Wave Anatomy: Amplitude, Wavelength, and Frequency
The main properties of a wave are its amplitude (maximum displacement from equilibrium), wavelength λ (distance between successive crests), frequency f (number of oscillations per second, in hertz) and period T (time for one oscillation, T = 1 ÷ f), and they are linked to the wave speed by v = fλ. In brief: the amplitude determines the energy of the wave (energy ∝ amplitude squared), the frequency is set by the source, and the speed is set by the medium. When a wave moves from one medium to another, the frequency stays the same, while speed and wavelength change.
To describe any wave, from a ripple on water to a radio signal, a few quantities are needed. This guide defines amplitude, wavelength, frequency, period, wave speed and phase, shows how to read them from graphs, compares the quantities in a table, works through calculations and corrects common mistakes.
What are the properties of a wave?
- Amplitude (A): the maximum displacement of a point on the wave from its rest (equilibrium) position. Measured in the unit of the displacement (meters for a rope, pascals for sound pressure). A larger amplitude means a more energetic wave: a louder sound, a brighter light, a bigger water wave.
- Wavelength (λ): the distance between two successive points in phase, such as two crests, two troughs or two compressions. Unit: meter.
- Frequency (f): the number of complete waves passing a point per second. Unit: hertz (Hz), 1 Hz = 1 s−1. It is determined by the source and does not change when the wave enters a new medium.
- Period (T): the time for one complete oscillation: T = 1 ÷ f.
- Wave speed (v): the speed at which the pattern moves, determined by the medium.
- Phase: the position within the cycle. Two points that are a whole number of wavelengths apart are in phase; points half a wavelength apart are in antiphase.
What is the wave equation?
In one period T, the wave travels one wavelength λ, so v = λ ÷ T = fλ. If the speed is fixed, a higher frequency means a shorter wavelength. For example, in air all sounds travel at the same speed, so a high note has a shorter wavelength than a low note. The angular frequency ω = 2πf is used in the equation of a harmonic wave: y = A sin(kx − ωt), with the wave number k = 2π ÷ λ.
How are waves shown in graphs?
There are two kinds of graphs, and they must be distinguished. A displacement-distance graph is a snapshot of the whole wave at one moment, and the horizontal distance between the crests gives the wavelength. A displacement-time graph shows how one point moves with time, and the horizontal distance between crests gives the period. In both, the height of a crest above the axis is the amplitude (not the crest-to-trough distance, which is twice the amplitude).
Wave quantities compared
| Quantity | Symbol | Unit | Determined by |
|---|---|---|---|
| Amplitude | A | m (or other) | Source energy |
| Wavelength | λ | m | Source and medium |
| Frequency | f | Hz | Source |
| Period | T | s | Source |
| Speed | v | m/s | Medium |
Worked example: calculations
- Water waves. A buoy rises and falls 12 times in 30 s, and the crests are 4.0 m apart. f = 12 ÷ 30 = 0.40 Hz; T = 1 ÷ 0.40 = 2.5 s; v = fλ = 0.40 × 4.0 = 1.6 m/s.
- Sound. A note of 440 Hz in air (343 m/s): λ = 343 ÷ 440 = 0.78 m. A note one octave higher, 880 Hz: λ = 343 ÷ 880 = 0.39 m, half as long.
- Light. Green light of wavelength 550 nm: f = c ÷ λ = 3.00 × 108 ÷ 550 × 10−9 = 5.45 × 1014 Hz.
- From a graph. A displacement-time graph shows 3 complete cycles in 0.060 s. T = 0.060 ÷ 3 = 0.020 s, so f = 50 Hz. A crest-to-trough height of 6.0 cm means an amplitude of 3.0 cm.
- Change of medium. A wave of frequency 200 Hz and speed 340 m/s in air enters water, where the speed is 1,480 m/s. The frequency remains 200 Hz, and the wavelength changes from 340 ÷ 200 = 1.7 m to 1,480 ÷ 200 = 7.4 m.
How were these quantities defined?
The unit hertz is named after Heinrich Hertz and was adopted in 1960, replacing “cycles per second”. The wave equation was applied by Marin Mersenne, who measured the speed of sound in the 1630s, and the relation between wavelength and frequency became central in Maxwell’s electromagnetic theory in the 1860s.
Which mistakes are common?
Mistake 1: Reading the amplitude as peak-to-peak
The amplitude is measured from the equilibrium line to a crest.
Mistake 2: Thinking that the frequency changes at a boundary
The frequency is fixed by the source. Wavelength and speed change.
Mistake 3: Confusing period and wavelength on graphs
Period is read on a time axis and wavelength on a distance axis.
Mistake 4: Thinking that a greater amplitude means a faster wave
Speed is determined by the medium, not by the amplitude.