In oscillations and waves, two closely related quantities show up everywhere: the period \(T\) and the
frequency \(f\). The period is the time for one complete cycle of repeating motion, measured in seconds.
The frequency is the number of cycles per second, measured in hertz (\(\text{Hz} = \text{s}^{-1}\)).
They are reciprocals:
\[
f=\frac{1}{T},\qquad T=\frac{1}{f}.
\]
Many oscillators are most naturally described using the angular frequency \(\omega\), measured in radians per second.
Angular frequency tracks the phase angle of a sinusoid, and it is related to \(T\) and \(f\) by
\[
\omega = 2\pi f,\qquad T=\frac{2\pi}{\omega},\qquad f=\frac{\omega}{2\pi}.
\]
The factor \(2\pi\) appears because one full cycle corresponds to a phase change of \(2\pi\) radians.
A standard way to represent ideal periodic motion is
\[
x(t)=A\cos(\omega t+\phi),
\]
where \(A\) is the amplitude (how large the motion is) and \(\phi\) is the phase shift (where in the cycle the motion starts).
While \(A\) and \(\phi\) affect the shape and starting point of the motion, the period and frequency are controlled by \(\omega\).
This tool focuses on computing \(\omega\), \(T\), and \(f\) from common physical models and then visualizing the cycle as a waveform.
Spring–mass oscillator. For a mass \(m\) attached to an ideal spring with spring constant \(k\), Hooke’s law gives a restoring force
\(F=-kx\). Newton’s second law \(mx''=F\) leads to the simple harmonic motion equation \(x''+\frac{k}{m}x=0\).
The solution is sinusoidal with angular frequency
\[
\omega=\sqrt{\frac{k}{m}}.
\]
Substituting into \(T=\frac{2\pi}{\omega}\) yields the well-known period formula
\[
T=2\pi\sqrt{\frac{m}{k}},
\]
and the frequency is \(f=\frac{1}{T}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\).
A stiffer spring (larger \(k\)) increases \(\omega\) and makes oscillations faster (smaller \(T\)), while a larger mass increases \(T\).
In the ideal model, the period does not depend on the amplitude.
Simple pendulum. A point mass suspended by a light string of length \(L\) oscillates as a pendulum.
For small angles, the tangential restoring component of gravity leads to the approximation \(\sin\theta\approx\theta\),
producing the linear equation \(\theta''+\frac{g}{L}\theta=0\).
The angular frequency is
\[
\omega=\sqrt{\frac{g}{L}},
\]
so the period is
\[
T=2\pi\sqrt{\frac{L}{g}},
\]
and the frequency is \(f=\frac{1}{2\pi}\sqrt{\frac{g}{L}}\).
Longer pendulums swing more slowly (larger \(T\)). The small-angle condition is important: for larger amplitudes,
the true period is slightly longer than \(2\pi\sqrt{L/g}\), and a more advanced treatment uses elliptic integrals.
University-level extensions include the physical pendulum, where the mass distribution and moment of inertia matter.
Why this matters for waves. In wave motion, frequency controls the time variation at a fixed point, while the wavelength \(\lambda\)
describes the spatial repetition. They are linked by the wave-speed relation \(v=\lambda f\).
Period and frequency therefore connect time behavior (how fast something oscillates) to spatial behavior (how far a pattern travels per cycle).
Whether you are analyzing vibrations, sound, or standing waves, computing \(T\), \(f\), and \(\omega\) accurately is often the first step.
Limits of ideal formulas. Real systems can include damping (friction) and driving forces.
Weak damping slightly changes the observed oscillation frequency and introduces exponential decay of amplitude.
Stronger damping can eliminate oscillations entirely. The formulas here assume ideal conditions (no damping, small pendulum angles, linear springs),
which is why they are so clean and widely used for baseline modeling.
