Waves

A wave is a disturbance that propagates through space and time. Waves are identified by particular patterns that move through a corresponding medium (like water), although some waves can travel through a vacuum. As a wave propagates, the medium itself does not necessarily move all that far. The key idea is that energy, momentum, and information are carried by the wave, while the medium oscillates around an equilibrium position.

Each wave begins with a local perturbation: a plucked string, a shifted tectonic plate, or a drop in a pond. If one part of the system is coupled to the next part, this disturbance can persist. In a string, tension pulls together neighboring segments. In air, pressure differences compress/rarefact adjacent regions. This coupling allows particular patterns to then propagate through the medium.

The definition of a wave is rather broad, so we will focus on periodic, traveling waves, in which the wave pattern repeats in both space and time.

Transverse & Longitudinal

Waves are often classified by how the medium moves in relation to the waveform.

In a transverse wave, the medium oscillates perpendicular to the direction of propagation. A wave on a taut string is the standard example.
In a longitudinal wave, the medium oscillates parallel to the direction of propagation. Sound in air is the classic case, with alternating compressions and rarefactions.

The geometry looks different, but the bookkeeping is the same: the pattern advances, while each local part of the medium mainly moves back and forth about equilibrium.

Particle Motion View

Both waves travel to the right. The highlighted particle shows what the medium does while the disturbance passes through it.

Transverse

The wave pattern moves right while the highlighted particle oscillates up and down.

Longitudinal

The wave pattern moves right while the highlighted particle oscillates left and right.

Amplitude, Wavelength, and Period

Three measurements organize most introductory wave problems:

\qquad A = \text{amplitude}, \qquad \lambda = \text{wavelength}, \qquad T = \text{period}.

Amplitude $A$ measures the maximum displacement from equilibrium. The wavelength $\lambda$ measures the spatial distance over which the pattern repeats, such as crest-to-crest or compression-to-compression. The period $T$ measures the time it takes one repetition of the pattern to pass by. Thinking locally, this is also the time it takes a particular oscillating piece of medium to complete one cycle.

Wave Guide

Many equivalent variable choices exist, but this trio is often the clearest place to start.

Amplitude

A larger amplitude means taller crests, deeper troughs, or stronger compressions. In a linear medium, it changes the energy carried by the wave much more than the propagation speed.

Wavelength

The wavelength is the distance between matching points on successive cycles, such as crest to crest or compression to compression. It is one convenient spatial variable for describing the pattern.

Period

Starting from the traveling pattern, the period is the time between one crest or compression passing a fixed point and the next one passing that same point. The local medium repeats on that same interval.

With these in mind, you can quantify the wave speed:

v = \frac{\lambda}{T}.

Note that the speed at which waves propagate is usually constant for each “type” of wave, being largely determined by properties of the medium rather than the wave itself (although longitudinal and transverse waves can have dramatically different speeds).

A Traveling Sinusoid

For a simple traveling wave moving in the $+x$ direction, one common model is

y(x,t) = A\sin(kx - \omega t + \phi),

where $A$ is the amplitude, $k = 2\pi/\lambda$ is the wave number, and $\omega = 2\pi/T = 2\pi f$ is the angular frequency. The minus sign tells us the pattern moves toward larger $x$ . Reversing that sign produces the same shape traveling the other way.

The important roles are:

$A$ sets the maximum displacement from equilibrium.
$\lambda$ sets the spatial spacing of the pattern.
$T$ , or equivalently $f = 1/T$ , sets how rapidly each point repeats its motion.
$v = \lambda/T = \lambda f$ then describes how quickly the overall pattern propagates.

As you explore the simulation below, try this sequence:

Keep the wavelength fixed and raise the frequency. Watch how the pattern arrives more often.
Hold the frequency steady and lengthen the wavelength. Notice the crests or compressions spread farther apart.
Switch between transverse and longitudinal modes and compare what the medium itself is doing.
Reverse the direction and confirm that the shape is the same even though the phase shifts the opposite way.

Wave Propagation Explorer

Compare transverse and longitudinal motion while changing amplitude, wavelength, and frequency.

What to Notice

The wave pattern moves through the medium, but the medium elements oscillate around equilibrium rather than drifting forward with the pattern.
One spatial variable and one temporal variable are enough to determine the propagation speed.
Transverse and longitudinal waves differ in the direction of particle motion, not in the core bookkeeping of amplitude, wavelength, and period.

Waves Checkpoint

Question 1 of 3

Which statement best describes what a mechanical wave transports through a medium?

Choose an answer to get instant feedback.