Amplitude and Phase

The Core Idea: What "Amplitude and Phase" Mean in a Wave

A progressive wave carries energy and information as it travels. To describe it fully, we need two numbers at every point: how big the disturbance is (the amplitude) and where in its cycle the disturbance is (the phase). The amplitude tells you the maximum displacement from equilibrium; the phase tells you whether the particle is at a crest, a trough, or somewhere in between.

The standard equation for a sinusoidal wave travelling along the positive $x$-direction is:

Here:

• $y(x,t)$ is the displacement of a particle at position $x$ at time $t$.
• $A$ is the amplitude — the maximum magnitude of $y$.
• $\omega$ is the angular frequency ($\omega = 2\pi f$).
• $k$ is the wave number ($k = 2\pi / \lambda$).
• $\phi$ is the initial phase (or phase constant).

The quantity inside the sine, $(\omega t - kx + \phi)$, is called the phase of the wave. It is a function of both position and time.

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Property 1: The Phase Difference Between Two Points at the Same Time

Consider two particles at positions $x_1$ and $x_2$ at the same instant $t$. Their phases are:

The phase difference $\Delta \Phi$ is the difference between these two:

So:

where $\Delta x = x_2 - x_1$ is the separation between the points.

What does this mean physically? If two particles are separated by a distance $\Delta x$, their motions are out of step by a phase $k\Delta x$. For example, if $\Delta x = \lambda$ (one full wavelength), then $k\Delta x = (2\pi/\lambda) \cdot \lambda = 2\pi$, so $\Delta \Phi = -2\pi$. A phase difference of $2\pi$ means the particles are exactly in step — they reach their maximum, minimum, and zero displacements simultaneously. If $\Delta x = \lambda/2$, then $\Delta \Phi = -\pi$, meaning they are exactly opposite in phase (one is at a crest when the other is at a trough).

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Property 2: The Phase Difference Between Two Instants at the Same Point

Now fix your attention on a single particle at position $x$. Look at its phase at two different times $t_1$ and $t_2$:

The phase difference is:

So:

where $\Delta t = t_2 - t_1$ is the time interval.

If $\Delta t = T$ (one time period), then $\omega T = (2\pi/T) \cdot T = 2\pi$, so $\Delta \Phi = 2\pi$. The particle has completed one full cycle and returned to the same state. If $\Delta t = T/2$, then $\Delta \Phi = \pi$ — the particle is exactly opposite to where it started.

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Property 3: The Phase Difference Between Two Points at Different Times (General Case)

This is the most general situation. Two particles at $x_1$ and $x_2$ are observed at times $t_1$ and $t_2$ respectively. Their phases are:

The phase difference is:

Or:

This is the master formula. The two special cases above (same time, same point) are just this formula with $\Delta t = 0$ or $\Delta x = 0$.

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