QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 14.1NCERT Class 11 · Physics · Chapter 14

Transverse & Longitudinal Waves, Wave Equation & Speed of Sound

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 4 min read
What is a Wave?

A wave is a disturbance that travels through a medium (or vacuum, in the case of electromagnetic waves), transporting energy and momentum from one point to another without the bulk transport of matter. Particles of the medium oscillate about their mean equilibrium positions.

1. Transverse vs. Longitudinal Waves

1. Transverse Waves

  • Particles of the medium oscillate perpendicular to the direction of wave propagation.
  • Propagate in the form of crests and troughs.
  • Require shear modulus (elasticity of shape). Can only propagate through solids and along the surface of liquids (due to surface tension). They cannot propagate inside fluids (liquids and gases).
  • Examples: Waves on a plucked guitar string, electromagnetic waves (light, radio), ripples on water surface.

2. Longitudinal Waves

  • Particles of the medium oscillate parallel to the direction of wave propagation.
  • Propagate in the form of alternating compressions (high pressure & density) and rarefactions (low pressure & density).
  • Require bulk modulus (volume elasticity). Can propagate through all states of matter: solids, liquids, and gases.
  • Examples: Sound waves in air, pressure waves inside fluids, compression waves in a slinky spring.

2. Mathematical Representation of a Travelling (Progressive) Wave

A simple harmonic wave travelling along the positive x-direction is described by the displacement function:

y(x, t) = A sin(k x − ω t + φ)   or   A sin(ω t − k x + φ')

Where:

  • A: Amplitude of the wave (maximum transverse/longitudinal displacement).
  • ω = 2π / T = 2π f: Angular frequency (rad/s).
  • k = 2π / λ: Angular wave number or propagation constant (rad/m).
  • λ: Wavelength (distance between two successive points in the same phase).
  • Phase of the wave: θ(x, t) = (k x − ω t + φ).
  • Wave speed (v): The speed at which a point of constant phase (crest/trough) moves:
    v = ω / k = f λ = λ / T
Direction of Propagation:
  • If signs of kx and ωt are opposite (e.g. kx − ωt or ωt − kx) → Wave travels in +x direction.
  • If signs of kx and ωt are same (e.g. kx + ωt or −kx − ωt) → Wave travels in −x direction.

3. Particle Velocity vs. Wave Velocity

Particle velocity vp is the velocity of an oscillating element of the medium at position x:

vp = ∂y / ∂t = −ω A cos(kx − ωt)

Wave slope: ∂y / ∂x = k A cos(kx − ωt). Combining the two:

vp = − v × (Slope of the wave profile, ∂y/∂x)

Maximum particle velocity: vp,max = ω A (occurs at y = 0, mean position).

4. Speed of Transverse Wave on a Stretched String

For a flexible string under tension T and having linear mass density μ (mass per unit length = m / L = ρ × Area):

v = √(T / μ) = √[ T / (ρ π r2) ]

5. Speed of Sound in a Gas: Newton's Formula & Laplace's Correction

Newton's Formula (Isothermal Assumption)

Newton assumed that compressions and rarefactions occur so slowly that heat flows freely to keep temperature strictly constant (Isothermal process: PV = constant ⇒ Bulk modulus B = P):

v = √(P / ρ)

At STP: P = 1.013 × 105 Pa, ρ = 1.293 kg/m3 ⇒ v ≈ 280 m/s.
Discrepancy: Experimental speed of sound in air is ~332 m/s. Newton's value had a ~16% error.

Laplace's Correction (Adiabatic Assumption)

Laplace pointed out that sound compressions and rarefactions occur so rapidly and air is such a poor conductor of heat that no heat exchange occurs (Adiabatic process: PVγ = constant ⇒ Bulk modulus B = γP):

v = √(γ P / ρ) = √(γ R T / M)

For air (diatomic γ ≈ 1.41): v ≈ 1.411/2 × 280 ≈ 332.5 m/s (matches experiment perfectly!).

6. Factors Affecting the Speed of Sound in a Gas

Physical Factor Effect on Speed of Sound v Mathematical Relation & Reason
Temperature (T) Increases with temperature v ∝ √T (in Kelvin). Approximately increases by 0.61 m/s per °C rise: vt ≈ v0 + 0.61 t.
Pressure (P) NO EFFECT (at constant temp) When P changes at constant T, density ρ changes in the exact same proportion (P / ρ = RT / M = constant).
Humidity / Moisture Speed increases in moist air Water vapor has lower molecular mass (MH2O = 18) than dry air (Mair ≈ 29). Damp air is less dense than dry air, so ρ decreases ⇒ v increases!
Wind Velocity (w) Direction dependent Sound in direction of wind: veff = v + w cos θ.
Your progress

Saved on this device only — no account, no sign-in.