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Angular frequency

Angular frequency (ω) measures how fast an angle changes in time, given in radians per second. It relates to ordinary frequency, period, angular velocity, and appears widely in oscillations, waves, and rotations.

Author: Leandro Alegsa Creation date: November 13, 2022 Last updated: May 21, 2026

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Overview

Angular frequency, commonly written as ω, expresses the rate at which an angle changes with respect to time. It is a central parameter in the description of rotational and oscillatory motion: a larger value of ω indicates faster rotation or more rapid oscillations. In many physics and engineering contexts it complements ordinary frequency by counting radians swept per unit time rather than complete cycles.

${\vec {\omega }}$

Definition and units

Formally, angular frequency is the time derivative of the angle (or phase) and has the dimensions of inverse time. In the International System of Units it is given in radians per second, often written as s-1 because radians are dimensionless. For a simple harmonic oscillation with ordinary frequency f (cycles per second) and period T (seconds), the standard relations are:

ω = 2πf — links angular frequency to cycles per second.
ω = 2π / T — expresses ω via the period of the motion.

Relation to angular velocity and phase

When describing rigid-body rotation, angular frequency is the magnitude of the angular velocity vector ω→, which also encodes the axis and direction of rotation. In wave and oscillation contexts, ω appears in the phase term of solutions, for example as cos(ωt + φ), where φ is a phase offset. Because it measures angle per time, it is especially convenient when the mathematics uses trigonometric functions and complex exponentials.

History, terminology and notation

The symbol ω is traditionally used for angular frequency and angular velocity in textbooks and research. Alternate names include angular speed, radial frequency, and radian frequency. While related concepts share notation, care is taken in each application to distinguish the scalar magnitude ω from the vector angular velocity in three-dimensional rotation.

Applications and examples

Angular frequency appears across science and technology. In mechanics it describes the spin of wheels and rotating machines; in acoustics and optics it parametrizes wave motion; in electrical engineering it is used to express sinusoidal voltages and currents and to compute impedances in circuits. Typical uses include the analysis of simple harmonic oscillators, normal modes, resonance behavior, and the spectral description of waves.

Important distinctions and notes

Although radians are dimensionless, treating ω as having units of "per second" clarifies its role as a rate. It is distinct from ordinary frequency by the factor 2π, and it should not be confused with angular momentum or torque, which are different physical quantities. For three-dimensional rotations one must also account for direction: the full angular velocity is a vector, while ω as commonly used in oscillation problems is a scalar magnitude. For further background see general introductions to physics and measurements in standard references.

Pointer model

Harmonic oscillations can be represented by the rotation of a pointer whose length corresponds to the amplitude of the oscillation. The instantaneous displacement is the projection of the pointer onto one of the coordinate axes. If the complex number plane is used to represent the pointer, either the real part or the imaginary part corresponds to the instantaneous displacement, depending on the definition.

The angular frequency ω $\omega$ is the rate of change of the phase angle φ $\varphi$ of the rotating pointer (see adjacent figure). In adaptation to the unit of the angular frequency, the angle should be given in radians.

$\omega ={\frac {{\text{d}}\varphi }{{\text{d}}t}}$

The pointer model is applicable to all types of oscillations (mechanical, electrical, etc.) and signals. Since one period of oscillation corresponds to one full revolution of the pointer, and the full angle $2\pi$ is , the angular frequency of a harmonic oscillation is always $2\pi$ times its frequency. Often, the specification of the angular frequency is preferred over the frequency, since many formulas in oscillation theory can be represented more compactly using the angular frequency due to the occurrence of trigonometric functions whose period $2\pi$ is by definition : e.g., in the case of a simple cosine, the period is 2 π {\displaystyle 2\pi }. For example, for a simple cosine oscillation: instead of $y={\hat {y}}\cdot \cos(\omega t)$ $y={\hat {y}}\cdot \cos(2\pi f\,t)$ .

In the case of circular frequencies which are not constant in time, the term instantaneous circular frequency is also used.

Use in vibration theory

A harmonic oscillation can be $\omega$ described in general terms as a function of the angular frequency ω

$x(t)=x_{0}\,\sin \left(\omega t+\varphi _{0}\right)$

It can be represented, as is common in electrical engineering, by the real and imaginary parts of a complex pointer ${\underline {x}}$ rotating at constant angular velocity in the Gaussian number plane as a function of angular frequency and time. The time-dependent angle φ $\varphi (t)=\omega t+\varphi _{0}$ of the complex pointer is called the phase angle.

${\underline {x}}(t)=x_{0}\,e^{i(\omega t+\varphi _{0})}=x_{0}\,(\cos(\omega t+\varphi _{0})+i\sin(\omega t+\varphi _{0}))$

The connection with sine and cosine results from Euler's formula.

Characteristic circuit frequency and natural circuit frequency

Oscillating systems are described by the characteristic angular frequency and the natural angular frequency. An undamped freely oscillating system oscillates with its characteristic angular frequency ω $\omega _{0}$ , a damped system with no external excitation oscillates at its natural angular frequency ω $\omega _{d}$ . The natural angular frequency of a damped system is always less than the characteristic angular frequency. In mechanics, the characteristic angular frequency is also referred to as the undamped natural angular frequency.

For the example of an electric resonant circuit, with the resistance the inductance and the capacitance for the characteristic frequency:

$\omega _{0}={\frac {1}{\sqrt {LC}}}$

For a spring pendulum with spring stiffness and mass following applies to the characteristic frequency:

$\omega _{0}={\sqrt {\frac {c}{m}}}$

and with the decay constant δ $\delta =R/{(2L)}$ or δ $\delta =d/{(2m)}$ for the natural angular frequency:

$\omega _{d}={\sqrt {\omega _{0}^{2}-\delta ^{2}}}$ .

For further examples see torsion pendulum, water pendulum, thread pendulum.

Complex angular frequency

From the complex pointer representation of a harmonic oscillation

${\underline {x}}(t)=x_{0}\,e^{\mathrm {i} \omega t}$

results with the usual approach

${\underline {x}}(t)=x_{0}\,e^{st}$

the generalization to the complex angular frequency $s=\sigma +\mathrm {i} \,\omega$ with the real part σ $\sigma$ and the angular frequency ω $\omega$ . By the complex angular frequency not only a constant harmonic oscillation with σ $\sigma =0$ represented, but also a damped oscillation with σ $\sigma <0$ and an excited oscillation with σ $\sigma >0$ . A classical application of the complex angular frequency is the extended symbolic method of AC.

A damped oscillation can be represented as a complex pointer with the constant complex angular frequency s as follows:

${\underline {x}}(t)=x_{0}\,e^{st}=x_{0}\,e^{(\sigma +\mathrm {i} \omega _{d})t}=x_{0}\,e^{\sigma t}e^{\mathrm {i} \omega _{d}t}=x_{0}\,e^{\sigma t}(\cos(\omega _{d}t)+\mathrm {i} \sin(\omega _{d}t))$

Here ω $\omega _{d}$ is the natural angular frequency of the oscillatory system and σ $\sigma$ is equal to the negative value of the decay constant: σ $\sigma =-\delta$ (see the previous section for details).

In the Laplace transform, the complex angular frequency $s=\sigma +\mathrm {i} \omega$ has a more general meaning as a variable in the image domain of the transform F(s) for representing arbitrary time functions and transfer functions in the complex frequency plane ("s-plane").

Author

AlegsaOnline.com - Angular frequency - Leandro Alegsa

Creation date: November 13, 2022
Last updated: May 21, 2026

URL: https://en.alegsaonline.com/art/4239