quantity

angle

The direction of wave propagation, varied here to average out interference.
19 related articles · 6 formulas

Description

In the context of wave propagation, the angle refers to the direction of travel of a wave relative to a reference axis or surface. It is a fundamental parameter in optics, acoustics, and electromagnetism, determining how waves reflect, refract, diffract, and interfere. When multiple waves arrive at a point from different angles, their relative phases depend on the angle, leading to constructive or destructive interference patterns. Varying the angle can average out interference effects, as in certain measurement techniques.

History

The quantitative study of angles in wave phenomena dates back to the 17th century with Willebrord Snellius's law of refraction (1621) and Christiaan Huygens's wave theory of light (1678). The modern treatment of wave interference and diffraction was developed by Thomas Young and Augustin-Jean Fresnel in the early 1800s.

How it works

The angle of propagation is typically measured with respect to the normal to a surface (angle of incidence) or with respect to a fixed axis (azimuthal angle). For a plane wave, the wave vector k points in the direction of propagation, and its components determine the phase variation in space. When waves from different angles superpose, the total field is the sum of each wave's amplitude and phase. By averaging over a range of angles, one can smooth out the rapid spatial oscillations that cause interference, yielding a more uniform intensity.

In practice

Controlling and averaging over angles is essential in antenna arrays for radar and wireless communication, in optical imaging systems to reduce speckle noise, and in acoustic beamforming for medical ultrasound.

Fun fact

In radio astronomy, the technique of 'angular averaging' is used to reduce interference from terrestrial sources by rotating the telescope's feed horn, which smears out the unwanted signals while preserving the astronomical ones.

Concept graph →

Formulas

$$|\theta, \phi\rangle = \begin{pmatrix} \cos\frac{\theta}{2} e^{-i\frac{\phi}{2}} \\ \sin\frac{\theta}{2} e^{i\frac{\phi}{2}} \end{pmatrix}$$
A qubit's pure state parameterized by polar and azimuthal angles on the Bloch sphere.
What is in the formula
θvariablePolar angle on Bloch sphere — angle radian
φvariableAzimuthal angle on Bloch sphere — angle radian
|θ,φ⟩variableQubit state vector — quantum state
$$2d \sin\theta = n\lambda$$
Constructive interference occurs when path difference equals integer wavelengths.
What is in the formula
dvariableInterplanar spacing of the crystal lattice — lattice spacing metre
θvariableGlancing angle of incidence (and reflection) relative to the planes — angle radian
nvariableOrder of reflection (positive integer) — diffraction order
λvariableWavelength of the incident wave — wavelength metre
sinoperatorTrigonometric sine of the angle — sine function
$$\Delta X \propto \cos[2(\theta - \theta_0)]$$
A physical quantity varies as the cosine squared of an angle offset.
What is in the formula
\Delta XvariableChange in a physical quantity — physical quantity change
\thetavariableAngle variable — angle radian
\theta_0variableReference angle offset — angle radian
\proptooperatorIndicates the left side scales with the right side — proportionality
\cosoperatorComputes the cosine of an angle — cosine function
$$1+z = rac{1}{ rac{ ext{v}}{c} ext{cos} heta ext{ sign} ext{ adjusted}}$$
Redshift due to relative motion between observer and source.
What is in the formula
zvariableredshift parameter — redshift
vvariablerelative velocity between source and observer — velocity metre per second
θvariableangle between velocity vector and line of sight — angle radian
cconstantspeed of light in vacuum — speed of light = 299792458 metre per second
cosoperatortrigonometric cosine function — cosine
$$\beta_{\mathrm{app}} = \frac{\beta \sin\theta}{1 - \beta \cos\theta}$$
Apparent speed exceeding light due to relativistic projection.
What is in the formula
β_appvariableapparent transverse speed in units of c — apparent speed
βvariableactual speed in units of c — velocity
θvariableangle between velocity and line of sight — angle radian
$$J(\theta) \propto \cos(2\theta)$$
Exchange coupling varies with angle between magnetic moments.
What is in the formula
\thetavariableAngle between the magnetic moments of two interacting spins — angle radian
JvariableExchange coupling strength between the two magnetic moments — exchange coupling joule

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