Waves
How disturbances travel through media — from ripples on water to the music of sound
Table of Contents
14.1 Introduction
In the previous chapter, we studied the motion of objects oscillating in isolation. What happens in a system which is a collection of such objects? A material medium provides such an example. Here, elastic forces bind the constituents to each other, and the motion of one affects that of the other. If you drop a little pebble in a pond of still water, the disturbance spreads out in circles.
A wave is a disturbance that propagates through a medium due to the repeated motion of particles about their mean positions without any net translation of the medium itself.
14.2 Types of Waves
Transverse Waves
Particle vibration is perpendicular to wave propagation. Forms crests and troughs. Examples: light, waves on string, water surface waves.
Longitudinal Waves
Particle vibration is parallel to propagation. Forms compressions and rarefactions. Examples: sound waves, ultrasound, seismic P-waves.
Electromagnetic Waves
Do not require a medium. Oscillating electric and magnetic fields perpendicular to each other and to propagation. Examples: light, radio waves, X-rays.
Mechanical Waves
Require a material medium for propagation. Energy transfers without net displacement of matter. Examples: sound, water waves, seismic waves.
14.3 Wavelength, Frequency and Wave Speed
For a wave on a stretched string: v = √(T/μ) where T = tension and μ = mass per unit length. For sound in air at 20°C: v ≈ 343 m/s.
14.4 Reflection of Waves
When a wave encounters a boundary, it reflects. The angle of incidence equals the angle of reflection. At a fixed end, the wave is reflected with a phase change of π (inverted). At a free end, there is no phase change.
Fixed end: Displacement node, phase change of π. Free end: Displacement antinode, no phase change. This determines the boundary conditions for standing waves.
14.5 Superposition and Standing Waves
The principle of superposition states that when two or more waves overlap, the resultant displacement is the algebraic sum of individual displacements.
When two identical waves traveling in opposite directions superpose, they form a standing wave — a wave pattern that appears to stand still, with nodes (zero displacement) and antinodes (maximum displacement).
14.6 Beats
When two sound waves of slightly different frequencies interfere, the resultant intensity periodically rises and falls. This phenomenon is called beats.
14.7 Doppler Effect
The apparent change in frequency of a wave due to relative motion between the source and observer is called the Doppler effect.
Ambulance Siren
Higher pitch as it approaches (compressed waves), lower pitch as it recedes (stretched waves). Same source frequency, different perceived frequency.
Red Shift / Blue Shift
Light from stars moving away is redshifted (lower frequency). Light from approaching stars is blueshifted (higher frequency). Evidence for expanding universe.
Radar and Speed Guns
Police radar uses Doppler effect to measure car speed. Emits microwaves, measures frequency shift of reflected waves.
Medical Ultrasound
Doppler ultrasound measures blood flow velocity. Used to detect blocked arteries and monitor fetal heartbeat.
The classical Doppler formula works for sound (needs medium). For light (EM waves), the relativistic Doppler formula must be used. At everyday speeds, the effect is negligible for light.