Class 11 Physics Notes Chapter 15 (Chapter 15) – Examplar Problems (English) Book

Examplar Problems (English)
Detailed Notes with MCQs of Chapter 15: Waves, from your NCERT Exemplar. This is a crucial chapter for many government exams, covering fundamental concepts of wave motion and sound. Pay close attention to the definitions, formulas, and the underlying principles.

Chapter 15: Waves - Detailed Notes for Exam Preparation

1. Introduction to Waves

  • Wave: A disturbance that propagates through space or a medium, transferring energy and momentum from one point to another without the bulk transport of the medium itself.
  • 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). They rely on the elastic properties and inertia of the medium.
      • Electromagnetic Waves: Do not require a material medium; can travel through vacuum (e.g., light waves, radio waves, X-rays).
    • Based on Direction of Particle Vibration:
      • Transverse Waves: Particles of the medium vibrate perpendicular to the direction of wave propagation.
        • Characterized by crests (maximum positive displacement) and troughs (maximum negative displacement).
        • Can propagate through solids and on the surface of liquids. Cannot propagate through ideal fluids (gases/liquids) as they lack shear strength.
        • Example: Waves on a stretched string, light waves.
      • Longitudinal Waves: Particles of the medium vibrate parallel to the direction of wave propagation.
        • Characterized by compressions (regions of high density/pressure) and rarefactions (regions of low density/pressure).
        • Can propagate through solids, liquids, and gases.
        • Example: Sound waves in air.

2. Key Wave Parameters & Terminology

  • Amplitude (A): Maximum displacement of a particle of the medium from its 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, or compressions or rarefactions). Unit: meter (m).
  • Time Period (T): Time taken for one complete oscillation of a particle or the time taken for the wave to travel a distance equal to one wavelength. Unit: second (s).
  • Frequency (ν or f): Number of complete oscillations per unit time, or the number of wavelengths passing a point per unit time. Unit: Hertz (Hz) or s⁻¹. Relation: ν = 1/T.
  • Angular Frequency (ω): Rate of change of phase. Relation: ω = 2πν = 2π/T. Unit: radian per second (rad s⁻¹).
  • Wave Number (k): Spatial frequency or propagation constant. Relation: k = 2π/λ. Unit: radian per meter (rad m⁻¹).
  • Wave Velocity (v): Speed at which the wave disturbance propagates through the medium. Relation: v = νλ = ω/k. This is the fundamental wave relation.

3. Displacement Relation for a Progressive Wave

  • A wave travelling along the positive x-axis can be represented by:
    y(x, t) = A sin(kx - ωt + φ)
    or y(x, t) = A cos(kx - ωt + φ)
  • A wave travelling along the negative x-axis can be represented by:
    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).

4. Speed of Travelling Waves

  • Speed of Transverse Wave on a Stretched String:
    v = √(T/μ)
    • T = Tension in the string (N)
    • μ = Linear mass density (mass per unit length) of the string (kg m⁻¹)
  • Speed of Longitudinal Wave (Sound) in a Medium:
    v = √(E/ρ)
    • E = Modulus of elasticity of the medium (Bulk modulus B for fluids, Young's modulus Y for solids in rod form).
    • ρ = Density of the medium (kg m⁻³)
  • Speed of Sound in Gases:
    • Newton's Formula (Isothermal assumption - incorrect): v = √(P/ρ), where P is pressure. This assumed temperature remains constant during compressions/rarefactions.
    • Laplace's Correction (Adiabatic assumption - correct): v = √(γP/ρ), where γ = Cp/Cv is the adiabatic index of the gas. This correctly assumes that compressions/rarefactions happen too quickly for heat exchange, making the process adiabatic.
    • Using the ideal gas equation (P = ρRT/M), the speed of sound can also be expressed as: 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 (Speed increases with temperature).
    • Humidity: Speed of sound is greater in humid air than in dry air (density of humid air is less than dry air at the same pressure).
    • Pressure: No effect on speed at constant temperature (as P/ρ remains constant).
    • Density: v ∝ 1/√ρ (Speed is lower in denser gases, assuming γ and T are constant).
    • Wind: Speed of sound is affected by the velocity component of the wind in the direction of sound propagation.

5. Principle of Superposition of Waves

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

6. Interference of Waves

  • Modification in the distribution of energy when two or more coherent waves superimpose.
  • Coherent Sources: Sources emitting waves of the same frequency (or wavelength) and having a constant phase difference between them.
  • Conditions:
    • Let y₁ = A₁ sin(kx - ωt) and y₂ = A₂ sin(kx - ωt + φ).
    • Resultant Amplitude: A_res = √(A₁² + A₂² + 2A₁A₂ cos φ)
    • Resultant Intensity: I ∝ A², so I_res = I₁ + I₂ + 2√(I₁I₂) cos φ
  • Constructive Interference: Intensity is 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: Intensity is minimum.
    • Condition: Phase difference φ = (2n + 1)π (where n = 0, 1, 2, ...)
    • Path difference Δx = (n + 1/2)λ or (2n+1)λ/2
    • A_min = |A₁ - A₂|
    • I_min = (√I₁ - √I₂)²

7. Reflection of Waves

  • When a wave encounters a boundary between two media, it is partly reflected and partly transmitted.
  • Reflection from a Rigid Boundary (Denser Medium): The reflected wave undergoes a phase change of π (or 180°). A crest reflects as a trough, and vice versa.
  • Reflection from a Free Boundary (Rarer Medium): There is no phase change in the reflected wave. A crest reflects as a crest, and a trough reflects as a trough.

8. Standing Waves (Stationary Waves)

  • Formed by the superposition of two identical waves (same amplitude, frequency, wavelength) travelling in opposite directions.
  • Characteristics:
    • Energy is confined within segments; no net energy transfer.
    • Nodes: Points where the amplitude of vibration is always zero. Displacement is always zero. Distance between two consecutive nodes = λ/2.
    • Antinodes: Points where the amplitude of vibration is maximum (2A). Distance between two consecutive antinodes = λ/2.
    • Distance between a node and an adjacent antinode = λ/4.
    • All particles between two consecutive nodes vibrate in phase, but particles in adjacent segments vibrate out of phase (by π).
  • Standing Waves on a String Fixed at Both Ends:
    • Ends must be nodes.
    • Possible wavelengths: L = n(λ_n / 2) => λ_n = 2L/n (where n = 1, 2, 3, ...)
    • Possible frequencies (Harmonics): f_n = v/λ_n = n(v/2L) = n f₁
    • n = 1: Fundamental frequency or First harmonic (f₁ = v/2L)
    • n = 2: Second harmonic or First overtone (f₂ = 2f₁)
    • n = 3: Third harmonic or Second overtone (f₃ = 3f₁)
    • All harmonics (integral multiples of the fundamental) are present.
  • Standing Waves in Organ Pipes:
    • Closed Organ Pipe (Closed at one end, open at the other):
      • Closed end is a node, open end is an antinode.
      • Possible wavelengths: L = (2n - 1)(λ_n / 4) => λ_n = 4L/(2n - 1) (where n = 1, 2, 3, ...)
      • Possible frequencies: f_n = v/λ_n = (2n - 1)(v/4L) = (2n - 1) f₁
      • n = 1: Fundamental frequency or First harmonic (f₁ = v/4L)
      • n = 2: Third harmonic or First overtone (f₃ = 3f₁)
      • n = 3: Fifth harmonic or Second overtone (f₅ = 5f₁)
      • Only odd harmonics are present.
    • Open Organ Pipe (Open at both ends):
      • Both ends are antinodes.
      • Possible wavelengths: L = n(λ_n / 2) => λ_n = 2L/n (where n = 1, 2, 3, ...)
      • Possible frequencies: f_n = v/λ_n = n(v/2L) = n f₁
      • n = 1: Fundamental frequency or First harmonic (f₁ = v/2L)
      • n = 2: Second harmonic or First overtone (f₂ = 2f₁)
      • n = 3: Third harmonic or Second overtone (f₃ = 3f₁)
      • All harmonics are present (similar to a string fixed at both ends).

9. Beats

  • Phenomenon of periodic variation (waxing and waning) in the intensity of sound resulting from the superposition of two sound waves having slightly different frequencies.
  • Beat Frequency (f_beat): The number of intensity maxima (or minima) heard per second.
    f_beat = |ν₁ - ν₂|
  • Used in tuning musical instruments.

10. Doppler Effect

  • The apparent change in the frequency (or pitch) of a wave perceived by an observer due to relative motion between the source of the wave and the observer.
  • General Formula:
    f' = f [(v ± v_o) / (v ∓ v_s)]
    • f' = Apparent frequency heard by the observer
    • f = Actual frequency emitted by the source
    • v = Speed of sound in the medium
    • v_o = Speed of the observer
    • v_s = Speed of the source
  • Sign Convention:
    • Use the upper sign (+ in numerator, - in denominator) when the observer or source moves towards the other.
    • Use the lower sign (- in numerator, + in denominator) when the observer or source moves away from the other.
    • All speeds are relative to the medium.
  • Special Cases:
    • Source moving towards stationary observer: f' = f [v / (v - v_s)]
    • Source moving away from stationary observer: f' = f [v / (v + v_s)]
    • Observer moving towards stationary source: f' = f [(v + v_o) / v]
    • Observer moving away from stationary source: f' = f [(v - v_o) / v]
  • Doppler effect is also observed for light waves, but the formula is different due to relativity.

Multiple Choice Questions (MCQs)

  1. When a sound wave travels from air into water, which of the following quantities remains unchanged?
    (a) Wavelength
    (b) Speed
    (c) Frequency
    (d) Amplitude

  2. A wave is represented by the equation y = 0.5 sin(10t - 2x), where x and y are in meters and t is in seconds. The speed of the wave is:
    (a) 5 m/s
    (b) 10 m/s
    (c) 2 m/s
    (d) 0.2 m/s

  3. Two sound waves with a phase difference of π/2 radian interfere. If the individual amplitudes are A₁ and A₂, the resultant amplitude will be:
    (a) A₁ + A₂
    (b) |A₁ - A₂|
    (c) √(A₁² + A₂²)
    (d) (A₁ + A₂)/√2

  4. The fundamental frequency of a closed organ pipe is 220 Hz. If one-third of the pipe is filled with water, the fundamental frequency will become:
    (a) 220 Hz
    (b) 330 Hz
    (c) 110 Hz
    (d) 660 Hz

  5. A string of length L is fixed at both ends. It is vibrating in its third harmonic. The wavelength of the wave is:
    (a) L/3
    (b) 2L/3
    (c) 3L/2
    (d) L

  6. Beats are produced by the superposition of two waves with frequencies 256 Hz and 260 Hz. The number of beats heard per second is:
    (a) 258
    (b) 516
    (c) 4
    (d) 2

  7. A train moving towards a stationary observer blows a whistle of frequency 400 Hz. If the speed of sound is 340 m/s and the speed of the train is 20 m/s, the apparent frequency heard by the observer is:
    (a) 400 Hz
    (b) 425 Hz
    (c) 378 Hz
    (d) 420 Hz

  8. When a wave reflects from a denser medium (rigid boundary), the change in phase is:
    (a) 0
    (b) π/2
    (c) π
    (d) 2π

  9. The speed of sound in a gas is proportional to:
    (a) Square root of absolute temperature (√T)
    (b) Absolute temperature (T)
    (c) Pressure (P)
    (d) 1/√T

  10. In which of the following do longitudinal waves not propagate?
    (a) Solids
    (b) Liquids
    (c) Gases
    (d) Vacuum


Answers to MCQs:

  1. (c) Frequency (Frequency is determined by the source and doesn't change when the medium changes. Speed and wavelength change.)
  2. (a) 5 m/s (Compare y = A sin(ωt - kx). ω = 10 rad/s, k = 2 rad/m. Speed v = ω/k = 10/2 = 5 m/s.)
  3. (c) √(A₁² + A₂²) (A_res = √(A₁² + A₂² + 2A₁A₂ cos φ). Here φ = π/2, so cos(π/2) = 0. Thus A_res = √(A₁² + A₂²).)
  4. (b) 330 Hz (Fundamental freq. of closed pipe f₁ = v/4L. When 1/3 is filled, new length L' = L - L/3 = 2L/3. New fundamental f₁' = v/(4L') = v/(4 * 2L/3) = (3/2) * (v/4L) = (3/2) * f₁ = (3/2) * 220 Hz = 330 Hz.)
  5. (b) 2L/3 (For nth harmonic on a string fixed at both ends, L = nλ/2. For n=3, L = 3λ/2 => λ = 2L/3.)
  6. (c) 4 (Beat frequency f_beat = |ν₁ - ν₂| = |260 - 256| = 4 Hz.)
  7. (b) 425 Hz (Source moving towards stationary observer. f' = f [v / (v - v_s)] = 400 [340 / (340 - 20)] = 400 [340 / 320] = 400 * (17/16) = 25 * 17 = 425 Hz.)
  8. (c) π (Reflection from a denser medium causes a phase reversal of π.)
  9. (a) Square root of absolute temperature (√T) (v = √(γRT/M), so v ∝ √T.)
  10. (d) Vacuum (Longitudinal waves, like sound, are mechanical waves and require a medium for propagation.)

Study these notes thoroughly, focusing on understanding the concepts behind the formulas. Practice solving problems related to each section. Good luck with your preparation!

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