Waves, resonance, and spectra
See how repeating disturbances carry energy and reveal hidden structure.
Learn amplitude, frequency, wavelength, interference, resonance, and spectra—the language needed for light, cavities, plasma, and quantum modes.
Before you begin
- • Course 1: Measurement
- • Course 2: Matter and fields
By the end, you can
- • Relate speed, frequency, and wavelength.
- • Predict constructive and destructive interference.
- • Explain resonance without implying energy appears from nowhere.
- • Read a spectrum as a physical fingerprint.
Interactive model
Explore before calculating
Live laboratory
Standing-wave atelier
Change the boundary length, mode number, and wave speed. The allowed patterns reveal why resonators select discrete frequencies.
Mode: n = 2
Wavelength: 2.00 m
Frequency: 60.0 Hz
The 3 amber points are nodes. Energy driving near 60.0 Hz can build this pattern, but the driver still supplies that energy.
Level 1 · Foundations teaching kit
Record the investigation. Teach the reasoning.
A learner-facing lab record and a course-specific instructor guide turn the live model into a repeatable classroom investigation.
Learner record
Standing-wave mode record
How do boundary length, wave speed, and integer mode number determine nodes, wavelength, and frequency?
Download learner recordInstructor guide
Teach for evidence, not button pushing
Learners connect boundary conditions with discrete modes and keep resonant response separate from energy creation.
Download instructor guideLesson 1 of 3
Amplitude, frequency, and wavelength
Which features of a repeating pattern determine what we observe?
Amplitude measures the size of an oscillation; frequency counts cycles per second; wavelength measures the spatial repeat distance.
For a wave traveling at speed v, the relationship is v = fλ. Changing frequency changes wavelength when the speed in the medium is fixed.
Worked example
A wave travels at 12 m/s with a frequency of 3 Hz. Find its wavelength.
- 1. Start with v = fλ.
- 2. Rearrange: λ = v/f.
- 3. Compute 12/3.
The wavelength is 4 m.
Try it
Human wave timer
Materials: A rope or spring and a stopwatch.
- 1. Create slow regular pulses.
- 2. Count ten cycles.
- 3. Divide ten by elapsed seconds to find frequency.
- 4. Increase frequency while keeping the disturbance controlled.
Notice: Higher frequency packs cycles closer together when wave speed is approximately unchanged.
Check your understanding: If wave speed stays constant and frequency doubles, what happens to wavelength?
Answer: It halves.
The product fλ must remain equal to the unchanged speed.
Lesson 2 of 3
Interference and resonance
Why can two small disturbances combine into a large response—or cancel?
When waves overlap, their displacements add. In phase they reinforce; out of phase they partly or fully cancel.
Resonance occurs when repeated driving matches a system's natural mode. The large response draws energy from the driver over many cycles.
Worked example
Two equal waves arrive with opposite displacement at the same time.
- 1. Apply superposition.
- 2. Add +A and −A.
- 3. The instantaneous sum is zero.
They destructively interfere at that place and time; energy accounting still requires the full wave field.
Try it
Find a pendulum resonance
Materials: A string, small weight, and a support.
- 1. Build a pendulum.
- 2. Push at random intervals and note the response.
- 3. Then push gently once per natural swing.
- 4. Compare the amplitude after equal numbers of pushes.
Notice: Correctly timed small inputs accumulate; resonance amplifies transferred energy rather than creating it.
Check your understanding: Does resonance create energy?
Answer: No.
It efficiently transfers energy from a periodic driver into a matching natural mode.
Lesson 3 of 3
Spectra and allowed modes
How can a pattern of frequencies reveal what a system is made of?
A spectrum shows how much signal exists at each frequency. Atoms, molecules, cavities, and vibrating objects allow characteristic modes.
Boundary conditions determine which wave patterns fit. This ordinary idea becomes essential when studying electromagnetic cavities and the Casimir effect.
Worked example
A string fixed at both ends has length 1 m. What is the longest standing-wave wavelength that fits?
- 1. The fundamental mode fits half a wavelength along the string.
- 2. Set L = λ/2.
- 3. Compute λ = 2L.
The fundamental wavelength is 2 m.
Try it
Sound spectrum explorer
Materials: A free spectrum-analyzer app or a browser audio visualizer.
- 1. Record a steady vowel.
- 2. Record a whistle at similar loudness.
- 3. Compare frequency peaks.
- 4. Change pitch and identify which peaks move.
Notice: Similar loudness can hide very different frequency structures.
Check your understanding: What selects the allowed standing-wave modes in a cavity?
Answer: Its boundary conditions and geometry.
Only patterns satisfying the constraints at the boundaries persist as modes.
Continue into the evidence
Source-linked next reading
Chapter 2: What Is the Vacuum?
Introduces field modes and the Casimir boundary-condition example.
Lecture 8: The Casimir effect, measured
Shows how mode physics becomes a precision-force measurement.
Chapter 10: Longitudinal Electrodynamics
Advanced material separating valid decomposition mathematics from unsupported extrapolations.