-i, d-iv
2. a-iii, b-i, c-ii, d-iv
3. a-i, b-iii, c-ii, d-iv
4. a-iv, b-i, c-ii, d-iii
Answer
B. a-iii, b-i, c-ii, d-iv
String √(T/μ); Newton √(P/ρ); Laplace √(γP/ρ); solid rod √(Y/ρ).
Match List I with List II. List I: a. Angular wavenumber b. Angular frequency c. Wave speed d. Period. List II: i. 2πf ii. ω/k iii. 1/f iv. 2π/λ.
1. a-i, b-iv, c-ii, d-iii
2. a-iv, b-i, c-ii, d-iii
3. a-ii, b-iv, c-i, d-iii
4. a-iv, b-i, c-iii, d-ii
Answer
B. a-iv, b-i, c-ii, d-iii
k = 2π/λ; ω = 2πf; v = ω/k; T = 1/f.
Match List I with List II. List I: a. Forks of 256 Hz and 260 Hz b. Forks of 300 Hz and 303 Hz c. Forks of 512 Hz and 520 Hz d. Forks of 440 Hz and 441 Hz. List II: i. 8 beats per second ii. 1 beat per second iii. 3 beats per second iv. 4 beats per second.
1. a-i, b-ii, c-iii, d-iv
2. a-iv, b-iii, c-ii, d-i
3. a-iii, b-iv, c-i, d-ii
4. a-iv, b-iii, c-i, d-ii
Answer
D. a-iv, b-iii, c-i, d-ii
Differences: 4, 3, 8 and 1.
Match List I with List II. List I: a. Node b. Antinode c. Reflection at a rigid boundary d. Reflection at a free boundary. List II: i. No phase change ii. Phase change of π iii. Zero amplitude iv. Maximum amplitude.
1. a-iv, b-iii, c-ii, d-i
2. a-iii, b-iv, c-ii, d-i
3. a-iii, b-iv, c-i, d-ii
4. a-iii, b-ii, c-iv, d-i
Answer
B. a-iii, b-iv, c-ii, d-i
Node: zero amplitude; antinode: maximum amplitude; rigid end: phase change π; free end: no change.
Match List I with List II. List I (speed of a transverse wave on a string): a. Tension made 4 times b. Mass per unit length made 4 times c. Tension and mass per unit length both made 4 times d. Tension made 9 times. List II: i. Tripled ii. Halved iii. Doubled iv. Unchanged.
1. a-iii, b-iv, c-ii, d-i
2. a-iii, b-ii, c-iv, d-i
3. a-iii, b-ii, c-i, d-iv
4. a-ii, b-iii, c-iv, d-i
Answer
B. a-iii, b-ii, c-iv, d-i
v = √(T/μ): √4 = 2; 1/√4 = 1/2; √(4/4) = 1; √9 = 3.
A wave of frequency 50 Hz travels at 20 m s⁻¹. What is the phase difference between two points on the wave that are 0.1 m apart along the direction of travel?
1. π/4
2. 2π
3. π/2
4. π
Answer
C. π/2
λ = v/f = 0.4 m. Δφ = 2πΔx/λ = 2π × 0.1/0.4 = π/2.
The fundamental frequency of a stretched string is 100 Hz. If the tension is made four times with the length and mass per unit length unchanged, what is the new fundamental frequency?
1. 141 Hz
2. 400 Hz
3. 50 Hz
4. 200 Hz
Answer
D. 200 Hz
f ∝ √T, so f′ = 100 × √4 = 200 Hz.
In a resonance experiment with a closed air column and a tuning fork of frequency 500 Hz, the first resonance occurs at a length of 17 cm and the second at 51 cm. What is the speed of sound?
1. 680 m s⁻¹
2. 340 m s⁻¹
3. 255 m s⁻¹
4. 170 m s⁻¹
Answer
B. 340 m s⁻¹
Successive resonant lengths differ by λ/2: λ/2 = 34 cm, λ = 0.68 m. v = fλ = 500 × 0.68 = 340 m s⁻¹.
A tuning fork A has frequency 256 Hz. A second fork B gives 4 beats per second with A. When B is loaded with a little wax, the number of beats per second becomes 6. What is the original frequency of B?
1. 260 Hz
2. 250 Hz
3. 252 Hz
4. 262 Hz
Answer
C. 252 Hz
B is 252 or 260 Hz. Wax lowers B's frequency. If B were 260 Hz, lowering it would reduce the beats; since the beats rise to 6, B is below A: B = 252 Hz (loaded: 250 Hz).
The speed of sound in air at 27 °C is v. At what temperature on the Celsius scale will the speed of sound in air be 2v?
1. 927 °C
2. 1200 °C
3. 54 °C
4. 600 °C
Answer
A. 927 °C
v ∝ √T, so T = 4 × 300 = 1200 K = 927 °C.
A uniform string of length 20 m has a mass of 5 kg and is held under a tension of 25 N. How long does a pulse take to travel the whole length of the string?
1. 0.5 s
2. 10 s
3. 4 s
4. 2 s
Answer
D. 2 s
μ = 5/20 = 0.25 kg m⁻¹; v = √(25/0.25) = 10 m s⁻¹; t = 20/10 = 2 s.
A string of length 1.2 m fixed at both ends vibrates in three loops at a frequency of 150 Hz. What is the speed of the wave on the string?
1. 180 m s⁻¹
2. 360 m s⁻¹
3. 60 m s⁻¹
4. 120 m s⁻¹
Answer
D. 120 m s⁻¹
Three loops: L = 3λ/2, λ = 0.8 m. v = fλ = 150 × 0.8 = 120 m s⁻¹. Check: f₁ = 50 Hz = v/2L, so v = 50 × 2.4 = 120.

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