Period, Angular Frequency and Frequency

Displacement Relation in a Progressive Wave

A wave travelling through a medium carries energy and momentum, but what exactly do we mean by "the wave" at a given instant? The answer lies in the displacement of each particle of the medium from its equilibrium position. For a wave on a string, this displacement is the vertical distance a point on the string has moved from its rest position. The key is that this displacement depends on two things: which particle you look at (its position $x$) and when you look at it (the time $t$).

So the displacement $y$ is a function of both $x$ and $t$, written as $y(x,t)$. The specific form of this function determines the shape and behaviour of the wave. For the simplest and most important case — a progressive wave travelling along the positive $x$-direction without changing shape — the displacement relation takes a particularly elegant form.

Here $v$ is the constant speed of the wave. The function $f$ describes the shape of the wave at $t=0$ (the initial profile). As time increases, the argument $x - vt$ remains constant if $x$ increases at exactly the rate $v$ — meaning the entire waveform slides to the right without distortion. If the wave travels in the negative $x$-direction, the relation becomes $y(x,t) = f(x + vt)$.

For a wave that is sinusoidal — the most fundamental periodic wave — the function $f$ is a sine or cosine. A sinusoidal wave travelling along the positive $x$-direction can be written as:

where:

• $a$ is the amplitude — the maximum displacement from equilibrium.
• $k$ is the angular wave number (or propagation constant).
• $\omega$ is the angular frequency.
• $\phi$ is the initial phase (phase constant).

The argument $(kx - \omega t + \phi)$ is called the phase of the wave. Two points on the wave that have the same phase (e.g., both at a crest) are separated by an integer number of wavelengths.

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Period, Angular Frequency and Frequency

A sinusoidal wave is periodic in both space and time. The time period $T$ is the time taken for one complete cycle of oscillation at any fixed point in the medium. If you sit at a fixed $x$, the displacement $y$ varies sinusoidally with time, repeating every $T$ seconds. The frequency $\nu$ (often denoted $f$ in many texts) is the number of cycles per second:

The angular frequency $\omega$ is related to the frequency by:

Why use $\omega$? Because the sine function's argument must be in radians, and $\omega$ directly gives the rate of change of phase with time. When time increases by $T$, the phase increases by $2\pi$ radians, so:

This is the fundamental link between angular frequency and period.

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Properties of Sinusoidal Waves

The textbook lists three key properties that follow directly from the displacement relation. Each one is derived step by step below.

›Proof

Property (I): For a fixed position $x$, the displacement $y$ varies sinusoidally with time, with angular frequency $\omega$.

Fix $x = x_0$. Then the displacement becomes:

$y(x_0, t) = a \sin(kx_0 - \omega t + \phi)$

This is a function of $t$ alone. Write the constant part as $\theta = kx_0 + \phi$, so:

$y(t) = a \sin(\theta - \omega t) = -a \sin(\omega t - \theta)$

The form $\sin(\omega t - \theta)$ is a standard sinusoidal oscillation with angular frequency $\omega$. The negative sign just represents a phase shift of $\pi$ (since $-\sin u = \sin(u + \pi)$), which does not change the frequency. Therefore, every particle in the medium executes simple harmonic motion with the same angular frequency $\omega$ as the source.

›Proof

Property (II): At any fixed time $t$, the shape of the wave is sinusoidal in $x$, with wavelength $\lambda$ related to $k$ by $k = 2\pi/\lambda$.

Fix $t = t_0$. Then:

$y(x, t_0) = a \sin(kx - \omega t_0 + \phi)$

This is a function of $x$ alone. Write the constant as $\theta' = -\omega t_0 + \phi$, so:

$y(x) = a \sin(kx + \theta')$

This is a sine wave in $x$ with angular wave number $k$. The wavelength $\lambda$ is the spatial period — the distance over which the sine function completes one full cycle. For a sine wave, $\sin(kx + \theta')$ repeats when $kx$ increases by $2\pi$, i.e., when $x$ increases by $\lambda$ such that $k\lambda = 2\pi$. Hence:

$k = \frac{2\pi}{\lambda}$

The angular wave number $k$ has units of radians per metre (rad/m). …