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Waves

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Chapter Notes

Overview
This chapter introduces the concept of waves as disturbances that propagate through a medium without the net transfer of matter, transporting energy and information. It covers the classification of waves into transverse and longitudinal types, the mathematical description of progressive harmonic waves, and the determination of wave speed in different media. The one big idea a student must walk away with is that waves are a fundamental mechanism for energy transfer, governed by principles like superposition, reflection, and the formation of standing waves, with specific boundary conditions leading to natural frequencies and normal modes.
Key Concepts (29)
Wave
A pattern of disturbance that moves without the actual physical transfer or flow of matter as a whole, transporting energy and information from one point to another.
e.g. When a pebble is dropped in a pond, circular ripples move outward, but cork pieces on the surface only move up and down, not outward with the waves.
Mechanical waves
Waves that require a material medium for propagation and cannot travel through vacuum; they involve oscillations of constituent particles and depend on the elastic properties of the medium.
e.g. Sound waves in air, waves on a string, water waves, and seismic waves.
Transverse wave
A wave in which the constituents of the medium oscillate perpendicular to the direction of wave propagation.
e.g. Waves on a stretched string when one end is given a periodic up-and-down jerk.
Longitudinal wave
A wave in which the constituents of the medium oscillate along the direction of wave propagation.
e.g. Sound waves in air generated by a piston moving back and forth in a pipe.
Progressive wave (Travelling wave)
A wave that travels from one part of the medium to another, transferring energy without the net motion of the medium as a whole.
e.g. A sinusoidal wave travelling along a stretched string.
Amplitude (a)
The maximum displacement of the constituents of the medium from their equilibrium position; it is always a positive constant.
e.g. In y(x,t) = 0.005 sin(80.0x – 3.0t), the amplitude is 0.005 m.
Phase
The argument (kx – ωt + φ) of the sine function in the wave equation, which determines the displacement of the wave at any position and at any instant.
e.g. For y = a sin(kx – ωt + φ), the phase is (kx – ωt + φ).
Initial phase angle (φ)
The phase at x = 0 and t = 0; it can be set to zero by suitable choice of origin and initial time.
e.g. In y = a sin(kx – ωt), φ = 0.
Wavelength (λ)
The minimum distance between two points having the same phase, such as the distance between two consecutive crests or troughs.
e.g. For a wave with k = 80.0 rad/m, λ = 2π/k = 7.85 cm.
Angular wave number (k)
The propagation constant defined as k = 2π/λ, representing 2π times the number of waves per unit length; SI unit is rad/m.
e.g. In y = 0.005 sin(80.0x – 3.0t), k = 80.0 rad/m.
Period (T)
The time taken for an element of the medium to complete one full oscillation; T = 2π/ω.
e.g. For ω = 3.0 rad/s, T = 2π/3.0 = 2.09 s.
Angular frequency (ω)
The rate of change of phase per unit time, given by ω = 2π/T; SI unit is rad/s.
e.g. In y = 0.005 sin(80.0x – 3.0t), ω = 3.0 rad/s.
Frequency (ν)
The number of oscillations per second; ν = 1/T = ω/(2π); measured in hertz (Hz).
e.g. For T = 2.09 s, ν = 0.48 Hz.
Speed of a travelling wave (v)
The speed at which a fixed phase point (e.g., a crest) moves; v = ω/k = λν.
e.g. For a wave with λ = 7.85 cm and ν = 0.48 Hz, v = 3.77 cm/s.
Principle of superposition of waves
When two or more waves overlap in a medium, the resultant displacement is the algebraic sum of the displacements due to each individual wave.
e.g. Two pulses of equal and opposite shapes crossing each other cancel out at the point of overlap.
Constructive interference
When two waves of equal amplitude and wavelength are in phase (φ = 0), the resultant wave has amplitude 2a, the largest possible value.
e.g. y = a sin(kx – ωt) + a sin(kx – ωt) = 2a sin(kx – ωt).
Destructive interference
When two waves of equal amplitude and wavelength are completely out of phase (φ = π), the resultant displacement is zero everywhere at all times.
e.g. y = a sin(kx – ωt) + a sin(kx – ωt + π) = 0.
Reflection of waves
When a wave meets a boundary, it is reflected; at a rigid boundary, the reflected wave suffers a phase change of π (180°), while at an open boundary, no phase change occurs.
e.g. A pulse on a string fixed at a rigid wall reflects with a phase change of π.
Standing waves (Stationary waves)
A wave pattern formed by the superposition of two identical waves travelling in opposite directions, where the amplitude varies from point to point and the pattern does not move.
e.g. y(x,t) = 2a sin(kx) cos(ωt) from y1 = a sin(kx – ωt) and y2 = a sin(kx + ωt).
Node
A point in a standing wave where the amplitude is zero (no motion); nodes are separated by λ/2.
e.g. In a string fixed at both ends, the fixed ends are nodes.
Antinode
A point in a standing wave where the amplitude is maximum; antinodes are separated by λ/2.
e.g. In a string fixed at both ends, the midpoint is an antinode for the fundamental mode.
Normal modes
The set of natural frequencies at which a system can oscillate, determined by boundary conditions; for a string fixed at both ends, ν_n = nv/(2L).
e.g. For a string of length L, the fundamental frequency is v/(2L), the second harmonic is v/L, etc.
Fundamental mode (First harmonic)
The lowest possible natural frequency of a system; for a string fixed at both ends, ν_1 = v/(2L).
e.g. A guitar string vibrating at its lowest pitch.
Harmonics
The integer multiples of the fundamental frequency; for a string fixed at both ends, ν_n = nν_1.
e.g. The second harmonic (n=2) has frequency 2ν_1.
Newton's formula for speed of sound
The speed of longitudinal waves in an ideal gas is given by v = √(P/ρ), assuming isothermal pressure variations.
e.g. For air at STP, this gives v ≈ 280 m/s, which is about 15% less than the experimental value.
Laplace correction
The modification to Newton's formula that accounts for adiabatic pressure variations, giving v = √(γP/ρ), where γ = C_p/C_v.
e.g. For air (γ = 7/5), v = √(7P/(5ρ)) gives 331.3 m/s at STP, matching experimental value.
Bulk modulus (B)
The elastic property that determines stress under compressional strain; B = –ΔP/(ΔV/V); used in the formula for speed of longitudinal waves in fluids.
e.g. For an ideal gas under adiabatic conditions, B_ad = γP.
Young's modulus (Y)
The relevant modulus for longitudinal waves in a solid bar where lateral expansion is negligible; v = √(Y/ρ).
e.g. Speed of longitudinal waves in a steel rod.
Linear mass density (μ)
The mass per unit length of a string; μ = m/L; used in the formula for speed of transverse waves on a string: v = √(T/μ).
e.g. For a steel wire of mass 5.0×10⁻³ kg and length 0.72 m, μ = 6.9×10⁻³ kg/m.
Formulas & Equations (14)
y(x,t) = a sin(kx – ωt + φ)
Reads as: y of x and t equals a times sine of k x minus omega t plus phi
Variables: y = displacement (m), a = amplitude (m), k = angular wave number (rad/m), x = position (m), ω = angular frequency (rad/s), t = time (s), φ = initial phase angle (rad)
Use when: Describing a sinusoidal progressive wave travelling in the positive x-direction.
⚠ Students often forget the initial phase φ or incorrectly use +ωt for a wave travelling in the positive x-direction.
y(x,t) = a sin(kx + ωt + φ)
Reads as: y of x and t equals a times sine of k x plus omega t plus phi
Variables: Same as above; the +ωt indicates wave travelling in the negative x-direction.
Use when: Describing a sinusoidal progressive wave travelling in the negative x-direction.
⚠ Confusing the sign of ωt with the direction of propagation.
k = 2π/λ
Reads as: k equals 2 pi over lambda
Variables: k = angular wave number (rad/m), λ = wavelength (m)
Use when: Relating angular wave number to wavelength.
⚠ Using λ = 2πk instead of the correct relation.
ω = 2π/T
Reads as: omega equals 2 pi over T
Variables: ω = angular frequency (rad/s), T = period (s)
Use when: Relating angular frequency to period.
⚠ Confusing ω with frequency ν; ω = 2πν, not ν.
ν = 1/T = ω/(2π)
Reads as: nu equals 1 over T equals omega over 2 pi
Variables: ν = frequency (Hz), T = period (s), ω = angular frequency (rad/s)
Use when: Calculating frequency from period or angular frequency.
⚠ Forgetting that ν is in hertz and ω in rad/s.
v = ω/k = λν
Reads as: v equals omega over k equals lambda nu
Variables: v = wave speed (m/s), ω = angular frequency (rad/s), k = angular wave number (rad/m), λ = wavelength (m), ν = frequency (Hz)
Use when: Calculating the speed of a progressive wave from its frequency and wavelength.
⚠ Using v = λ/T incorrectly; v = λν is correct, and ν = 1/T.
v = √(T/μ)
Reads as: v equals the square root of T over mu
Variables: v = speed of transverse wave on a string (m/s), T = tension in the string (N), μ = linear mass density (kg/m)
Use when: Calculating the speed of transverse waves on a stretched string.
⚠ Using mass instead of linear mass density; μ = m/L, not just m.
v = √(B/ρ)
Reads as: v equals the square root of B over rho
Variables: v = speed of longitudinal wave in a fluid (m/s), B = bulk modulus (Pa), ρ = density (kg/m³)
Use when: Calculating the speed of sound in a fluid (liquid or gas).
⚠ Using Young's modulus instead of bulk modulus for fluids.
v = √(Y/ρ)
Reads as: v equals the square root of Y over rho
Variables: v = speed of longitudinal wave in a solid bar (m/s), Y = Young's modulus (Pa), ρ = density (kg/m³)
Use when: Calculating the speed of longitudinal waves in a solid bar.
⚠ Using bulk modulus instead of Young's modulus for a solid bar.
v = √(P/ρ) (Newton's formula)
Reads as: v equals the square root of P over rho
Variables: v = speed of sound in an ideal gas (m/s), P = pressure (Pa), ρ = density (kg/m³)
Use when: Calculating the speed of sound in an ideal gas under isothermal conditions.
⚠ Using this formula without Laplace correction; it gives a value about 15% lower than experimental.
v = √(γP/ρ) (Laplace correction)
Reads as: v equals the square root of gamma P over rho
Variables: v = speed of sound in an ideal gas (m/s), γ = ratio of specific heats (C_p/C_v), P = pressure (Pa), ρ = density (kg/m³)
Use when: Calculating the speed of sound in an ideal gas under adiabatic conditions (correct formula).
⚠ Forgetting to use γ = 7/5 for diatomic gases like air.
y(x,t) = 2a sin(kx) cos(ωt)
Reads as: y of x and t equals 2 a times sine of k x times cosine of omega t
Variables: y = displacement (m), a = amplitude of each travelling wave (m), k = angular wave number (rad/m), x = position (m), ω = angular frequency (rad/s), t = time (s)
Use when: Describing a standing wave formed by superposition of two identical waves travelling in opposite directions.
⚠ Thinking this represents a travelling wave; it is a stationary wave with nodes and antinodes.
λ_n = 2L/n, ν_n = nv/(2L) for string fixed at both ends
Reads as: lambda n equals 2 L over n, nu n equals n v over 2 L
Variables: λ_n = wavelength of nth harmonic (m), L = length of string (m), n = harmonic number (1,2,3,...), ν_n = frequency of nth harmonic (Hz), v = wave speed (m/s)
Use when: Finding the allowed wavelengths and frequencies (normal modes) for a string fixed at both ends.
⚠ Using n starting from 0; n = 1 is the fundamental mode.
λ_n = 4L/(2n+1), ν_n = (2n+1)v/(4L) for pipe closed at one end
Reads as: lambda n equals 4 L over (2 n plus 1), nu n equals (2 n plus 1) v over 4 L
Variables: λ_n = wavelength of nth mode (m), L = length of pipe (m), n = 0,1,2,3,..., ν_n = frequency (Hz), v = speed of sound (m/s)
Use when: Finding the allowed wavelengths and frequencies for an air column with one end closed and the other open.
⚠ Using n starting from 1; the fundamental corresponds to n = 0.
How It Works
Propagation of a disturbance in a medium
  1. A disturbance (e.g., a jerk or push) is applied to one part of the medium.
  2. Elastic forces between constituents (e.g., springs connecting atoms) cause the disturbance to be transmitted to neighbouring parts.
  3. Each constituent oscillates about its equilibrium position, but the disturbance moves forward.
  4. The wave pattern travels without net motion of the medium as a whole.
💡 The medium does not flow with the wave; only the disturbance propagates.
Determining wave speed from phase
  1. Fix attention on a point of constant phase, e.g., a crest, where kx – ωt = constant.
  2. Differentiate with respect to time: k(dx/dt) – ω = 0.
  3. Solve for speed: v = dx/dt = ω/k.
  4. Relate to wavelength and frequency: v = λν.
💡 The speed v is the speed of a fixed phase point, not the speed of the particles of the medium.
Superposition of two harmonic waves
  1. Consider two waves y1 = a sin(kx – ωt) and y2 = a sin(kx – ωt + φ).
  2. Add them using the principle of superposition: y = y1 + y2.
  3. Use trigonometric identity: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2).
  4. Obtain y = 2a cos(φ/2) sin(kx – ωt + φ/2).
  5. The amplitude is A = 2a cos(φ/2).
💡 Constructive interference occurs when φ = 0 (amplitude 2a); destructive when φ = π (amplitude 0).
Reflection of a wave at a boundary
  1. A travelling wave or pulse meets a boundary.
  2. If the boundary is rigid, the reflected wave suffers a phase change of π (180°).
  3. If the boundary is open (free to move), the reflected wave has no phase change.
  4. At a rigid boundary, the net displacement is always zero; at an open boundary, the net displacement is twice the amplitude.
💡 Rigid boundary → phase change of π; open boundary → no phase change.
Formation of standing waves
  1. A wave travelling in the positive x-direction (y1 = a sin(kx – ωt)) is reflected at a boundary.
  2. The reflected wave travels in the negative x-direction (y2 = a sin(kx + ωt)).
  3. Superpose the two waves: y = y1 + y2 = 2a sin(kx) cos(ωt).
  4. The amplitude 2a sin(kx) varies with position, creating nodes (zero amplitude) and antinodes (maximum amplitude).
  5. The pattern does not move; it is a stationary wave.
💡 Standing waves have nodes and antinodes; the distance between successive nodes or antinodes is λ/2.
Determining normal modes of a string fixed at both ends
  1. Apply boundary condition: displacement is zero at x = 0 and x = L (nodes).
  2. From standing wave equation, nodes occur where sin(kx) = 0, i.e., kx = nπ.
  3. At x = L, kL = nπ, so k = nπ/L.
  4. Since k = 2π/λ, we get λ = 2L/n.
  5. Using v = λν, the frequency is ν_n = nv/(2L) for n = 1,2,3,...
💡 The fundamental frequency (n=1) is v/(2L); higher harmonics are integer multiples.
Determining normal modes of an air column with one end closed
  1. Apply boundary conditions: closed end (x=0) is a node; open end (x=L) is an antinode.
  2. Node condition: sin(kx) = 0 at x=0 is automatically satisfied.
  3. Antinode condition: |sin(kx)| = 1 at x=L, so kL = (n + 1/2)π.
  4. Thus k = (2n+1)π/(2L), and λ = 4L/(2n+1).
  5. Frequency: ν_n = (2n+1)v/(4L) for n = 0,1,2,3,...
💡 Only odd harmonics are present; fundamental frequency is v/(4L).
Exam Traps — Don't Make These Mistakes
⚠ Confusing wave speed with particle speed
✗ The speed of the wave is the speed at which the particles of the medium move.
→ The wave speed is the speed at which the disturbance propagates, not the speed of the oscillating particles. Particles oscillate about their mean position with a different speed (e.g., maximum particle speed = aω).
✓ The wave speed v = ω/k is the speed at which a fixed phase point (like a crest) moves; the particles of the medium oscillate with simple harmonic motion and do not travel with the wave.
⚠ Using Newton's formula without Laplace correction
✗ The speed of sound in air at STP is v = √(P/ρ) = 280 m/s.
→ Newton assumed isothermal pressure variations, but sound propagation is adiabatic because the compressions and rarefactions occur too rapidly for heat exchange. The correct formula includes the adiabatic index γ.
✓ The speed of sound in air is given by v = √(γP/ρ) with γ = 7/5, giving 331.3 m/s at STP, which matches experimental value.
⚠ Misidentifying nodes and antinodes in standing waves
✗ In a standing wave, the points of maximum displacement are nodes, and points of zero displacement are antinodes.
→ This is the opposite of the correct definitions. Nodes are points of zero displacement (no motion), and antinodes are points of maximum displacement.
✓ Nodes are points where the amplitude is zero (no motion), and antinodes are points where the amplitude is maximum.
⚠ Incorrect sign in wave equation for direction of propagation
✗ y = a sin(kx + ωt) represents a wave travelling in the positive x-direction.
→ For a wave travelling in the positive x-direction, the phase must be of the form (kx – ωt). The plus sign (kx + ωt) indicates a wave travelling in the negative x-direction.
✓ A wave travelling in the positive x-direction is described by y = a sin(kx – ωt); a wave travelling in the negative x-direction is described by y = a sin(kx + ωt).
⚠ Confusing harmonics for open and closed pipes
✗ For a pipe closed at one end, the frequencies are ν_n = nv/(2L) for n = 1,2,3,...
→ That formula is for a string fixed at both ends or a pipe open at both ends. For a pipe closed at one end, only odd harmonics are present: ν_n = (2n+1)v/(4L) for n = 0,1,2,...
✓ For a pipe closed at one end, the fundamental frequency is v/(4L), and the allowed frequencies are odd multiples of the fundamental: v/(4L), 3v/(4L), 5v/(4L), etc.
⚠ Forgetting that transverse waves cannot propagate in fluids
✗ Sound waves in air are transverse waves.
→ Sound waves in air are longitudinal waves because fluids cannot sustain shearing stress; they can only sustain compressive strain. Transverse waves require a medium that can sustain shearing stress, such as solids.
✓ Sound waves in air are longitudinal waves; transverse waves can propagate only in solids (and on the surface of liquids, but not in the bulk of fluids).
Compare & Contrast
Direction of particle oscillation relative to wave propagation
Transverse wave
Longitudinal wave
In transverse waves, particles oscillate perpendicular to the direction of wave propagation; in longitudinal waves, particles oscillate parallel to the direction of wave propagation.
Amplitude of oscillation in a standing wave
Node
Antinode
A node is a point of zero amplitude (no motion), while an antinode is a point of maximum amplitude.
Assumption about thermodynamic process during wave propagation
Newton's formula for speed of sound
Laplace correction for speed of sound
Newton assumed isothermal pressure variations (v = √(P/ρ)), giving a value 15% lower than experimental; Laplace corrected it to adiabatic variations (v = √(γP/ρ)), which matches experimental values.
Allowed frequencies (normal modes)
String fixed at both ends
Pipe closed at one end
For a string fixed at both ends, frequencies are ν_n = nv/(2L) for n = 1,2,3,... (all harmonics); for a pipe closed at one end, frequencies are ν_n = (2n+1)v/(4L) for n = 0,1,2,... (only odd harmonics).
Exam Focus
Marks pattern: Typically 1-2 MCQs (1 mark each), 1 short answer question (2-3 marks) on definitions or derivations, 1 long answer question (5 marks) on standing waves or speed of sound, totalling around 8-10 marks.
Most asked: Speed of a travelling wave (v = ω/k = λν) · Speed of transverse waves on a string (v = √(T/μ)) · Newton's formula and Laplace correction for speed of sound · Principle of superposition and interference (constructive/destructive) · Standing waves in strings and pipes (normal modes, harmonics) · Reflection of waves (phase change at rigid/open boundaries)
Don't skip: Derivation of the speed of transverse waves on a string using dimensional analysis or Newton's laws.; Derivation of standing waves and calculation of normal modes for a string fixed at both ends.; Numerical problems on wave speed, frequency, wavelength, and harmonics.; Comparison of Newton's and Laplace's formulas for speed of sound.
One-Line Revision (read aloud before the exam)
  1. A wave is a disturbance that propagates through a medium without net transfer of matter.
  2. Mechanical waves require a medium; electromagnetic waves do not.
  3. In transverse waves, particles oscillate perpendicular to wave direction; in longitudinal waves, they oscillate parallel.
  4. The displacement of a progressive wave is given by y(x,t) = a sin(kx – ωt + φ).
  5. Angular wave number k = 2π/λ, angular frequency ω = 2π/T, and frequency ν = 1/T.
  6. Wave speed v = ω/k = λν.
  7. Speed of transverse waves on a string: v = √(T/μ).
  8. Speed of longitudinal waves in a fluid: v = √(B/ρ).
  9. Newton's formula for speed of sound: v = √(P/ρ); Laplace correction: v = √(γP/ρ).
  10. The principle of superposition states that net displacement is the algebraic sum of individual displacements.
  11. Constructive interference occurs when waves are in phase (φ=0); destructive when out of phase (φ=π).
  12. At a rigid boundary, reflected wave suffers a phase change of π; at an open boundary, no phase change.
  13. Standing waves are formed by superposition of two identical waves travelling in opposite directions.
  14. Nodes are points of zero displacement; antinodes are points of maximum displacement in standing waves.
  15. For a string fixed at both ends, allowed wavelengths are λ_n = 2L/n and frequencies ν_n = nv/(2L).
  16. For a pipe closed at one end, only odd harmonics are present: ν_n = (2n+1)v/(4L).
  17. The fundamental frequency of a string is v/(2L); of a closed pipe is v/(4L).
  18. Transverse waves can propagate only in solids; longitudinal waves can propagate in solids, liquids, and gases.
  19. The speed of a mechanical wave depends on the elastic and inertial properties of the medium, not on frequency or wavelength.
  20. In a standing wave, the amplitude varies with position but each element oscillates with the same angular frequency ω.

MCQs — 10 Questions with Answers

Q1. A cork piece floats on water. When a wave passes through the water, the cork:

  • A. moves outward with the wave away from the disturbance centre
  • B. oscillates up and down but remains approximately at the same horizontal position ✓
  • C. sinks deeper into the water as the wave passes
  • D. rotates in circles following the wave motion

Answer: B — Waves transport energy and pattern, not matter; the cork oscillates with local particle motion but the water mass does not flow outward, so the cork stays at nearly the same position.

Q2. Which of the following is NOT a property that distinguishes transverse from longitudinal waves?

  • A. Direction of particle oscillation relative to wave propagation
  • B. Type of medium through which the wave travels ✓
  • C. Presence of crests and troughs in the wave pattern
  • D. Whether compressions and rarefactions occur

Answer: B — Both transverse and longitudinal waves can travel through the same medium (e.g., seismic P and S waves in Earth's crust); the key difference is particle motion direction, not medium type.

Q3. A wave travels along a stretched string with wavelength 0.5 m and frequency 4 Hz. The wave speed is:

  • A. 0.125 m/s
  • B. 2 m/s ✓
  • C. 4 m/s
  • D. 8 m/s

Answer: B — Using v = λf = 0.5 m × 4 Hz = 2 m/s.

Q4. Why do mechanical waves require a material medium for propagation?

  • A. Because the wave must carry matter with it as it travels
  • B. Because wave propagation depends on oscillations of particles and elastic restoring forces present in the medium ✓
  • C. Because the speed of mechanical waves must always be less than the speed of light
  • D. Because mechanical waves always travel slower in vacuum than in any medium

Answer: B — Mechanical waves arise from particle oscillations coupled by elastic forces; these properties are only present in material media, not in vacuum.

Q5. Sound is a longitudinal wave in air. Which statement correctly describes the particle motion?

  • A. Particles move perpendicular to the direction of sound propagation
  • B. Particles oscillate back and forth parallel to the direction of sound propagation, creating alternating compressions and rarefactions ✓
  • C. Particles move in circles as the sound wave passes
  • D. Particles permanently move outward from the source

Answer: B — In longitudinal waves like sound, particles oscillate along the direction of wave travel, causing local density and pressure variations (compressions and rarefactions).

Q6. A spring is stretched and then released, creating a disturbance that travels along a chain of connected springs. This happens because:

  • A. the springs physically carry the disturbance as they move forward
  • B. elastic forces in each spring pull or push the neighbouring spring out of equilibrium, and inertia causes it to overshoot, propagating the disturbance ✓
  • C. the material of the springs flows from one end to the other
  • D. gravity pulls all springs downward equally, causing synchronized motion

Answer: B — Elasticity provides restoring forces and inertia allows particles to overshoot equilibrium; this back-and-forth coupling transfers the disturbance along the chain without bulk matter flow.

Q7. If the frequency of a wave is doubled while the wave speed remains constant, the wavelength:

  • A. doubles
  • B. halves ✓
  • C. remains the same
  • D. increases fourfold

Answer: B — From v = λf, if v is constant and f doubles, then λ must halve: λ_new = v/(2f) = λ_old/2.

Q8. Which pair correctly matches wave type with an example?

  • A. Transverse wave: sound in air; Longitudinal wave: ripples on water surface
  • B. Transverse wave: seismic S-waves; Longitudinal wave: seismic P-waves ✓
  • C. Transverse wave: compression in a spring; Longitudinal wave: vibration of a guitar string
  • D. Transverse wave: density changes in air; Longitudinal wave: light in vacuum

Answer: B — Seismic S-waves cause particle motion perpendicular to wave direction (transverse); P-waves cause parallel motion (longitudinal), creating compressions and rarefactions in Earth's layers.

Q9. A student observes waves on a pond created by dropping stones. She notes that when the distance between consecutive wave crests increases, the frequency of waves (as measured by a fixed observer) stays the same. Which conclusion is correct? (Assume waves are not being absorbed.)

  • A. The wave speed has decreased because wavelength increased while frequency stayed constant
  • B. The wave speed has increased because wavelength increased while frequency stayed constant ✓
  • C. The wave speed has remained constant even though wavelength increased
  • D. The amplitude of the wave has increased, so the speed must increase

Answer: B — Using v = λf: if λ increases and f remains constant, then v must increase proportionally; the wave speed depends on the medium properties, which can change (e.g., depth of pond affecting water wave speed).

Q10. A sinusoidal wave is described by y = 0.02 sin(50x − 400t), where y and x are in metres and t is in seconds. The wavelength and frequency of this wave are approximately:

  • A. λ = 0.126 m, f = 63.7 Hz ✓
  • B. λ = 0.126 m, f = 400 Hz
  • C. λ = 25 m, f = 6.37 Hz
  • D. λ = 50 m, f = 400 Hz

Answer: A — From y = A sin(kx − ωt), k = 50 rad/m and ω = 400 rad/s. λ = 2π/k = 2π/50 ≈ 0.126 m; f = ω/(2π) = 400/(2π) ≈ 63.7 Hz.

Flashcards

What is a mechanical wave?

A disturbance that propagates through a medium by oscillation of particles, requiring elasticity and inertia of the medium.

How do you distinguish transverse from longitudinal waves?

In transverse waves particles oscillate perpendicular to wave propagation; in longitudinal waves they oscillate parallel to it.

Why does a cork piece bob up-down but not move away when a wave passes?

Waves transport energy and pattern, not matter; the cork oscillates with the medium locally but the medium itself does not flow outward.

Give one example each of transverse and longitudinal waves.

Transverse: waves on a string or water surface; Longitudinal: sound waves in air or compression waves in a spring.

What is the relationship between wave speed v, wavelength λ, and frequency f?

v = λf, where v is speed, λ is wavelength, and f is frequency.

Why do mechanical waves require a medium?

They depend on oscillations of particles and elastic restoring forces, which only exist in a material medium.

What is the speed of electromagnetic waves in vacuum?

c = 299,792,458 m/s ≈ 3 × 10⁸ m/s, the same for all EM waves.

How does the disturbance propagate in connected springs?

One spring compresses or stretches, pulling the next spring out of equilibrium, which then pulls the next, creating wave propagation along the chain.

Describe how sound propagates in air using compression and rarefaction.

A compressed air region pushes molecules outward creating compression in the next region and rarefaction in its own; this alternating pattern moves as sound.

What role do elastic forces play in wave propagation?

Elastic forces provide restoring forces that accelerate particles back toward equilibrium after disturbance, enabling energy transfer through the medium.

Important Board Questions

Define a mechanical wave and explain why it requires a medium for propagation. Give one example. [2 marks]

State that a wave is a disturbance carrying energy without bulk matter flow; explain that elasticity and inertia of particles (only in medium) enable propagation; example: sound in air, water ripples, or string waves.

Distinguish between transverse and longitudinal waves with one example each. Why are water surface waves considered transverse while sound waves in air are longitudinal? [5 marks]

Transverse: particle motion ⊥ to wave direction (e.g., string); Longitudinal: particle motion ∥ to wave direction (e.g., spring compression). For water and sound: explain particle motion relative to disturbance propagation in each case; water surface particles move up-down as waves move horizontally; air molecules oscillate back-forth along sound direction creating compression-rarefaction.

A wave with frequency 10 Hz travels along a string with speed 20 m/s. (a) Calculate the wavelength. (b) If the frequency is doubled to 20 Hz while the speed remains 20 m/s, how does the wavelength change? (c) Explain this relationship using the equation v = λf and discuss the physical meaning of this result. [6 marks]

(a) Use v = λf directly to find λ = 2 m. (b) New λ = v/(2f) = 1 m; wavelength halves. (c) Show that v depends on medium properties (tension, mass density) which do not change; when f increases, λ must decrease proportionally to keep v constant; this reflects that a given medium cannot transmit all frequencies at the same speed in all scenarios—connect to dispersive media if applicable.

Assertion-Reason Questions for Waves

Q1. Assertion (A):

Water waves in a pond are mechanical waves.

Reason (R):

Water waves require a material medium for propagation and involve elastic forces binding water constituents together.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains why water waves are mechanical. Mechanical waves depend on medium presence and elastic properties, which are both characteristics of water waves.

Q2. Assertion (A):

Cork pieces floating on a disturbed water surface move away from the center of disturbance.

Reason (R):

Waves transport the actual mass of the medium from one location to another.

Answer: D — A is false but R is true — Both the assertion and reason are false. Cork pieces oscillate up and down but remain at approximately the same location. Waves transport a pattern of disturbance and energy, not the bulk motion of the medium. This is a critical misconception the chapter explicitly addresses.

Q3. Assertion (A):

Light waves can travel through a perfect vacuum, while sound waves cannot.

Reason (R):

Light is an electromagnetic wave; sound is a mechanical wave that requires a medium with elastic properties.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true, and the reason correctly explains the assertion. The chapter clearly distinguishes electromagnetic waves (like light) that can propagate through vacuum from mechanical waves (like sound) that require a material medium.

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Case-Based Questions (CBQ) for Waves

Case Study:

During a seismic event in India, scientists monitor both primary (P) waves and secondary (S) waves from the epicenter. P-waves are longitudinal waves traveling at approximately 6 km/s through the earth's crust, while S-waves are transverse waves traveling at approximately 3.5 km/s. A seismograph station records the P-wave arriving at 8:15:30 AM and the S-wave arriving at 8:15:45 AM. The difference in arrival times helps determine the epicenter distance. Additionally, the ground motion recorded shows vertical oscillations for P-waves and horizontal oscillations for S-waves.

Q1. What is the reason for the different arrival times of P-waves and S-waves at the same location?

  • A. P-waves and S-waves travel at different speeds due to different properties of wave propagation in the medium ✓
  • B. S-waves are weaker and lose energy faster than P-waves
  • C. P-waves and S-waves originate from different points in the epicenter
  • D. The distance from epicenter increases with time, so later waves travel farther

Answer: A

Q2. Based on the passage, which statement correctly describes the characteristic motion of ground particles for each wave type?

  • A. P-waves cause vertical oscillations; S-waves cause vertical oscillations
  • B. P-waves cause horizontal oscillations; S-waves cause vertical oscillations
  • C. P-waves cause vertical oscillations; S-waves cause horizontal oscillations ✓
  • D. Both P-waves and S-waves cause oscillations in all directions equally

Answer: C

Q3. If the time difference between P-wave and S-wave arrival is 15 seconds, approximately how far is the seismograph station from the epicenter?

  • A. 22.5 km
  • B. 52.5 km
  • C. 90 km
  • D. 157.5 km ✓

Answer: D

Q4. Why are seismologists able to use both P-waves and S-waves to detect earthquakes, whereas some detection devices might use only one type?

  • A. P-waves and S-waves have different frequencies that allow independent verification
  • B. S-waves are more damaging and arrive second, giving warning time
  • C. The two different wave types provide complementary information about motion direction and enable more accurate epicenter location ✓
  • D. P-waves and S-waves interact destructively to amplify signals

Answer: C

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Class 11 Physics Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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