Introduction to Waves
What a wave is and what it carries. Mechanical and non-mechanical waves, transverse and longitudinal waves, the travelling-wave function \(y = f(x - vt)\), the wave equation, and the harmonic wave \(y = A\sin(kx - \omega t)\).
Why this chapter
Ripples on a pond, a pulse on a rope, the sound of a voice and the light of the Sun are all waves. Each one carries energy from one place to another, yet no material makes the journey. This chapter says exactly what a wave is and sorts waves into their kinds. It then builds the mathematics of a wave on a string, one step at a time. A single function of \(x - vt\) describes any pulse, and a single equation, the wave equation, governs every travelling wave. The chapter ends with the harmonic wave, \(y = A\sin(kx - \omega t)\), the building block of all the waves in this unit.
You will learn
- Say what a wave carries and what it does not, and why a rope is the simplest wave to study.
- Sort waves into mechanical and non-mechanical, and into transverse and longitudinal, with examples of each.
- Write a travelling pulse as \(y = f(x - vt)\) or \(g(x + vt)\), and read its speed and direction from the function.
- Derive the particle velocity \(-v \times \text{slope}\) and the wave equation, and test a function against the conditions for a valid wave.
- Derive \(y = A\sin(kx - \omega t)\) and show that each particle performs SHM with \(\lambda = vT\).
- Find every quantity of a harmonic wave from its equation.
Before you begin
From Motion in One Dimension you should know velocity as a derivative and a slope. From the chapter Kinematics of SHM you should know simple harmonic motion, \(x = A\sin(\omega t + \phi)\). Its amplitude, period, frequency, angular frequency and phase should be familiar. You should be able to differentiate a sine and use the chain rule.
Lessons
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%Glossary
The key words of this chapter, with the meaning this chapter gives them.
- amplitude
- The maximum displacement of a particle of the medium from its equilibrium position, \(A\).
- angular frequency
- \(\omega = 2\pi/T\), the rate of change of the phase of a wave with time, in rad/s.
- compression
- A region of a longitudinal wave where the particles of the medium are crowded closer together than at rest.
- crest
- A point of greatest positive displacement of a wave at one instant, a highest point of the wave.
- Fourier series
- A way of writing any periodic function as a sum of sines and cosines, so any periodic wave as a sum of harmonic waves.
- harmonic wave
- A travelling wave whose displacement is a sine function of position and time, \(y = A\sin(kx - \omega t)\). Every particle of the medium performs simple harmonic motion.
- longitudinal wave
- A wave in which the particles of the medium oscillate parallel to the direction in which the wave travels, as in sound in air.
- mechanical wave
- A wave that needs a material medium to travel. Sound, water waves, seismic waves and waves on a string are mechanical waves.
- medium
- The material through which a mechanical wave travels, such as air, water, a rope or the rock of the Earth. Each part of the medium only oscillates about its own place.
- non-mechanical wave
- A wave that needs no medium and can travel through a vacuum at the speed of light. Electromagnetic waves and gravitational waves are non-mechanical.
- partial derivative
- The derivative of a function of two or more variables with respect to one of them, the others held fixed. It is written with a curly \(\partial\), as in \(\partial y/\partial t\).
- partial differential equation
- An equation that relates partial derivatives of an unknown function of two or more variables. The wave equation is one.
- phase
- The argument of the sine in the equation of a harmonic wave, \(\Phi = kx - \omega t\). It fixes where in its cycle each particle is.
- pulse
- A single disturbance, such as one bump, that travels along a medium.
- rarefaction
- A region of a longitudinal wave where the particles of the medium are spread farther apart than at rest.
- transverse wave
- A wave in which the particles of the medium oscillate perpendicular to the direction in which the wave travels, as on a rope.
- trough
- A point of greatest negative displacement of a wave at one instant, a lowest point of the wave.
- wave function
- The function \(y(x, t)\) that gives the displacement of the medium at every position \(x\) and every time \(t\).
- wave number
- \(k = 2\pi/\lambda\), the rate of change of the phase of a wave with position, in rad/m.
- wavelength
- The distance between two consecutive crests, or troughs, of a wave at a fixed time, \(\lambda\). The spatial period of the wave.