Short on time? This page is the recap. For explanations, Figure 1 and the worked examples, use the full study guide.
Recap
- A periodic wave repeats in time at each place and in space at each instant.
- Period T: time for one full oscillation. Frequency f: oscillations per second (Hz).
- Wavelength λ: distance between successive matching points (crest to crest).
- Amplitude A: maximum displacement from equilibrium. It is independent of T and f.
- Displacement–position graph (snapshot): repeat distance = λ. Displacement–time graph (one point): repeat time = T.
- Pitch goes with frequency; loudness goes with amplitude.
- The energy a wave carries increases with amplitude and with frequency.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| T = 1/f | T in s, f in Hz | switch between period and frequency |
| v = fλ = λ/T | v in m/s, λ in m | link speed, frequency and wavelength |
| x(t) = A cos(ωt) = A cos(2πft) | at one location; x(t) is the displacement, not a position; radians | find displacement at a given time |
| y(x) = A cos(2πx/λ) | at one instant; x is position along the wave; radians | find displacement at a given position |
| ω = 2πf | rad/s | switch between A cos(ωt) and A cos(2πft) |
| Change (same medium unless stated) | f | λ | v |
|---|---|---|---|
| Source frequency doubled | × 2 | × ½ | same |
| Faster medium, same source | same | longer | faster |
| Amplitude doubled | same | same | same |
Assumptions behind the numbers
- The wave is sinusoidal, with constant amplitude.
- In x(t) = A cos(2πft), x stands for displacement; in y(x) = A cos(2πx/λ), x stands for position.
- Cosine is used when y = +A at t = 0 (or x = 0); sine when y = 0 and increasing.
- The speed is set by the medium only, so it is the same for every frequency.
Mistakes to avoid
- Reading λ from a time axis (or T from a position axis).
- Crest to trough as λ. That is λ/2.
- Peak-to-peak height as A. That is 2A.
- “Higher f means faster.” Same medium, same v; λ shrinks.
- Degrees mode for cos(2πft) or cos(2πx/λ).
- Linking pitch to amplitude. Pitch follows frequency.
Quick self-check
- What is the period of a 200 Hz wave? (5.0 × 10⁻³ s)
- Sound at 550 Hz travels at 330 m/s. What is its wavelength? (0.60 m)
- A wave on a string has λ = 2.0 m and v = 12 m/s. What is its frequency? (6.0 Hz)
- The frequency of a source is tripled in the same medium. What happens to λ? (It becomes one third.)
Next: practice questions.