class 11 notes

Class 11 Physics Notes Chapter 16 (Chapter 16) – Examplar Problems (English) Book

Philoid

Philoid

— 11 min read

Examplar Problems (English)
Alright class, let's delve into Chapter 16: Waves, from your NCERT Exemplar. This chapter is crucial for understanding many physical phenomena and frequently appears in various government exams. Pay close attention to the concepts and formulas.

Chapter 16: Waves - Detailed Notes for Government Exam Preparation

1. Introduction to Waves:

  • Wave: A disturbance that propagates through a medium or vacuum, transferring energy and momentum without the bulk transport of the medium itself.
  • 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, radio waves, X-rays). Discussed in detail in Class 12.
  • Matter Waves: Associated with moving particles (electrons, protons, etc.). Studied in Quantum Mechanics. (Beyond the primary scope of this chapter but good to know).

2. Types of Mechanical Waves:

  • Transverse Waves: Particle oscillations are perpendicular to the direction of wave propagation.
    • Form crests (maximum positive displacement) and troughs (maximum negative displacement).
    • Can propagate through solids and on the surface of liquids.
    • Cannot propagate through fluids (liquids and gases) because fluids lack shear strength.
    • Example: Waves on a stretched string, light waves (EM waves).
  • Longitudinal Waves: Particle oscillations are parallel to the direction of wave propagation.
    • Form 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.

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 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. Unit: Hertz (Hz) or s⁻¹. Relation: ν = 1/T.
  • Angular Frequency (ω): Rate of change of phase. Unit: radian per second (rad/s). Relation: ω = 2πν = 2π/T.
  • Wave Number (k): Spatial frequency or propagation constant. Number of wavelengths per unit distance (sometimes defined as 2π/λ). Unit: radian 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 = νλ = ω/k.

4. Displacement Relation for a Progressive Wave:

  • A wave traveling along the +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 along the -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): Velocity of the oscillating particle in the medium.
    • vp = ∂y/∂t = -Aω cos(kx - ωt + φ) (for sine function)
    • Maximum particle velocity (vp)max = Aω
  • Wave Velocity (v): Constant speed of the wave profile. v = ω/k.
    • Important: Particle velocity (vp) is generally not the same as wave velocity (v). vp varies with time, while v is constant for a given medium.

5. Speed of Travelling 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) in a Medium:
    • v = √(B/ρ) (For liquids and solids)
      • B = Bulk Modulus of the medium
      • ρ = Density of the medium
    • v = √(Y/ρ) (For solids, specifically in a thin rod)
      • Y = Young's Modulus
      • ρ = Density
    • Speed of Sound in Gases:
      • Newton's Formula (Isothermal assumption - incorrect): v = √(P/ρ), where P is pressure. Predicts lower speed.
      • Laplace Correction (Adiabatic assumption - correct): v = √(γP/ρ)
        • γ = Adiabatic index (Cp/Cv) of the gas.
        • Using ideal gas law (P = ρRT/M), v = √(γRT/M), where R is the universal gas constant, T is absolute temperature, M is molar mass.
      • Factors affecting speed of sound in gas:
        • Temperature: v ∝ √T (in Kelvin). Speed increases with temperature.
        • Humidity: Speed increases with humidity (moist air is less dense than dry air at the same pressure and temperature).
        • Pressure: No effect at constant temperature (as P increases, ρ also increases proportionally).
        • Frequency/Wavelength: Speed is generally independent of frequency (non-dispersive medium).

6. Principle of Superposition of Waves:

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

7. Interference of Waves:

  • Superposition of two waves of the same frequency (and preferably same amplitude) traveling in the same direction.
  • Let y₁ = A₁ sin(kx - ωt) and y₂ = A₂ sin(kx - ωt + φ)
  • Resultant wave: y = A_res sin(kx - ωt + δ)
  • Resultant Amplitude (A_res): A_res² = A₁² + A₂² + 2A₁A₂ cos(φ)
    • Where φ is the phase difference between the waves.
  • Resultant Intensity (I): I ∝ A²
    • I_res = I₁ + I₂ + 2√(I₁I₂) cos(φ)
  • Constructive Interference:
    • Phase difference φ = 2nπ (where n = 0, 1, 2, ...)
    • Path difference Δx = nλ
    • A_res = A₁ + A₂ (Maximum amplitude)
    • I_res = (√I₁ + √I₂)² (Maximum intensity)
  • Destructive Interference:
    • Phase difference φ = (2n + 1)π (where n = 0, 1, 2, ...)
    • Path difference Δx = (n + 1/2)λ
    • A_res = |A₁ - A₂| (Minimum amplitude)
    • I_res = (√I₁ - √I₂)² (Minimum intensity)
  • For interference to be sustained, sources must be coherent (constant phase difference).

8. Reflection of Waves:

  • When a wave encounters a boundary between two media, it is partly reflected and partly transmitted.
  • Reflection from a Rigid Boundary (Fixed End):
    • The reflected wave undergoes a phase change of π (180°).
    • A crest is reflected as a trough, and vice versa.
    • If incident wave: y_i = A sin(kx - ωt)
    • Reflected wave: y_r = A sin(kx + ωt + π) = -A sin(kx + ωt)
  • Reflection from a Free Boundary (Open End):
    • The reflected wave undergoes no phase change (0°).
    • A crest is reflected as a crest, a trough as a trough.
    • If incident wave: y_i = A sin(kx - ωt)
    • Reflected wave: y_r = A sin(kx + ωt)

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).
  • Equation: Superposing y₁ = A sin(kx - ωt) and y₂ = A sin(kx + ωt) gives:
    • y(x, t) = y₁ + y₂ = [2A sin(kx)] cos(ωt)
  • The term [2A sin(kx)] represents the amplitude of oscillation at position x, which varies with x.
  • The term cos(ωt) shows that all particles oscillate in simple harmonic motion with the same frequency ω.
  • Nodes: Points where the amplitude is always zero (2A sin(kx) = 0).
    • sin(kx) = 0 => kx = nπ (where n = 0, 1, 2, ...)
    • x = nπ/k = n(λ/2)
    • Positions: x = 0, λ/2, λ, 3λ/2, ...
    • Distance between consecutive nodes = λ/2.
  • Antinodes: Points where the amplitude is maximum (2A sin(kx) = ±2A).
    • sin(kx) = ±1 => kx = (n + 1/2)π (where n = 0, 1, 2, ...)
    • x = (n + 1/2)π/k = (n + 1/2)λ/2 = (2n+1)λ/4
    • Positions: x = λ/4, 3λ/4, 5λ/4, ...
    • Distance between consecutive antinodes = λ/2.
  • Distance between a node and the adjacent antinode = λ/4.
  • Energy: In a standing wave, energy is not transported; it remains confined between nodes.

10. Standing Waves in Strings (Fixed at Both Ends):

  • Boundary conditions: Nodes must exist at both fixed ends (x=0 and x=L).
  • Allowed wavelengths: L = n(λ_n / 2) => λ_n = 2L/n (where n = 1, 2, 3, ...)
  • Allowed frequencies (Normal modes or Harmonics):
    • ν_n = v/λ_n = n(v/2L)
    • ν_n = n ν₁
    • Where v = √(T/μ) is the wave speed on the string.
  • n = 1: Fundamental Frequency or First Harmonic: ν₁ = v/2L
  • n = 2: Second Harmonic or First Overtone: ν₂ = 2ν₁ = v/L
  • n = 3: Third Harmonic or Second Overtone: ν₃ = 3ν₁ = 3v/2L
  • All harmonics (integral multiples of the fundamental frequency) are present.

11. Standing Waves in Organ Pipes:

  • Closed Organ Pipe (Closed at one end, open at the other):
    • Boundary conditions: Node at the closed end (x=0), Antinode at the open end (x=L). (Approximately, antinode is slightly outside the open end - end correction usually ignored in basic problems).
    • Allowed wavelengths: L = (n + 1/2)λ_n / 2 = (2n+1)λ_n / 4 (where n = 0, 1, 2, ...)
      • Alternatively, L = (m)λ_m / 4 where m = 1, 3, 5... (odd numbers)
    • λ_m = 4L/m (m = 1, 3, 5, ...)
  • Allowed frequencies:
    • ν_m = v/λ_m = m(v/4L) (m = 1, 3, 5, ...)
    • ν_m = m ν₁ (where m is odd)
  • m = 1: Fundamental Frequency or First Harmonic: ν₁ = v/4L
  • m = 3: Third Harmonic or First Overtone: ν₃ = 3ν₁ = 3v/4L
  • m = 5: Fifth Harmonic or Second Overtone: ν₅ = 5ν₁ = 5v/4L
  • Only odd harmonics are present.
  • Open Organ Pipe (Open at both ends):
    • Boundary conditions: Antinodes at both open ends (x=0 and x=L).
    • Allowed wavelengths: L = n(λ_n / 2) => λ_n = 2L/n (where n = 1, 2, 3, ...)
    • Allowed frequencies:
      • ν_n = v/λ_n = n(v/2L)
      • ν_n = n ν₁
  • n = 1: Fundamental Frequency or First Harmonic: ν₁ = v/2L
  • n = 2: Second Harmonic or First Overtone: ν₂ = 2ν₁ = v/L
  • n = 3: Third Harmonic or Second Overtone: ν₃ = 3ν₁ = 3v/2L
  • All harmonics are present (similar to the string fixed at both ends).
  • Note: Fundamental frequency of an open pipe (v/2L) is twice that of a closed pipe of the same length (v/4L).

12. Beats:

  • Superposition of two waves of slightly different frequencies (ν₁ and ν₂) traveling in the same direction.
  • Resultant sound intensity varies periodically with time (waxing and waning of sound).
  • Beat Frequency (ν_beat): The number of intensity maxima (or minima) per second.
    • ν_beat = |ν₁ - ν₂|
  • Used for tuning musical instruments.

13. Doppler Effect (Sound):

  • Apparent change in the frequency of sound heard by an observer due to relative motion between the source of sound and the observer.
  • General Formula:
    • ν' = ν [(v ± v₀) / (v ∓ v<0xE2><0x82><0x95>)]
    • Where:
      • ν' = Apparent frequency heard by the observer
      • ν = Actual frequency emitted by the source
      • v = Speed of sound in the medium
      • v₀ = Speed of the observer
      • v<0xE2><0x82><0x95> = 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.
    • Use -v₀ if observer moves away from the source.
    • Use -v<0xE2><0x82><0x95> if source moves towards the observer.
    • Use +v<0xE2><0x82><0x95> if source moves away from the observer.
  • Alternative Sign Convention (Commonly Used): Assume direction from Source (S) to Observer (O) is positive.
    • ν' = ν [(v + v₀) / (v - v<0xE2><0x82><0x95>)] (For relative approach)
    • ν' = ν [(v - v₀) / (v + v<0xE2><0x82><0x95>)] (For relative recession)
    • Be consistent with the convention you choose.
  • Special Cases (Using 2nd convention):
    • Source moves towards stationary observer: ν' = ν [v / (v - v<0xE2><0x82><0x95>)] (ν' > ν)
    • Source moves away from stationary observer: ν' = ν [v / (v + v<0xE2><0x82><0x95>)] (ν' < ν)
    • Observer moves towards stationary source: ν' = ν [(v + v₀) / v] (ν' > ν)
    • Observer moves away from stationary source: ν' = ν [(v - v₀) / v] (ν' < ν)
  • Wind Effect: If wind blows with speed w, replace v with (v+w) if wind supports wave propagation from S to O, and (v-w) if it opposes.
  • Doppler effect is also applicable to light (EM waves), but the formula is different due to relativity (discussed in Class 12).

Multiple Choice Questions (MCQs):

  1. A wave travelling along the string is described by y(x, t) = 0.005 sin(80.0x - 3.0t), in which the numerical constants are in SI units. The wavelength is:
    a) 0.0785 m
    b) 0.157 m
    c) 0.005 m
    d) 80.0 m

  2. Speed of sound in air is 332 m/s. The speed of sound in air will be doubled when its absolute temperature is:
    a) Doubled
    b) Halved
    c) Made four times
    d) Made one-fourth

  3. 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

  4. A string of length L is fixed at both ends. It is vibrating in its 3rd overtone. What is the wavelength of the wave?
    a) L/2
    b) L/3
    c) L/4
    d) 2L/3

  5. An open organ pipe has a fundamental frequency ν. If one end is closed, the fundamental frequency will become:
    a) ν/4
    b) ν/2
    c) ν
    d) 2ν

  6. When a wave is reflected from a rigid boundary, the reflected wave has:
    a) Same phase and amplitude as the incident wave.
    b) Opposite phase and same amplitude as the incident wave.
    c) Same phase but different amplitude.
    d) Opposite phase and different amplitude.

  7. The principle of superposition is fundamental to the phenomenon of:
    a) Reflection
    b) Refraction
    c) Interference
    d) Diffraction (also relies on superposition)
    Choose the most direct consequence listed.

  8. 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

  9. Which of the following wave characteristics is independent of the others?
    a) Speed
    b) Wavelength
    c) Frequency
    d) Amplitude

  10. In a stationary wave, the distance between a node and the next antinode is:
    a) λ
    b) λ/2
    c) λ/4
    d) 2λ

Answer Key for MCQs:

  1. a) 0.0785 m (Comparing with y = A sin(kx - ωt), k = 80.0. Since k = 2π/λ, λ = 2π/k = 2π/80.0 ≈ 0.0785 m)
  2. c) Made four times (v ∝ √T. To double v, T must become 4 times its initial value.)
  3. d) 6 (ν₁ = v/λ₁ = 330/5.0 = 66 Hz. ν₂ = v/λ₂ = 330/5.5 = 60 Hz. ν_beat = |ν₁ - ν₂| = |66 - 60| = 6 Hz)
  4. c) L/4 (3rd overtone for a string fixed at both ends corresponds to the 4th harmonic (n=4). L = n(λ/2) => L = 4(λ/2) => λ = 2L/4 = L/2. Correction: 3rd Overtone means n=4. L = n(λ_n/2) => L = 4(λ/2) => λ = 2L/4 = L/2. Let me recheck the question and standard definition. String fixed at both ends: n=1 (Fundamental), n=2 (1st Overtone), n=3 (2nd Overtone), n=4 (3rd Overtone). So n=4. L = 4(λ/2) => λ = 2L/4 = L/2. The options seem incorrect based on this. Let me re-evaluate. Ah, the question asks for the wavelength in that mode. For n=4, L = 4(λ₄/2). Therefore, λ₄ = 2L/4 = L/2. My previous calculation was correct, the options provided might be typical distractors or there's a misunderstanding. Let's assume the question intended to ask for something else or has incorrect options. Let's re-examine the standard modes. n=1, λ₁=2L; n=2, λ₂=L; n=3, λ₃=2L/3; n=4, λ₄=2L/4=L/2. The wavelength for the 3rd overtone (n=4) is L/2. Option (a) is L/2. Let's select (a). Self-correction: My initial calculation λ=L/2 was correct. Option (a) is L/2. )
    Final Answer after re-evaluation: a) L/2
  5. b) ν/2 (Fundamental frequency of open pipe ν = v/2L. Fundamental frequency of closed pipe ν' = v/4L. Therefore, ν' = (v/4L) = (1/2) * (v/2L) = ν/2.)
  6. b) Opposite phase and same amplitude as the incident wave. (Phase change of π occurs at a rigid boundary. Amplitude ideally remains the same if no energy is lost.)
  7. c) Interference (Interference is a direct result of superposition. While reflection and diffraction also involve wave principles, interference is the most direct manifestation of adding displacements.)
  8. c) 5000 Hz (Step 1: Frequency received by stationary object: ν' = ν [v / (v - v<0xE2><0x82><0x95>)] = 1000 [330 / (330 - 220)] = 1000 [330 / 110] = 3000 Hz. Step 2: Object acts as a source emitting 3000 Hz. Train (observer) is moving towards this source. ν'' = ν' [(v + v₀) / v] = 3000 [(330 + 220) / 330] = 3000 [550 / 330] = 3000 * (5/3) = 5000 Hz.)
  9. d) Amplitude (Amplitude determines the energy of the wave but is generally independent of speed, frequency, and wavelength, which are related by v = νλ. Frequency is determined by the source, speed by the medium.)
  10. c) λ/4 (Distance between consecutive nodes is λ/2. An antinode lies exactly midway between two nodes.)

Make sure you understand the derivation of formulas, especially for standing waves and the Doppler effect, as questions often test variations or specific conditions. Good luck with your preparation!

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