Phase, Wavelength & Frequency

Waves are disturbances that carry energy through space or matter without transporting matter itself. To describe a wave precisely, physicists use a set of spatial and temporal parameters. The most fundamental are wavelength, frequency, and phase. Wavelength tells us about the wave's spatial repetition — how far it travels before the pattern repeats. Frequency tells us about its temporal repetition — how many complete cycles pass a fixed point each second. Together, these parameters allow us to characterise every wave, from light and sound to water ripples and seismic tremors. Phase is perhaps the subtler concept. It describes the position of a wave within its cycle at a particular point in space and time. Imagine two identical pendulums swinging: if they reach their highest points simultaneously, they are 'in phase'; if one is at its peak while the other is at its trough, they are 'out of phase' by half a cycle, or 180°. Phase is crucial for understanding interference — the phenomenon responsible for noise-cancelling headphones, the colours in soap bubbles, and the operation of radio antennas. These three quantities — phase, wavelength, and frequency — are not independent. They are linked through the wave speed equation: the speed of a wave equals its frequency multiplied by its wavelength. This elegant relationship means that for a wave travelling at a fixed speed (such as light in a vacuum), increasing the frequency necessarily decreases the wavelength, and vice versa. Understanding these parameters is foundational to all of wave physics.

Key takeaways

Wavelength (λ): The Spatial Period
Frequency (f): The Temporal Period
Phase (φ): Position Within a Cycle
Wave Number (k): Phase per Unit Length
The Wave Speed Equation

Wavelength (λ): The Spatial Period

Wavelength is the distance between two consecutive points on a wave that are in exactly the same stage of their cycle — for example, crest to crest, or trough to trough. It is measured in metres (m) and given the symbol λ (lambda). A sound wave in air at middle C has a wavelength of about 1.3 m, whereas visible light has wavelengths between roughly 380 nm (violet) and 700 nm (red). A longer wavelength corresponds to a 'stretched-out' wave pattern.

Frequency (f): The Temporal Period

Frequency is the number of complete wave cycles that pass a fixed point per second. Its unit is the hertz (Hz), where 1 Hz = 1 cycle per second. The closely related quantity, period T, is the time for one complete cycle: T = 1/f. Human hearing detects frequencies from about 20 Hz to 20,000 Hz. Radio waves used in FM broadcasting have frequencies near 100 MHz (100 million cycles per second). Higher frequency means the wave oscillates more rapidly.

Phase (φ): Position Within a Cycle

Phase describes where a particular point on a wave is within its repeating cycle at a given instant. It is measured in radians or degrees. A complete cycle corresponds to 2π radians (360°). The phase of a wave is encoded in its mathematical description: y(x, t) = A sin(kx − ωt + φ₀), where φ₀ is the initial phase. Two waves with a phase difference of π radians (180°) are perfectly out of phase — their displacements always cancel, producing destructive interference.

Wave Number (k): Phase per Unit Length

The wave number k = 2π/λ is the spatial equivalent of angular frequency. It tells you how many radians of phase the wave goes through per metre. Similarly, angular frequency ω = 2πf tells you how many radians of phase pass per second. The full wave equation combines both: y(x, t) = A sin(kx − ωt + φ₀). The speed of the wave emerges naturally as v = ω/k = fλ, the universal wave speed relationship.

The Wave Speed Equation

The relationship v = fλ connects all three core descriptors. For mechanical waves (sound, water waves), the speed depends on the medium — temperature, density, elasticity. For electromagnetic waves in a vacuum, v = c ≈ 3 × 10⁸ m/s, fixed by nature. This means that a radio wave at 100 MHz has wavelength λ = c/f = (3 × 10⁸)/(10⁸) = 3 m, while a 300 MHz wave has λ = 1 m. The speed is constant; frequency and wavelength trade off against each other.

The Key Equation

v = f\lambda

v — Wave speed (m s⁻¹): The speed at which the wave pattern travels through the medium.
f — Frequency (Hz (hertz)): The number of complete cycles passing a fixed point per second.
\lambda — Wavelength (m (metres)): The distance between two consecutive points in phase, such as crest to crest.

Worked Example

Problem. A loudspeaker emits a sound wave at a frequency of 440 Hz (concert pitch A). The speed of sound in air at room temperature is 340 m/s. (a) Calculate the wavelength of the sound wave. (b) Calculate the period of the wave. (c) Two identical speakers emit this note, but one has its signal delayed so that it is exactly half a wavelength behind. What is the phase difference between the two waves, and what effect does this have?

\lambda = \frac{v}{f} = \frac{340}{440} \approx 0.773 \text{ m};\quad T = \frac{1}{f} \approx 2.27 \times 10^{-3} \text{ s};\quad \Delta\phi = \pi \text{ rad}

Answer. Wavelength ≈ 0.773 m; Period ≈ 2.27 ms; Phase difference = π radians (180°), producing destructive interference.

Figure

Diagram

Origins · 1 minAugustin-Jean Fresnel and how this was discovered

Augustin-Jean Fresnel

1788–1827 · French

Fresnel placed wave optics on a rigorous mathematical foundation in the early 19th century. He developed the mathematical treatment of wave phase and interference that explained diffraction and polarisation — phenomena that the earlier particle theory of light could not account for. His zone construction used phase relationships to calculate the intensity of light at any point, and his equations for the reflection and transmission of light at interfaces are still used today. Fresnel was a civil engineer by training and worked on French roads and canals. He pursued optics entirely in his spare time, yet his contributions transformed physics. He died of tuberculosis at just 39, but the Fresnel lens he invented — used in lighthouses — saved countless ships before radar was invented.

Transverse and Longitudinal Waves — Understanding wave types is prerequisite to applying phase and wavelength — the physical meaning of displacement differs between transverse and longitudinal waves.
Superposition and Interference — Interference is entirely governed by phase differences between waves; understanding phase is essential to predict whether two waves constructively or destructively interfere.
Standing Waves and Resonance — Standing waves form when waves of the same frequency and wavelength travelling in opposite directions superpose — wavelength determines the resonant frequencies of strings and pipes.
The Doppler Effect — The Doppler effect describes how observed frequency changes with relative motion between source and observer, directly applying the relationship between wave speed, frequency, and wavelength.