Waves and Their Types

Waves

• Waves are patterns that move without the physical transfer of matter.
• They transport energy and information.

Types of Waves

• Electromagnetic Waves:
  • Do not require a medium for propagation.
  • Examples: Light waves, radio waves.
• Mechanical Waves:
  • Require a medium for propagation.
  • Involve oscillations of particles and depend on the medium's elastic properties.
  • Examples: Water waves, seismic waves, sound waves.
• Matter Waves:
  • Associated with matter constituents: electrons, protons, neutrons, atoms, molecules.
  • Found in modern technology, such as electron microscopes.

Propagation of Sound Waves

• Sound waves travel through air by compressing and expanding it.
• This creates changes in density (δρ) and pressure (δρ) in the air.
• Compressions and rarefactions move through the air, propagating the disturbance.

Transverse and Longitudinal Waves

• Transverse Waves:
  • Constituents of the medium oscillate perpendicular to the wave's direction of propagation.
• Longitudinal Waves:
  • Constituents of the medium oscillate parallel to the wave's direction of propagation.

Displacement Relation in a Progressive Wave

• Described by the function y(x, t), where:
  • y is the displacement of the medium.
  • x is the position.
  • t is the time.
• This function describes the wave's shape at any given time, and the motion of the medium's constituents.
• Sinusoidal waves are described by y(x, t) = a sin(kx − ωt + ϕ), where:
  • a is the amplitude.
  • ω is the angular frequency.
  • k is the angular wave number.
  • ϕ is the initial phase angle.

Crest and Trough

• Crest: Point of maximum positive displacement.
• Trough: Point of maximum negative displacement.

Amplitude and Phase

• Amplitude (a): Maximum displacement of the medium from its equilibrium position.
• Phase (kx − ωt + ϕ): Determines the displacement at any position and time, given the amplitude.

Wavelength and Angular Wave Number

• Wavelength (λ): The minimum distance between two points with the same phase.
• Angular Wave Number (k): Related to wavelength by k = 2π/λ.

Period, Angular Frequency, and Frequency

• Time Period (T): Time for an element to complete one full oscillation.
• Angular Frequency (ω): Related to period by ω = 2π/T.
• Frequency (f): Number of oscillations per second, related to angular frequency by f = ω/2π.

The Principle of Superposition of Waves

• When waves overlap, the resultant displacement is the algebraic sum of the individual displacements.
• Each wave acts as if the others are not present.
• The net displacement is given by y(x, t) = y1(x, t) + y2(x, t), where y1 and y2 are the individual wave functions.

Reflection of Waves

• When a wave encounters a boundary, it can be reflected.
• Rigid Boundary: Reflected pulse has opposite phase to the incident pulse.
• Free Boundary: Reflected pulse has the same phase as the incident pulse.

Standing Waves and Normal Modes

• When a wave is reflected and interferes with the incident wave, it can form a standing wave.
• Standing waves appear stationary, with fixed points of zero displacement called nodes.
• The points of maximum displacement are called antinodes.

Speed of Sound in Air

• The speed of sound in air is dependent on:
  • Temperature: Increases with temperature.
  • Humidity: Increases with humidity.
  • Pressure: Independent of pressure at constant temperature.
  • The speed of sound in air can be calculated using the formula v = √(γP/ρ):
    • γ is the adiabatic index (1.4 for air).
    • P is the pressure.
    • ρ is the density.
• Using ideal gas law, the speed of sound can be expressed as v = √(γRT/M):
  • R is the ideal gas constant.
  • T is the temperature.
  • M is the molar mass of the gas.