Wave Pulses & Periodic Waves
A wave is a disturbance that transfers energy from one place to another without permanently displacing the medium through which it travels. Think of dropping a stone into a still pond: ripples spread outward, carrying energy to the shore, yet the water molecules themselves simply bob up and down rather than travelling across the pond. This distinction — energy travels, matter stays — is the defining principle of wave motion and underpins everything from sound to light to seismic activity. Waves come in two broad varieties relevant here. A wave pulse is a single, localised disturbance — one ripple on a rope — that travels through a medium and then is gone. A periodic wave is a continuous, repeating series of such disturbances, like the steady hum of a guitar string or the regular crests of ocean swells. Periodic waves have well-defined characteristics: wavelength, frequency, period, and amplitude, all of which are mathematically related and physically meaningful. Understanding how pulses and periodic waves transport energy, how they reflect at boundaries, and how they superpose when they meet is essential for analysing sound, optics, and electromagnetic radiation. This article builds the rigorous foundations you need for advanced wave physics, including the wave equation and energy-transport relationships.
- What Makes a Wave Pulse
- Periodic Waves and Their Characteristics
- Wave Speed Depends on the Medium
- Energy Transport by Waves
- Reflection, Transmission, and Superposition
What Makes a Wave Pulse
A wave pulse is created by a single, brief disturbance of a medium. Flick one end of a stretched rope sharply upward and release it: a hump travels along the rope at a definite speed determined by the medium's properties (tension and linear density for a rope), not by how hard you flicked. The pulse shape is preserved as it travels in an ideal, non-dispersive medium. Crucially, the rope particles move perpendicular to the direction of energy travel — this is a transverse pulse. In contrast, a sound pulse pushes air molecules parallel to its direction of travel — a longitudinal pulse.
Periodic Waves and Their Characteristics
If you oscillate the rope end continuously with simple harmonic motion, each completed oscillation sends out one wavelength λ of wave. The frequency f (oscillations per second, unit: Hz) and period T (time per oscillation, T = 1/f) are set by the source. The wavelength λ is the distance between successive identical points (crest to crest, or trough to trough). The amplitude A is the maximum displacement from equilibrium. These quantities combine in the fundamental wave equation: v = fλ, where v is the wave's propagation speed through the medium.
Wave Speed Depends on the Medium
The speed of a wave is a property of the medium, not the source. For a transverse wave on a stretched string, v = √(T/μ), where T is tension (N) and μ is linear mass density (kg m⁻¹). For sound in air at temperature θ°C, v ≈ 331 + 0.6θ m s⁻¹. Changing the frequency of your source does not change v; instead, the wavelength adjusts so that fλ = v remains constant. This is why a higher-pitched sound has a shorter wavelength than a lower-pitched one travelling through the same air.
Energy Transport by Waves
Waves carry energy, not matter. The power (energy per unit time) carried by a periodic transverse wave is proportional to the square of both the frequency and the amplitude: P ∝ f²A². This means doubling the amplitude quadruples the power — a crucial relationship in acoustics (loudness) and optics (intensity). As a wave spreads out from a point source in three dimensions, its intensity I (power per unit area) follows the inverse-square law: I ∝ 1/r², so doubling the distance from the source reduces intensity by a factor of four.
Reflection, Transmission, and Superposition
When a pulse reaches a boundary between two media, part of its energy is reflected and part is transmitted. The nature of the reflection depends on the boundary: a pulse on a rope meeting a fixed (rigid) end reflects inverted (a phase change of π radians, or 180°), because the boundary exerts an upward restoring force. Meeting a free end, the pulse reflects without inversion. When two pulses occupy the same region simultaneously, the principle of superposition applies: the net displacement at any point is the algebraic sum of the individual displacements. After passing through each other, each pulse continues unchanged — a remarkable property that distinguishes waves from colliding particles.
The Key Equation
- v — Wave speed (m s⁻¹): The speed at which the wave pattern (and energy) propagates through the medium. Determined by the medium's physical properties, not the source.
- f — Frequency (Hz (s⁻¹)): The number of complete oscillations (cycles) passing a fixed point per second. Set by the source of the wave.
- \lambda — Wavelength (m): The spatial distance between two successive points that are in phase — for example, crest to crest or trough to trough.
Worked Example
Problem. A guitar string vibrates at 440 Hz (concert A). The wave speed along the string is 220 m s⁻¹. (a) Calculate the wavelength of the wave on the string. (b) If the tension in the string is doubled while keeping its length and linear density the same, what is the new frequency of vibration? (Assume the wavelength is fixed by the string length.)
Answer. Wavelength = 0.50 m; new frequency ≈ 622 Hz
Origins · 1 minChristiaan Huygens and how this was discovered
Huygens developed the first rigorous wave theory of light in his 1690 work Traité de la Lumière, proposing that every point on a wavefront acts as a secondary source of spherical wavelets — a principle now called Huygens' Principle. He also formulated the correct mathematical description of pendulum motion and made precise measurements of the speed of propagating wave phenomena, laying essential groundwork for understanding periodic waves and pulse propagation. Huygens invented the pendulum clock in 1656, reducing timekeeping error from about 15 minutes per day to less than 15 seconds — a 60-fold improvement — by applying his own wave and oscillation mathematics to engineering.
Related Concepts
- Simple Harmonic Motion — Periodic waves are generated by SHM at the source; the frequency and amplitude of the wave are directly determined by the oscillator's SHM parameters.
- Superposition and Interference — When two wave pulses or periodic waves overlap, the superposition principle governs the resultant displacement, leading directly to constructive and destructive interference patterns.
- Standing Waves and Resonance — Standing waves arise from the superposition of two periodic waves travelling in opposite directions — typically a wave and its reflection — making pulse reflection behaviour directly relevant.
- The Doppler Effect — The Doppler effect describes how the observed frequency of a periodic wave changes when source or observer moves, building directly on the wave equation v = fλ.