Characteristics of a Wave and Wave Speed: MDCAT Physics notes
Characteristics of a Wave and Wave Speed notes for MDCAT: crest, trough, amplitude, wavelength, frequency, v = fλ, phase, pitch, loudness, Doppler shift.
Wave terms
- Crest: portion of a transverse wave above the mean level; its highest point. Trough: portion below the mean level; its lowest point.
- Amplitude (A): maximum displacement of a particle on either side of its mean position.
- Wavelength ($\lambda$): distance between two successive particles in the same state (phase) of vibration, i.e. the shortest distance between two points in phase; equal to the distance between two consecutive crests or troughs, and the distance a wave travels in one cycle.
- Time period (T): time for one complete vibration.
- Frequency (f): vibrations per second; unit hertz (Hz). $f = 1/T$, so $fT = 1$.
Wave speed
In one period the wave moves one wavelength, so
$$v = \frac{\lambda}{T} = f\lambda$$
| Data | Working | Answer |
|---|---|---|
| 40 waves/s, $\lambda = 5\ \text{cm}$ | $40 \times 0.05$ | $2\ \text{m s}^{-1}$ |
| 54 waves/min, $\lambda = 10\ \text{m}$ | $(54/60) \times 10$ | $9\ \text{m s}^{-1}$ |
| 20 waves in 2 s, $v = 5\ \text{m s}^{-1}$ | $5/10$ | $\lambda = 0.5\ \text{m}$ |
| 500 Hz, $\lambda = 0.6\ \text{m}$ | $500 \times 0.6$ | $300\ \text{m s}^{-1}$ |
| $T = 20\ \mu\text{s}$ | $1/(20\times10^{-6})$ | $f = 50\,000\ \text{Hz}$ |
Always convert "per minute" to per second and cm to m.
Change of medium
When a wave passes into another medium, its frequency stays the same (fixed by the source). Speed and wavelength change together: $\lambda \propto v$. If speed doubles, wavelength doubles.
Phase and path difference
A path difference of $\lambda$ corresponds to a phase difference of $2\pi$:
$$\Delta\phi = \frac{2\pi}{\lambda}\Delta x$$
- Phase $3\pi$ over 15 cm: $15 = 1.5\lambda$, so $\lambda = 10\ \text{cm}$.
- Phase $\pi/3$ over 5 cm: $5 = \lambda/6$, so $\lambda = 30\ \text{cm}$.
- From the 2nd to the 5th crest is $3\lambda$: 15 cm gives $\lambda = 5\ \text{cm}$.
Sound: pitch, loudness, audible range
- Pitch distinguishes a shrill (high) from a grave (low) sound; it depends on, and increases with, frequency.
- Loudness depends on intensity, which increases with amplitude. Increasing amplitude does not change wavelength, frequency or period.
- Human hearing: 20 Hz to 20 kHz. Below 20 Hz: infrasonic; above 20 kHz: ultrasonic.
- Longest audible wavelength (in air, $v \approx 340\ \text{m s}^{-1}$): $340/20 = 17\ \text{m}$; shortest: 17 mm.
Doppler effect (moving source)
A source moving with speed $u$ towards a stationary listener: $f' = \dfrac{v}{v-u}f$ (higher). Moving away: $f' = \dfrac{v}{v+u}f$ (lower). With $u = v/3$ towards, $f' = 1.5f$: 100 Hz is heard as 150 Hz.
Key formulas
- $v = f\lambda$; $f = 1/T$
- $\Delta\phi = 2\pi\Delta x/\lambda$
- Doppler, source approaching: $f' = vf/(v-u)$
Common MDCAT traps
- Frequency is unchanged when a wave changes medium.
- Amplitude affects loudness only.
- Wavelength is between consecutive crests, not alternate ones.
- Convert waves per minute to per second (54/min = 0.9 Hz).
- Ultrasonic means above 20 kHz, not above 20 Hz.
Quick revision
- $fT = 1$.
- Pitch ∝ frequency.
- Audible range 20 Hz – 20 kHz.
- Crest above, trough below the mean level.
- Unit of frequency: hertz.