Waves & oscillations

Oscillations

Start

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

Resonance

Here

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

Waves

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.

Resonance

To efficiently push your friend on a swing, you must match the natural rhythm of their swings. Many small pushes can lead to a relatively large amplitude in your friend’s motion. Driving a system at one of its natural frequencies is known as resonance.

A single oscillator has one natural frequency at which it can exhibit resonance. Once you start connecting oscillators together, shared patterns of motion begin to emerge, each with its own characteristic frequency. These patterns and corresponding frequencies are the bridge between simple oscillations and waves.

Driven Response

A poorly timed driving force can still move a system, but the response stays comparatively small. A drive that lines up with one of the system’s natural rhythms can build a much larger motion, especially when damping is weak.

The system’s response to a periodic driving force is often represented with a resonance curve. Higher values represent a more efficient transfer of energy into the system at that frequency. The resonance peak occurs when the driving frequency matches a natural frequency of the system. Without damping, energy can be added to the system without limit and the peak is theoretically infinite in height. In real systems with damping, the peak is finite and widened over a range of frequencies.

Sweep along the resonance curve

Flexible rod on a shake table, first bending mode

Drive: 0.92 Hz

Response: 2.96x

Phase lag: 47°

damping

Below resonance, the rod mostly follows the table and only bends modestly before the next shove arrives.

System Response

The rod tip moves about 2.96 times the base amplitude.

Normal Modes

For two or more coupled oscillators, resonance is no longer about a single favorite rhythm. The system has multiple normal modes: patterns of motion in which the pieces keep a fixed phase relationship while oscillating together.

In the lowest frequency normal mode, neighboring masses tend to move in the same direction at the same time. In higher modes, interesting phase relationships form between the oscillators. When the driving frequency lands near one of those mode frequencies, that pattern becomes the easiest one for the system to support.

Shared Modes to Waves

Now add more neighbors. A longer chain of coupled oscillators still has normal modes, but it also starts to look like a medium that can pass a disturbance along one link at a time. That is the key wave idea hiding inside the oscillator story.

As you explore the simulation below, try this sequence:

Start in 1 oscillator mode and click the resonance preset. This gives you the clean single-oscillator version of the story.
Switch to 2 oscillators, then compare lower mode and higher mode. The pair should flip between mostly same-direction and opposite-direction motion.
Move to 5 oscillators and reuse the presets. Notice how the same coupling idea now organizes several neighbors at once.
Finish in chain mode and wait for the larger pattern to emerge.

Coupled Resonance Explorer

Drive one or more coupled oscillators and compare how resonance picks out different shared patterns.

Problem Solving

Identify the Resonant Two-Mass Mode

Problem

A pair of identical coupled oscillators has a lower shared mode at $0.68\,\text{Hz}$ and a higher shared mode at $1.17\,\text{Hz}$ . The system is driven at $1.10\,\text{Hz}$ with light damping. In steady motion, the two masses move mostly in opposite directions. Which mode is being excited most strongly?

Given

lower mode: $0.68\,\text{Hz}$
higher mode: $1.17\,\text{Hz}$
drive frequency: $1.10\,\text{Hz}$
neighboring masses move mostly opposite directions

Use

Resonance is strongest near a mode frequency, and the phase pattern tells you which shared mode the system is favoring.

Solve

The drive frequency $1.10\,\text{Hz}$ lies much closer to the higher-mode value than to the lower-mode value:

|1.10 - 1.17| = 0.07\,\text{Hz}, \qquad |1.10 - 0.68| = 0.42\,\text{Hz}.

That already points toward the higher mode. The observed motion confirms it: opposite-direction motion is the signature of the higher shared mode in this two-mass system.

Check

Both the frequency match and the phase pattern tell the same story, so the higher mode is the dominant resonant response.

Interpret a Short Chain Pattern

Problem

A short fixed-end chain has a first mode near $0.35\,\text{Hz}$ and a second mode near $0.68\,\text{Hz}$ . It is driven at $0.66\,\text{Hz}$ . The middle of the chain stays relatively quiet while neighboring sections swing with opposite signs. Which pattern is the drive selecting?

Given

first chain mode: $0.35\,\text{Hz}$
second chain mode: $0.68\,\text{Hz}$
drive frequency: $0.66\,\text{Hz}$
the chain shows an internal sign change

Use

Higher modes have more internal structure. A sign change or near-node inside the chain means the pattern is no longer the smooth first mode.

Solve

The drive sits very close to the second mode:

|0.66 - 0.68| = 0.02\,\text{Hz}.

It is far from the first mode:

|0.66 - 0.35| = 0.31\,\text{Hz}.

The extra internal structure in the chain also matches the second mode rather than the first. So the drive is selecting the second resonant pattern.

Check

A first-mode response would look smoother across the whole chain. The extra turning point is the clue that a higher pattern has taken over.

Resonance Checkpoint

Question 1 of 3

Why does coupling change the resonance story compared with a single isolated oscillator?

Choose an answer to get instant feedback.