Class 11 Physics · Chapter 14 NotesWaves

A wave is a disturbance that propagates through a medium or space, transferring energy and information without the bulk movement of matter. When a pebble is dropped in still water, the disturbance spreads outward in circles, but a cork floating on the surface only moves up and down, showing that water itself does not flow outward. Mechanical waves such as sound, water waves and seismic waves require a material medium and depend on the elastic properties of that medium. Electromagnetic waves such as light, radio waves and X-rays can travel through vacuum, all with the same speed c = 299,792,458 m s⁻¹. Matter waves are associated with electrons, protons and other particles and are used in devices such as electron microscopes. In this chapter, the focus is on mechanical waves and their behaviour.

• Mechanical waves need a medium; electromagnetic waves do not
• Speed of light in vacuumc = 299,792,458 m s⁻¹
• Waves transport energy, not matter as a whole

In a transverse wave, the particles of the medium oscillate perpendicular to the direction of wave propagation. A pulse or a sinusoidal wave on a stretched string is a common example. In a longitudinal wave, the particles oscillate along the direction of propagation, as in sound waves travelling through air in the form of compressions and rarefactions. Transverse waves can propagate only in media that can sustain shearing stress, such as solids; fluids cannot support transverse waves. Longitudinal waves can travel through solids, liquids and gases because all these media can sustain compressive strain. In the same medium, transverse and longitudinal waves generally travel with different speeds. Water waves are a combination of both types, and their restoring force may be surface tension (capillary waves) or gravity (gravity waves).

• Transverse waveparticle motion perpendicular to wave direction
• Longitudinal waveparticle motion parallel to wave direction
• Transverse waves need shear modulus; longitudinal waves need bulk modulus

A sinusoidal travelling wave moving along the positive x-direction is described by y(x, t) = a sin (kx – ωt + φ). Here a is the amplitude, k = 2π/λ is the angular wave number, ω = 2π/T = 2πν is the angular frequency, and (kx – ωt + φ) is the phase. The quantity φ is the initial phase angle. A wave travelling in the negative x-direction is written as y(x, t) = a sin (kx + ωt + φ). At a fixed time, the wave has a sinusoidal shape in space; at a fixed position, the displacement varies sinusoidally with time, so each element executes simple harmonic motion. The wavelength λ is the distance between two consecutive crests or troughs, and the period T is the time for one complete oscillation. The speed of a progressive wave is v = ω/k = λ/T = λν.

• y(x, t) = a sin (kx – ωt + φ) for a wave along +x
• k = 2π/λ, ω = 2π/T = 2πν
• v = ω/k = λν

The speed of a mechanical wave is determined by the inertial and elastic properties of the medium, not by the wavelength or frequency. For a transverse wave on a stretched string with tension T and linear mass density µ, the speed is v = √(T/µ). For a longitudinal wave in a fluid with bulk modulus B and density ρ, the speed is v = √(B/ρ). In a solid bar, the relevant modulus is Young's modulus Y, giving v = √(Y/ρ). For sound in an ideal gas, Newton assumed isothermal pressure changes and obtained v = √(P/ρ), which gives about 280 m s⁻¹ at STP, lower than the experimental value of 331 m s⁻¹. Laplace corrected this by assuming adiabatic changes, giving v = √(γP/ρ), which agrees with experiment. Here γ is the ratio of specific heats.

• Transverse wave on stringv = √(T/µ)
• Longitudinal wave in fluidv = √(B/ρ)
• Sound in gas (Laplace correction)v = √(γP/ρ)

When two or more waves travel through the same medium, the net displacement of any element is the algebraic sum of the displacements due to each wave. This is the principle of superposition: y(x, t) = y₁(x, t) + y₂(x, t). For two harmonic waves of equal amplitude a and frequency travelling in the same direction with a phase difference φ, the resultant wave has amplitude A = 2a cos(φ/2). If φ = 0, the waves are in phase and the interference is constructive, giving maximum amplitude 2a. If φ = π, the waves are out of phase and the interference is destructive, giving zero displacement everywhere. Interference is a direct consequence of the superposition principle and is observed in many wave phenomena.

• Superpositiony = y₁ + y₂ + ... + yₙ
• Resultant amplitudeA = 2a cos(φ/2)
• Constructiveφ = 0; destructive: φ = π

When a wave meets a boundary, it may be reflected, transmitted or both. At a rigid boundary, the reflected wave undergoes a phase change of π (180°), so that the displacement at the boundary is always zero. At an open boundary, there is no phase change. When a wave reflects back and forth between two boundaries, a standing wave is formed. For a string fixed at both ends, the standing wave is y(x, t) = 2a sin kx cos ωt. Nodes are points of zero displacement, separated by λ/2; antinodes are points of maximum displacement, also separated by λ/2. The allowed wavelengths for a string of length L fixed at both ends are λ = 2L/n, giving frequencies ν = nv/(2L), where n = 1, 2, 3, ... These are the normal modes. For an air column closed at one end and open at the other, only odd harmonics are present, with frequencies ν = (n + ½)v/(2L), n = 0, 1, 2, ...

• Rigid boundaryphase change of π; open boundary: no phase change
• Standing wave on string fixed at both endsy = 2a sin kx cos ωt
• String fixed at both endsν = nv/(2L), n = 1, 2, 3, ...
• Pipe closed at one endν = (n + ½)v/(2L), n = 0, 1, 2, ...

Beats arise when two harmonic waves of slightly different frequencies ν₁ and ν₂ and comparable amplitudes are superposed. The resultant wave has an average frequency but its amplitude varies periodically with time. The intensity waxes and wanes, and the number of beats heard per second is the beat frequency, given by ν_beat = ν₁ – ν₂. For example, if two waves of frequencies 11 Hz and 9 Hz superpose, the beat frequency is 2 Hz. Musicians use beats to tune instruments: they adjust the tension until the beat frequency becomes zero. The phenomenon is a direct consequence of the principle of superposition and is used in various applications, including tuning and detecting small frequency differences.

• Beat frequencyν_beat = ν₁ – ν₂
• Beats are heard as periodic variations in loudness
• Used for tuning musical instruments