Waves & oscillations

Oscillations

Start

Use springs and pendulums to connect equilibrium, sinusoidal motion, and energy exchange.

Resonance

Use coupled oscillators to connect driven resonance, normal modes, and the first hints of wave motion.

Waves

Here

Build intuition for wave propagation by comparing what the pattern does with what the medium does.

Standing Waves and Harmonics

Finish

Connect reflected waves, nodes, antinodes, harmonics, and musical pitch on strings and air columns.

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.

Note that the dependent variable $y$ could represent many different things depending on the type of wave. It may be the displacement of a small segment along a vibrating guitar string, the height of a lilypad riding on a water wave, or the local pressure of air at a music concert. For simplicity, we’re starting with 1D waves (i.e. one spatial dimension $x$ ). A solid understanding of these will allow you to generalize to higher dimensions.

As you explore the simulation below, try the following:

Keep the wavelength fixed and raise the frequency. Watch how the pattern arrives more often.
Hold the frequency constant and increase the wavelength. Notice the crests or compressions spread farther apart.
Switch between transverse and longitudinal modes and compare what the medium itself is doing.
Careful: wavelength and frequency affect the wave speed, but it’s usually the speed that’s constant IRL.

Wave Propagation Explorer

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

Problem Solving

Frequency and Wave Speed

Problem

A periodic wave on a rope has wavelength $\lambda = 0.80\,\text{m}$ and period $T = 0.20\,\text{s}$ . Find the frequency and the wave speed.

Given

$\lambda = 0.80\,\text{m}$
$T = 0.20\,\text{s}$

Use

f = \frac{1}{T}, \qquad v = \frac{\lambda}{T} = f\lambda.

Solve

First compute the frequency:

f = \frac{1}{0.20} = 5.0\,\text{Hz}.

Now compute the speed:

v = \frac{0.80}{0.20} = 4.0\,\text{m/s}.

Check

A wave that repeats five times each second and stretches less than a meter from crest to crest should travel a few meters per second, so the result is reasonable.

Constant Wave Speed

Problem

A rope wave travels at $12\,\text{m/s}$ . If the frequency is increased from $3.0\,\text{Hz}$ to $6.0\,\text{Hz}$ while the rope and its tension stay the same, find the initial and final wavelengths.

Given

$v = 12\,\text{m/s}$
$f_1 = 3.0\,\text{Hz}$
$f_2 = 6.0\,\text{Hz}$

Use

In the same medium, the speed stays fixed, so

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

Solve

Initially,

\lambda_1 = \frac{12}{3.0} = 4.0\,\text{m}.

After the frequency doubles,

\lambda_2 = \frac{12}{6.0} = 2.0\,\text{m}.

Check

The wavelength is cut in half because the wave speed stayed fixed while the oscillations arrived twice as often. That is the same tradeoff you can see in the simulator when the medium stays the same.

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.