February 23, 2026

Waves

Class 11 Physics: Waves | Physics Q&A
CBSE Class 11 › Unit X: Oscillations and Waves

Waves

NCERT Chapter 14 • Wave Motion, Superposition, Beats, Standing Waves & Doppler Effect

NCERT 2025–26 Unit X • ~10 Marks JEE Main • 1-2 Questions
Artistic representation of sound wave propagation and wave equation.
Waves: The transport of energy without the transport of matter.

1. Introduction & Wave Motion

A wave is a disturbance that propagates in space, transporting energy and momentum from one point to another without the transport of matter. In mechanical waves (like sound or water waves), this requires a material medium.

Elasticity and Inertia: The medium must possess elasticity (to return to equilibrium) and inertia (to store kinetic energy) for wave propagation.

2. Transverse & Longitudinal Waves

  • Transverse Waves: Particles vibrate perpendicular to the direction of wave propagation. Example: Light, waves on a string. Form Crests and Troughs.
  • Longitudinal Waves: Particles vibrate parallel to the direction of wave propagation. Example: Sound waves in air. Form Compressions and Rarefactions.
Diagram comparing particle motion in Transverse vs Longitudinal waves.
In transverse waves, particles move perpendicular to propagation; in longitudinal waves, they move parallel.

3. Progressive Wave Equation

A harmonic traveling wave moving in the positive x-direction is described by:

y(x, t) = A sin(kx – ωt + φ)
  • A: Amplitude
  • k: Angular Wave Number (k = 2π/λ)
  • ω: Angular Frequency (ω = 2πν)
  • λ: Wavelength, ν: Frequency, T: Period
  • v: Wave Speed (v = ω/k = λν)

4. Speed of Traveling Waves

4.1 Speed on a Stretched String

v = √(T/μ)

where T is Tension and μ is mass per unit length (linear mass density).

4.2 Speed of Sound in Fluids (Newton-Laplace Formula)

v = √(γP/ρ)

where P is pressure, ρ is density, and γ is the adiabatic index (Cp/Cv). Newton originally assumed the process was isothermal (v=√(P/ρ)), which gave a wrong value. Laplace corrected it to adiabatic.

4.3 Speed of Sound in Solids

For longitudinal waves in a solid rod, the speed is given by:

v = √(Y/ρ)

where Y is Young’s modulus of the material and ρ is its density.

Problem 1: Calculate the speed of sound in air at STP. (γ=1.4, ρ=1.29 kg/m³, P=1.01 × 10⁵ Pa)
Show Answer v = √(1.4 × 1.01 × 10⁵ / 1.29) = √(1.414 × 10⁵ / 1.29) ≈ √109612 ≈ 331 m/s.

5. Principle of Superposition

When two waves meet, the net displacement is the algebraic sum of individual displacements.

ynet = y₁ + y₂
Visualizing constructive and destructive interference of waves.
Superposition: When waves meet, their displacements add up algebraically.

6. Reflection of Waves

  • Rigid Boundary: Wave reflects with a phase change of π (180°). Crest becomes Trough.
  • Open Boundary: Wave reflects with no phase change. Crest returns as Crest.

7. Standing Waves & Normal Modes

Formed by the superposition of two identical waves traveling in opposite directions.

y(x, t) = 2A sin(kx) cos(ωt)
  • Nodes: Points of zero amplitude (sin kx = 0).
  • Antinodes: Points of maximum amplitude (sin kx = ±1).
Diagram of standing waves showing Nodes and Antinodes for first three harmonics.
Standing waves on a string fixed at both ends.

Frequencies for Stretched String (Fixed Ends)

νn = n v / (2L) = (n / 2L) √(T/μ),    n = 1, 2, 3…

All harmonics (odd and even) are present.

Organ Pipes

  • Open Pipe: Open at both ends. νn = nv/(2L). All harmonics present.
  • Closed Pipe: Closed at one end. νn = (2n-1)v/(4L). Only odd harmonics present (1, 3, 5…).
Pressure vs. Displacement in Sound Waves: In a sound wave, a displacement node is a pressure antinode, and vice versa. This is because pressure is maximum where the density change is maximum, which occurs where particles are not moving (node).
Problem 2: A pipe, 30.0 cm long, is open at both ends. What is the fundamental frequency? (Speed of sound = 340 m/s)
Show Answer L = 0.3 m. Open pipe fundamental: ν₁ = v/(2L).
ν₁ = 340 / (2 × 0.3) = 340 / 0.6 ≈ 566.7 Hz.

8. Beats

The periodic variation in intensity of sound when two waves of slightly different frequencies (ν₁, ν₂) superimpose.

νbeat = |ν₁ – ν₂|

Used in tuning musical instruments.

Problem 3: Two sitar strings produce a beat frequency of 6 Hz. The frequency of string A is 324 Hz. If string B is slightly tightened, the beat frequency reduces to 3 Hz. Find original frequency of B.
Show Answer Possible frequencies for B: 324 ± 6 ⇒ 330 Hz or 318 Hz.
Tightening B increases tension ⇒ increases νB.
If νB = 330 and increases, beat freq (330+δ – 324) increases (>6). Incorrect.
If νB = 318 and increases, beat freq (324 – (318+δ)) decreases (<6). Correct.
So, original frequency of B was 318 Hz.

9. Doppler Effect

The apparent change in frequency of sound due to the relative motion between the source and the observer.

Illustration of Doppler Effect showing frequency change due to source motion.
Source approaching observer: Frequency increases. Source receding: Frequency decreases.
ν’ = ν ( (v ± vo) / (v ∓ vs) )
  • v: Speed of sound
  • vo: Speed of observer (Top sign + if moving towards source)
  • vs: Speed of source (Top sign – if moving towards observer)
Key Cases:
• Source moving towards stationary observer: ν’ = ν [v / (v – vs)]
• Observer moving towards stationary source: ν’ = ν [(v + vo) / v]
Problem 4: A train moves at 40 m/s towards a stationary observer, blowing a whistle of frequency 500 Hz. Speed of sound is 340 m/s. Find the apparent frequency.
Show Answer Observer stationary (vo = 0). Source approaching (vs = 40). Use ‘-‘ in denominator.
ν’ = 500 [340 / (340 – 40)] = 500 [340 / 300]
ν’ = 500 × 1.133 ≈ 566.7 Hz.
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