Atlas MCAT Course
Physical Foundations → Waves & Harmonics
Waves & Harmonics
Oscillations that propagate.
01The idea
Oscillations (SHM) describe local energy exchange. Waves describe what happens when oscillations are coupled so energy moves through space.
THREAD · Gradients and flow
02Build the model
AWhat a wave is (formally)
A wave is a repeating disturbance characterized by amplitude (A, the size of the disturbance), wavelength (λ, the spatial period), frequency (f, the temporal repetition rate), and wave speed (v, how fast the pattern travels). These quantities are related, not independent.
BThe fundamental wave equation
Speed equals “how often” × “how far each cycle goes”. If one variable changes, at least one other must adjust. This equation applies to sound waves, water waves, and light (with caveats).
CFrequency vs. wavelength: who sets what?
| Quantity | Set by |
|---|---|
| Frequency (f) | The source |
| Wavelength (λ) | The medium |
| Wave speed (v) | The medium |
DTransverse vs. longitudinal waves
| Transverse | Longitudinal |
|---|---|
| Oscillation ⟂ direction of propagation | Oscillation ∥ direction of propagation |
| Examples: strings, light | Example: sound in air |
This distinction affects how energy is stored and how waves interact with boundaries.
EEnergy transport in waves
For mechanical waves, energy ∝ amplitude²: larger amplitude carries more energy. Frequency affects the energy delivery rate (power), not the energy per cycle in simple mechanical systems.
FSuperposition: waves add
When waves overlap, the displacement is the sum of the individual displacements. Consequences: constructive interference (reinforcement) and destructive interference (cancellation).
GStanding waves: when propagation stops
Standing waves form when a wave reflects, interferes with itself, and creates fixed patterns. Key features: nodes (zero displacement) and antinodes (maximum displacement). Energy oscillates locally but does not propagate.
HHarmonics: quantized wave patterns
Only specific wavelengths “fit” in bounded systems. For a string fixed at both ends:
Boundaries enforce discrete modes. Higher harmonics have higher frequency and shorter wavelength.
IFundamental frequency
The fundamental (first harmonic) has the longest wavelength, the lowest frequency, and the simplest standing wave pattern. All other harmonics are integer multiples.
JResonance revisited (now with waves)
Resonance occurs when the driving frequency matches a natural harmonic. This produces large amplitudes and efficient energy transfer — the same principle as SHM, now spatially extended.
KWaves in media: what sets speed?
Wave speed depends on elastic restoring forces and the inertial properties of the medium. The general trend: a stiffer medium gives faster waves; a denser medium gives slower waves. This unifies sound in solids vs gases, and waves on tight vs loose strings.
LConnecting waves to prior topics
Sound is a longitudinal pressure wave; hearing maps frequency in the cochlea; vision (preview) uses EM waves with the same formal structure; and SHM is the local oscillator that builds waves. Nothing here is new — it’s a reorganization.
03See the connections
The lesson names these links itself. Read the map left to right.
Describe
- A, λ, f, v
- v = fλ
Set by
- Source: frequency
- Medium: speed and wavelength
Behaviors
- Superposition
- Standing waves and harmonics
- Resonance
Next: sound and acoustics.
04Reason with it
Four questions. Answer in your head first, then choose. You’ll see the reasoning either way.
Think first.
A wave passes from one medium into another. Which quantity stays the same?
Think first.
A source doubles its frequency in the same medium. What happens?
Think first.
A string fixed at both ends has length L. What is the wavelength of its third harmonic?
Think first.
The amplitude of a wave increases, in the same medium. What changes?
05MCAT lens
Equation-level checks
v = fλ; frequency is set by the source, speed by the medium
Frequency stays constant across boundaries
Energy ∝ amplitude²
Harmonics on a fixed string: λₙ = 2L/n, fₙ = nv/2L
06Common traps
Trap 1: Increasing frequency increases speed
Misconception
Not necessarily.
Correct model
Speed is set by the medium.
Trap 2: Amplitude affects speed
Misconception
Amplitude affects energy, not speed.
Correct model
Bigger waves carry more energy at the same speed.
Trap 3: Frequency and wavelength are always inversely related
Misconception
Only if speed is fixed.
Correct model
In one medium, v is fixed, so f and λ trade off.
07Dependency check
Can you reconstruct this concept without looking at the page? You should now be able to reason through:
How wave speed relates to frequency and wavelength
Check against 02 · B: v = fλ.
Why frequency stays constant across media
Check against 02 · C: the source sets frequency.
How standing waves form
Check against 02 · G: reflection and self-interference.
Why harmonics are discrete
Check against 02 · H: boundaries allow only certain wavelengths.
If something feels unclear, the missing idea is usually thisWaves are oscillations constrained by space and medium.
08Remember
REMEMBER · if you keep only a few things
- Waves carry energy and information, not matter.
- v = fλ.
- Source sets f; medium sets v and λ.
- Energy ∝ amplitude².
- Superposition: displacements add.
- Fixed string: λₙ = 2L/n, fₙ = nf₁.
09Forward hook
Now that waves are formalized, the next natural step is sound and acoustics, the formal physics layer. We’ll connect wave equations, pressure variations, and intensity and decibels.
How do wave equations describe sound in a medium?
That leads to sound and acoustics.
Next lessonSound & Acoustics→