Class 11 Physics Notes Chapter 15 (Chapter 15) – Examplar Problems (English) Book
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.
- Transverse Waves: Particles of the medium vibrate perpendicular to the direction of wave propagation.
- Based on Medium Requirement:
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).
- Closed Organ Pipe (Closed at one end, open at the other):
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)
-
When a sound wave travels from air into water, which of the following quantities remains unchanged?
(a) Wavelength
(b) Speed
(c) Frequency
(d) Amplitude -
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 -
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 -
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 -
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 -
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 -
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 -
When a wave reflects from a denser medium (rigid boundary), the change in phase is:
(a) 0
(b) π/2
(c) π
(d) 2π -
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 -
In which of the following do longitudinal waves not propagate?
(a) Solids
(b) Liquids
(c) Gases
(d) Vacuum
Answers to MCQs:
- (c) Frequency (Frequency is determined by the source and doesn't change when the medium changes. Speed and wavelength change.)
- (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.)
- (c) √(A₁² + A₂²) (A_res = √(A₁² + A₂² + 2A₁A₂ cos φ). Here φ = π/2, so cos(π/2) = 0. Thus A_res = √(A₁² + A₂²).)
- (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.)
- (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.)
- (c) 4 (Beat frequency f_beat = |ν₁ - ν₂| = |260 - 256| = 4 Hz.)
- (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.)
- (c) π (Reflection from a denser medium causes a phase reversal of π.)
- (a) Square root of absolute temperature (√T) (v = √(γRT/M), so v ∝ √T.)
- (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!