Physics of Waves: Concepts, Properties, and Energy

Wave Motion Fundamentals

What is Wave Motion?

Wave motion is a phenomenon involving the transmission of a disturbance from one point in space to another without any net transport of material between them.

Key Features of Waves

• An initial disturbance, often a vibration, originates at a source and is transmitted.
• Waves transport energy without transporting matter.
• There is a delay between the point where the disturbance is produced and its arrival at other distant points.
• Waves can be produced by an instantaneous disturbance (a pulse) or a continuous disturbance (a train of waves).

Understanding the Wavefront

A wavefront is the locus of all points in the medium affected by the disturbance at the same instant.

Types of Wavefronts:

• Flat Wavefronts: Propagate in one direction (e.g., plane waves).
• Circular Wavefronts: Propagate in two dimensions (e.g., ripples on water).
• Spherical Wavefronts: Propagate in three dimensions (e.g., sound from a point source).

Wave Rays

A wave ray (or simply ray) represents each of the directions perpendicular to the wavefront, indicating the direction and sense in which the disturbance progresses.

Classification of Waves

By Medium of Propagation:

• Mechanical Waves: These waves require a material medium (which must be elastic) to propagate.
  • Longitudinal Waves: The direction of wave propagation coincides with the direction of vibration of the particles in the medium through which it spreads (e.g., sound waves, waves in a spring).
  • Transverse Waves: The direction of wave propagation is perpendicular to the direction of vibration of the particles in the medium through which it spreads (e.g., waves on a string, light waves in some contexts).
• Electromagnetic Waves: These waves do not require a material medium to propagate (e.g., light, radio waves, X-rays). They can travel through a vacuum.

Key Wave Properties and Variables

• Amplitude (A): The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
• Wavelength (λ): The spatial period of a periodic wave – the distance over which the wave's shape repeats.
• Period (T): The time it takes for one complete wave cycle to pass a given point.
• Frequency (f): The number of complete wave cycles that pass a given point per unit of time. It is the reciprocal of the period (f = 1/T).
• Phase (φ): The position of a point on a waveform cycle.
• Wave Number (k): Often represented as k, it is related to the spatial frequency of a wave, k = 2π/λ.
• Wave Velocity (v): The speed at which a wave propagates through a medium. It is related to wavelength and frequency by v = λf.

The Wave Function

A wave function describes the propagation of a disturbance, often represented as a sinusoidal function for simple harmonic waves. The wave equation represents the value of displacement (elongation) for each point in the medium as a function of both position and time.

Energy in Wave Motion

When a particle is affected by a wave disturbance, it undergoes simple harmonic vibration. The mechanical energy of such a particle is the sum of its kinetic energy and potential energy.

Energy Formulas:

• Potential Energy (E_p): E_p = ½kx² (where k is the spring constant and x is displacement)
  • Maximum Potential Energy (E_p,max): E_p,max = ½kA² (where A is amplitude)
• Kinetic Energy (E_c): E_c = ½mv² (where m is mass and v is velocity)
  • Maximum Kinetic Energy (E_c,max): E_c,max = ½m(Aω)² = ½mA²(2πf)² (where ω is angular frequency and f is frequency)
• Total Mechanical Energy (E_m): For a simple harmonic oscillator, the total mechanical energy is constant and equal to the maximum kinetic or potential energy.
  • E_m = ½mA²(2πf)²

Wave Intensity

Intensity (I) is defined as the energy transported per unit time through a unit area perpendicular to the direction of wave propagation. It is often measured in Watts per square meter (W/m²).

Wave Attenuation

Attenuation refers to the natural decrease in transported energy as a wave propagates further from its source. As a wave attenuates, its amplitude decreases, and the particles in the medium vibrate with less energy. This occurs because the energy spreads out over a larger area or is dissipated within the medium.

Wave Absorption

Absorption is the decrease in intensity experienced by a wave as it passes through a medium, where part of the wave's energy is converted into other forms of energy (e.g., heat) within the medium itself. This process is distinct from attenuation due to spreading.