Wave Physics: Mechanics, Electromagnetic Radiation, and Quantum Theory

In mathematics and physical science, a wave is defined as a propagating dynamic disturbance—a change from equilibrium—of one or more quantities. These disturbances move through space and time, transporting energy, momentum, and information. While some waves oscillate repeatedly about a resting value at a specific frequency, others may appear as single, solitary pulses.

Waves are generally categorized by how they travel. A traveling wave occurs when the entire waveform moves in one direction. Conversely, a standing wave is formed by the superposition of two identical periodic waves traveling in opposite directions, resulting in specific positions called nulls where the amplitude is zero or significantly reduced.

Surface waves in water showing water ripples

Key Facts

Example of biological waves expanding over the brain cortex, an example of spreading depolarizations[3]

Mechanical waves require a physical medium to propagate via local stress and strain.
Electromagnetic waves can travel through a vacuum and consist of oscillating electric and magnetic fields.
Gravitational waves are disturbances in the curvature of spacetime predicted by general relativity.
Wave-particle duality in quantum mechanics suggests that particles with momentum possess a wavelength.
The Doppler effect describes the change in wave frequency relative to an observer moving toward or away from the source.

Types of Waves

Sound pressure standing wave in a half-open pipe playing the 7th harmonic of the fundamental (n = 4)

Mechanical Waves

Mechanical waves are local deformations (strain) in a physical medium. They propagate from particle to particle by creating local stresses that affect neighboring particles. Common examples include:

Acoustic waves: Variations in local pressure and particle motion (sound).
Seismic waves: Vibrations traveling through the Earth's crust, including P-waves and SV-waves.
Surface waves: Perturbations propagating across the surface of liquids, such as ocean waves.
Shock waves: High-pressure waves often formed by supersonic motion.

Seismic wave propagation in 2D modelled using FDTD method in the presence of a landmine

Electromagnetic Waves

Electromagnetic waves consist of coupled electric and magnetic fields oscillating at right angles to the direction of propagation. Unlike mechanical waves, they do not require a medium and can travel through a vacuum at the speed of light. Depending on their frequency and wavelength, they are classified into various bands:

Radio waves and Microwaves
Infrared radiation and Terahertz waves
Visible light (the only band visible to the human eye)
Ultraviolet, X-rays, and Gamma rays

Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism

Quantum Mechanical Waves

In the realm of quantum mechanics, Louis de Broglie postulated that all particles with momentum have an associated wavelength. These are described by a wave function, often modeled using the Schrödinger or Dirac equations. A localized wave packet is created by a narrow range of wavelengths; the more localized the envelope of the packet, the larger the spread of required wavelengths.

A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.[30]

Wave Properties and Behaviors

Wavelength λ can be measured between any two corresponding points on a waveform.

Waveforms and Modulation

Waves can take various shapes, including sine, square, triangle, and sawtooth waveforms. Amplitude modulation occurs when the amplitude of a carrier wave is varied by another signal, creating an envelope that defines the overall shape of the wave.

Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.

Velocity: Phase vs. Group

The speed of a wave can be described in two ways: phase velocity, the rate at which the phase of a single frequency component travels, and group velocity, the speed at which the overall envelope (or wave packet) propagates. In some media, these two velocities can differ or even move in opposite directions.

The red square moves with the phase velocity, while the green circles propagate with the group velocity.

Standard Wave Phenomena

Waves interact with their environment and other waves in predictable ways:

Reflection: Bouncing back when hitting a boundary.
Refraction: Changing direction and wavelength when entering a medium with a different velocity.
Diffraction: Bending around obstacles or spreading through openings.
Interference: The superposition of two waves to form a resultant wave, which can be constructive (in phase) or destructive (out of phase).
Dispersion: The separation of waves into different frequencies, often seen when light passes through a prism.

Identical waves from two sources undergoing interference. Observed at the bottom one sees 5 positions where the waves add in phase, but in between which they are out of phase and cancel.

Schematic of light being dispersed by a prism. Click to see animation.

Summary of Wave Classifications

Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x, t) + g(x, t) = u(x, t).

Wave Type	Medium Required	Primary Mechanism	Examples
Mechanical	Yes	Stress and Strain	Sound, Seismic, Water waves
Electromagnetic	No	Electric/Magnetic Fields	Light, X-rays, Radio
Gravitational	No (Spacetime)	Spacetime Curvature	Black hole mergers
Quantum	N/A	Probability Amplitude	Electron wave functions

Frequently Asked Questions

Amplitude modulation can be achieved through f(x,t) = 1.00×sin(2π/0.10×(x−1.00×t)) and g(x,t) = 1.00×sin(2π/0.11×(x−1.00×t)). Only the resultant is visible to improve clarity of waveform.

A wave with the group and phase velocities going in different directions

Tracing the y component of a circle while going around the circle results in a sine wave (red). Tracing the x component results in a cosine wave (blue). Both waves are sinusoids of the same frequency but different phases.

Standing wave. The red dots represent the wave nodes.

Solitary wave in a laboratory wave channel

The propagation of SV-wave in a homogeneous half-space (the horizontal displacement field)

The propagation of SV-wave in a homogeneous half-space (The vertical displacement field)[clarification needed]

Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results

ภาพประกอบบทความ — ภาพประกอบจากบทความต้นฉบับ

Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

What is the difference between a traveling wave and a standing wave?

A traveling wave moves its entire waveform through a medium in a specific direction. A standing wave is created when two identical waves move in opposite directions and overlap, creating fixed points of zero amplitude called nodes.

Can electromagnetic waves travel through a vacuum?

Yes, electromagnetic waves do not require a physical medium to propagate and can travel through the vacuum of space, which is how light from stars reaches Earth.

What is the Doppler effect?

The Doppler effect is the change in the observed frequency of a wave when there is relative motion between the wave source and the observer, such as the change in pitch of a passing siren.

What are gravitational waves?

Gravitational waves are disturbances in the curvature of spacetime that propagate at the speed of light, as predicted by Albert Einstein's theory of general relativity. Their first observation was announced on February 11, 2016.

How does wave-particle duality work?

Wave-particle duality is the concept that every particle or quantum entity may be described as either a particle or a wave. For example, electrons exhibit wave-like properties such as interference and diffraction.