Class 11 Physics Notes Chapter 17 (Chapter 17) – Examplar Problems (English) Book

Alright class, let's begin our detailed study of Chapter 17, 'Waves', from the NCERT Physics Exemplar for Class 11. This chapter is crucial for understanding many physical phenomena and frequently appears in various government examinations. Pay close attention to the concepts, formulas, and their applications.

Chapter 17: Waves - Detailed Notes

1. Introduction to Waves:

• A wave is a disturbance that propagates through a medium or vacuum, transferring energy and momentum without the bulk transport of matter.
• Examples: Ripples on water, sound waves, light waves, waves on a string.

2. Types of Waves:

• Based on Medium Requirement:
  • Mechanical Waves: Require a material medium for propagation (e.g., sound waves, waves on a string, water waves). Properties depend on the medium's elasticity and inertia.
  • Non-mechanical Waves (Electromagnetic Waves): Do not require a medium; can travel through vacuum (e.g., light, radio waves, X-rays).
• Based on Particle Vibration Direction:
  • Transverse Waves: Particles of the medium oscillate perpendicular to the direction of wave propagation (e.g., waves on a stretched string, light waves). Can propagate through solids and on the surface of liquids. Characterized by crests (maximum positive displacement) and troughs (maximum negative displacement).
  • Longitudinal Waves: Particles of the medium oscillate parallel to the direction of wave propagation (e.g., sound waves in air, waves in a spring compressed and released along its length). Propagate through solids, liquids, and gases. Characterized by compressions (regions of high density/pressure) and rarefactions (regions of low density/pressure).

3. Key Wave Parameters & Terminology:

• Amplitude (A): Maximum displacement of a particle from its mean (equilibrium) position. Unit: meter (m).
• Wavelength (λ): Minimum distance between two consecutive points in the same phase of vibration (e.g., distance between two consecutive crests or troughs). Unit: meter (m).
• Time Period (T): Time taken for one complete oscillation of a particle or time taken for the wave to travel a distance equal to one wavelength. Unit: second (s).
• Frequency (f or ν): Number of complete oscillations per second by a particle or number of waves passing a point per second. Unit: Hertz (Hz) or s⁻¹. Relation: f = 1/T.
• Angular Frequency (ω): Rate of change of phase. Unit: radians per second (rad/s). Relation: ω = 2πf = 2π/T.
• Wave Number (k): Spatial frequency or number of wavelengths per unit distance multiplied by 2π. Unit: radians per meter (rad/m). Relation: k = 2π/λ.
• Wave Speed (v): Distance covered by the wave per unit time. Unit: meter per second (m/s). Relation: v = λ/T = fλ = ω/k.

4. Mathematical Description of a Progressive Wave (Harmonic Wave):

• A wave traveling in the positive x-direction can be represented as:
  y(x, t) = A sin(kx - ωt + φ)
  or y(x, t) = A cos(kx - ωt + φ)
• A wave traveling in the negative x-direction can be represented as:
  y(x, t) = A sin(kx + ωt + φ)
  or y(x, t) = A cos(kx + ωt + φ)
• Where:
  • y(x, t) is the displacement of the particle at position x and time t.
  • A is the amplitude.
  • k is the wave number.
  • ω is the angular frequency.
  • (kx ± ωt + φ) is the phase of the wave.
  • φ is the initial phase angle (phase at x=0 and t=0).
• Particle Velocity (vp): vp = ∂y/∂t = -Aω cos(kx - ωt + φ) (for sine function form)
• Particle Acceleration (ap): ap = ∂²y/∂t² = -Aω² sin(kx - ωt + φ) = -ω²y (Simple Harmonic Motion)
• Relationship between Wave Velocity and Particle Velocity: vp = -v × (slope of wave curve) i.e., ∂y/∂t = -v (∂y/∂x)

5. Speed of Waves:

• Speed of Transverse Wave on a Stretched String:
  v = √(T/μ)
  Where T is the tension in the string and μ is the linear mass density (mass per unit length) of the string.
• Speed of Longitudinal Wave (e.g., Sound):
  v = √(E/ρ)
  Where E is the appropriate modulus of elasticity of the medium and ρ is the density of the medium.
  • In Solids: v = √(Y/ρ) (for thin rods, Y = Young's Modulus)
  • In Liquids: v = √(B/ρ) (B = Bulk Modulus)
  • In Gases:
    • Newton's Formula (Isothermal assumption - incorrect): v = √(P/ρ) (P = Pressure). Gave a lower value for air.
    • Laplace's Correction (Adiabatic assumption - correct): Sound propagation is fast, so heat exchange is negligible (adiabatic). v = √(γP/ρ), where γ = Cp/Cv is the adiabatic index.
    • Using Ideal Gas Law (P/ρ = RT/M): v = √(γRT/M), where R is the universal gas constant, T is the absolute temperature, and M is the molar mass.
• Factors affecting speed of sound in gas:
  • Temperature: v ∝ √T (in Kelvin).
  • Molar Mass: v ∝ 1/√M. Lighter gases transmit sound faster.
  • Humidity: Air with water vapour is less dense than dry air (M_water < M_air), so sound travels faster in humid air. v_moist > v_dry.
  • Pressure: No effect at constant temperature (as P/ρ remains constant).
  • Wind: Speed adds or subtracts depending on direction relative to sound propagation.
  • Amplitude/Frequency: Speed is independent of amplitude and frequency (for moderate levels).

6. Principle of Superposition:

• When two or more waves travel through a medium simultaneously, the resultant displacement of any particle is the vector sum of the displacements produced by the individual waves.
  y_resultant = y₁ + y₂ + y₃ + ...
• This principle leads to phenomena like interference and beats.

7. Interference of Waves:

• Modification in the distribution of energy when two or more coherent waves (same frequency, constant phase difference) superimpose.
• Let y₁ = A₁ sin(ωt) and y₂ = A₂ sin(ωt + φ).
• Resultant wave: y = y₁ + y₂ = A_res sin(ωt + δ)
• Resultant Amplitude: A_res = √(A₁² + A₂² + 2A₁A₂ cos φ)
• Resultant Intensity: I_res = I₁ + I₂ + 2√(I₁I₂) cos φ (Since Intensity I ∝ A²)
• Constructive Interference: Amplitude and intensity are maximum.
  • Condition: Phase difference φ = 2nπ (where n = 0, 1, 2, ...)
  • Path difference Δx = nλ
  • A_max = A₁ + A₂
  • I_max = (√I₁ + √I₂)²
• Destructive Interference: Amplitude and intensity are minimum.
  • Condition: Phase difference φ = (2n + 1)π (where n = 0, 1, 2, ...)
  • Path difference Δx = (2n + 1)λ/2
  • A_min = |A₁ - A₂|
  • I_min = (√I₁ - √I₂)²
• If A₁ = A₂ = A (and I₁ = I₂ = I₀):
  • A_res = 2A |cos(φ/2)|
  • I_res = 4I₀ cos²(φ/2)
  • A_max = 2A, I_max = 4I₀
  • A_min = 0, I_min = 0

8. Reflection of Waves:

• When a wave encounters a boundary between two media, a part of it is reflected back into the original medium.
• Reflection from a Rigid Boundary (Fixed End): The reflected wave undergoes a phase change of π (180°). A crest reflects as a trough and vice versa.
• Reflection from a Free Boundary (Open End): The reflected wave undergoes no phase change (0°). A crest reflects as a crest.

9. Standing Waves (Stationary Waves):

• Formed by the superposition of two identical waves (same amplitude, frequency, wavelength) traveling in opposite directions (usually an incident wave and its reflection).
• Energy is confined within segments; it does not propagate.
• Nodes: Points where the amplitude of vibration is always zero. Displacement is always zero. Distance between consecutive nodes = λ/2.
• Antinodes: Points where the amplitude of vibration is maximum (2A). Distance between consecutive antinodes = λ/2.
• Distance between a node and the adjacent antinode = λ/4.
• All particles between two consecutive nodes vibrate in phase, but particles in adjacent segments vibrate in opposite phases.
• Equation: If y₁ = A sin(kx - ωt) and y₂ = A sin(kx + ωt) (reflection from fixed end involves phase change, implicitly handled by superposition setup or explicit phase addition), the resultant is:
  y(x, t) = y₁ + y₂ = [2A sin(kx)] cos(ωt)
  • Amplitude at position x: A(x) = 2A sin(kx). This amplitude varies with x.
  • Nodes occur where A(x) = 0 => sin(kx) = 0 => kx = nπ => x = nλ/2 (n=0, 1, 2...).
  • Antinodes occur where A(x) is max => |sin(kx)| = 1 => kx = (n + 1/2)π => x = (2n+1)λ/4 (n=0, 1, 2...).

10. Standing Waves in Specific Systems:

• String Fixed at Both Ends (Length L):
  • Ends must be nodes. L = n(λ/2) where n = 1, 2, 3... (number of loops)
  • Allowed wavelengths: λn = 2L/n
  • Allowed frequencies (Harmonics): fn = v/λn = nv/(2L) = n f₁
  • f₁ = v/(2L) is the fundamental frequency or first harmonic (n=1).
  • f₂ = 2f₁ is the second harmonic or first overtone.
  • f₃ = 3f₁ is the third harmonic or second overtone.
  • All harmonics are present.
• String Fixed at One End, Free at the Other (Length L):
  • Fixed end is a node, free end is an antinode. L = (2n-1)λ/4 where n = 1, 2, 3...
  • Allowed wavelengths: λn = 4L/(2n-1)
  • Allowed frequencies: fn = v/λn = (2n-1)v/(4L) = (2n-1) f₁
  • f₁ = v/(4L) is the fundamental frequency or first harmonic (n=1).
  • f₃ = 3f₁ is the third harmonic or first overtone (n=2).
  • f₅ = 5f₁ is the fifth harmonic or second overtone (n=3).
  • Only odd harmonics are present.
• Organ Pipes (Air Columns):
  • Closed Organ Pipe (Closed at one end, open at the other): Similar to string fixed at one end. Closed end is a displacement node (pressure antinode), open end is a displacement antinode (pressure node).
    • L = (2n-1)λ/4
    • Frequencies: fn = (2n-1)v/(4L) (Only odd harmonics)
    • f₁ = v/(4L) (Fundamental)
  • Open Organ Pipe (Open at both ends): Similar to string fixed at both ends. Both ends are displacement antinodes (pressure nodes).
    • L = n(λ/2)
    • Frequencies: fn = nv/(2L) (All harmonics)
    • f₁ = v/(2L) (Fundamental)
  • End Correction: The antinode at the open end is slightly outside the pipe. Effective length L_eff = L + e, where e ≈ 0.6r (r = radius of pipe). This needs to be considered for accurate calculations. For a pipe open at both ends, L_eff = L + 2e.

11. Beats:

• The periodic variation in the intensity of sound (waxing and waning) heard when two sound waves of slightly different frequencies superimpose.
• Caused by interference that alternates between constructive and destructive over time.
• Beat Frequency (f_beat): The number of intensity maxima (beats) heard per second.
  f_beat = |f₁ - f₂|
  Where f₁ and f₂ are the frequencies of the two sources.
• Maximum number of beats detectable by the human ear is around 10-15 Hz.

12. Doppler Effect:

• The apparent change in the frequency (and wavelength) of a wave perceived by an observer due to relative motion between the source of the wave and the observer.
• General Formula (Sound):
  f' = f₀ [ (v ± v₀) / (v ∓ v_s) ]
  Where:
  • f' = Apparent frequency heard by the observer
  • f₀ = Actual frequency emitted by the source
  • v = Speed of sound in the medium
  • v₀ = Speed of the observer
  • v_s = Speed of the source
• Sign Convention (Crucial!):
  • Assume direction from Observer (O) to Source (S) is positive.
  • Use +v₀ if observer moves towards the source; -v₀ if observer moves away.
  • Use -v_s if source moves towards the observer; +v_s if source moves away.
  • (Alternative convention: Assume direction from Source (S) to Observer (O) is positive. Then use +v₀ if O moves towards S, -v₀ if O moves away; +v_s if S moves towards O, -v_s if S moves away. The formula structure might change slightly: f' = f₀ [ (v ± v₀) / (v ± v_s) ] - check formula used with its convention). The first convention given above is common in NCERT.
• Effect of Medium Wind: If wind blows with speed w from S to O, replace v with (v+w). If wind blows from O to S, replace v with (v-w). The speeds v₀ and v_s are relative to the ground/medium.
• Special Cases:
  • Source moves towards stationary observer: f' = f₀ [ v / (v - v_s) ] (f' > f₀)
  • Source moves away from stationary observer: f' = f₀ [ v / (v + v_s) ] (f' < f₀)
  • Observer moves towards stationary source: f' = f₀ [ (v + v₀) / v ] (f' > f₀)
  • Observer moves away from stationary source: f' = f₀ [ (v - v₀) / v ] (f' < f₀)
• Doppler Effect in Light: Similar concept, but the formula is different due to relativity and independence of light speed from source/observer motion in vacuum. For low speeds (v << c), Δf/f ≈ ± v_relative / c.

Multiple Choice Questions (MCQs)

1. A wave travelling along the x-axis is described by the equation y(x, t) = 0.005 cos(αx - βt). If the wavelength and time period of the wave are 0.08 m and 2.0 s respectively, then α and β in appropriate units are:
   (a) α = 25.00 π, β = π
   (b) α = 0.08/π, β = 2.0/π
   (c) α = 0.04/π, β = 1.0/π
   (d) α = 12.50 π, β = π/2.0

2. The speed of sound in oxygen (O₂) at a certain temperature is 460 m/s. The speed of sound in helium (He) at the same temperature will be (assuming both gases to be ideal, Molar mass of O₂ = 32 g/mol, Molar mass of He = 4 g/mol):
   (a) 460 m/s
   (b) 500 m/s
   (c) 650 m/s
   (d) 1301 m/s (approx)

3. A string of length L is fixed at both ends. It is vibrating in its 3rd harmonic. The points along the string which have maximum amplitude are located at distances (from one end):
   (a) L/6, L/2, 5L/6
   (b) L/3, 2L/3
   (c) L/4, L/2, 3L/4
   (d) L/2

4. Two sound waves with wavelengths 5.0 m and 5.5 m respectively, each propagate in a gas with velocity 330 m/s. We expect the following number of beats per second:
   (a) 12
   (b) 0
   (c) 1
   (d) 6

5. When a wave reflects from a rigid boundary, the reflected wave has:
   (a) Same phase and amplitude as the incident wave.
   (b) A phase change of π and same amplitude.
   (c) Same phase but different amplitude.
   (d) A phase change of π/2 and same amplitude.

6. A closed organ pipe and an open organ pipe of same length produce 2 beats/sec while vibrating in their fundamental modes. The length of the open organ pipe is halved and that of closed pipe is doubled. The number of beats produced per second will be:
   (a) 2
   (b) 7
   (c) 0
   (d) 3.5

7. A train moving at a speed of 220 m/s towards a stationary object, emits a sound of frequency 1000 Hz. Some of the sound reaching the object gets reflected back to the train as an echo. The frequency of the echo as detected by the driver of the train is (Speed of sound in air is 330 m/s):
   (a) 3500 Hz
   (b) 4000 Hz
   (c) 5000 Hz
   (d) 6000 Hz

8. Which of the following statements is incorrect for a stationary wave?
   (a) Every particle has a fixed amplitude which is different from the amplitude of its nearest particle.
   (b) All the particles cross their mean position at the same time.
   (c) All the particles are oscillating with the same frequency.
   (d) Energy is uniformly distributed throughout the wave.

9. The principle of superposition applies to:
   (a) Only transverse waves
   (b) Only longitudinal waves
   (c) Both transverse and longitudinal waves
   (d) Only electromagnetic waves

10. A wave disturbance in a medium is described by y(x,t) = 0.02 cos(50πt + π/2) cos(10πx). Which of the following is correct?
    (a) This represents a travelling wave of frequency 25 Hz.
    (b) This represents a stationary wave with nodes at x = 0.05 m, 0.15 m, 0.25 m, ...
    (c) This represents a stationary wave with antinodes at x = 0, 0.1 m, 0.2 m, ...
    (d) The wavelength of the component waves is 0.1 m.

Answer Key for MCQs:

1. (a) [α = k = 2π/λ = 2π/0.08 = 25π; β = ω = 2π/T = 2π/2 = π]
2. (d) [v ∝ √(γ/M). Assuming γ is same (or considering monatomic He γ=5/3, diatomic O₂ γ=7/5 doesn't change the proportionality significantly for MCQs usually, but let's use the M dependence primarily: v_He / v_O₂ = √(M_O₂ / M_He) = √(32/4) = √8 = 2√2. v_He = 460 * 2√2 ≈ 460 * 2.828 ≈ 1301 m/s]
3. (a) [3rd harmonic means n=3. L = 3(λ/2) => λ = 2L/3. Antinodes are at x = λ/4, 3λ/4, 5λ/4... Substituting λ: x = (2L/3)/4 = L/6; x = 3(2L/3)/4 = L/2; x = 5(2L/3)/4 = 5L/6]
4. (d) [f₁ = v/λ₁ = 330/5.0 = 66 Hz. f₂ = v/λ₂ = 330/5.5 = 60 Hz. f_beat = |f₁ - f₂| = |66 - 60| = 6 Hz]
5. (b) [Standard result for reflection from a denser/rigid medium.]
6. (b) [Let initial lengths be L. f_open = v/2L, f_closed = v/4L. Given |f_open - f_closed| = |v/2L - v/4L| = |v/4L| = 2 Hz. New lengths: L'_open = L/2, L'_closed = 2L. New frequencies: f'_open = v/(2L'_open) = v/(2(L/2)) = v/L. f'_closed = v/(4L'_closed) = v/(4(2L)) = v/8L. New beat freq = |f'_open - f'_closed| = |v/L - v/8L| = |(7/8) (v/L)| = (7/2) * (v/4L) = (7/2) * 2 = 7 Hz.]
7. (c) [Step 1: Frequency received by stationary object (observer). Source moving towards. f_obj = f₀ [ v / (v - v_s) ] = 1000 [ 330 / (330 - 220) ] = 1000 [ 330 / 110 ] = 3000 Hz. Step 2: Object reflects this frequency (f_obj = 3000 Hz), acting as a stationary source. Train (observer) moves towards this source. f_echo = f_obj [ (v + v₀) / v ] = 3000 [ (330 + 220) / 330 ] = 3000 [ 550 / 330 ] = 3000 * (5/3) = 5000 Hz.]
8. (d) [Energy is not uniformly distributed; it's zero at nodes and maximum at antinodes. Energy is confined between nodes.]
9. (c) [The principle of superposition is fundamental to wave behaviour and applies generally.]
10. (b) [The equation is of the form y = A(x) cos(ωt + φ'), where A(x) = 0.02 cos(10πx). This is a stationary wave. ω = 50π => f = ω/2π = 25 Hz. The term cos(10πx) determines amplitude. Nodes occur when A(x) = 0 => cos(10πx) = 0. => 10πx = (n + 1/2)π => x = (n + 1/2)/10 = (2n+1)/20. For n=0, x=1/20=0.05m. For n=1, x=3/20=0.15m. For n=2, x=5/20=0.25m. So (b) is correct. Antinodes occur when |cos(10πx)| = 1 => 10πx = nπ => x = n/10 = 0, 0.1, 0.2.... Wavelength from k: The term cos(10πx) implies k = 10π. λ = 2π/k = 2π/(10π) = 1/5 = 0.2 m for the component waves.]

Make sure you understand the derivation of formulas and the conditions under which they apply. Practice problems from the Exemplar book itself, as they often test deeper conceptual understanding. Good luck with your preparation!