Wave Motion & Pulses
A disturbance travels; the medium stays where it is.
Lesson at a glanceShow the key points
- A wave carries energy from place to place without carrying the medium along (sound in air, a jolt down a queue, ripples on water).
- Waves that need a material medium are mechanical; light needs none and is nonmechanical.
- A small, brief shake of a stretched string sends a pulse along it at constant speed without changing shape; repeated shaking sends a wave train.
- If the end x = 0 moves as f(t), the point x moves as y = f(t − x/v) (15.1): a wave travelling toward +x. Toward −x: y = f(t + x/v) (15.2).
- Equivalently y = g(x − vt) (15.3), where g(x) is the shape at t = 0; t and x must appear only in the combination t − x/v.
Say “hello” to a friend across the room. No air from your mouth reaches their ear, yet your voice does. Energy has travelled through the air while the air itself stayed put. That is a wave.
Energy without the stuff
A moving particle carries its energy with it. A wave carries energy in a different way. When you speak you squeeze and stretch the air next to your lips; that air pushes and pulls the next layer, and so on, until the air by your friend’s ear is disturbed. The disturbance travels; the air molecules near you stay near you.
Picture a restless queue at a closed ticket window. The person at the back leans forward and shoves the next person, then stands straight again. That person stumbles into the one in front, and recovers. The jolt runs all the way to the front, carrying energy with it, while everyone ends up roughly where they started. Raindrops on a pond send out circular ripples in the same way.
A pulse on a string
Tie one end of a long string to a wall, pull the other end tight and flick your hand quickly up and down once. A bump forms near your hand and runs along the string toward the wall. For a uniform, elastic string and a small bump, it travels at a constant speed without changing shape. No part of the string moves toward the wall: each bit simply moves up and back down as the bump passes, receiving energy from the bit behind it and handing it on.
A disturbance confined to a small region at any moment is a wave pulse. Keep shaking for a while (say ten times) and a wave train of ten loops runs down the string.
The equation of a travelling wave
Put the hand end at x = 0, the string along +x, and let the hand’s displacement be . The disturbance travels at speed v, so whatever the end does at time t reaches the point x at time t + x/v. Turned around: the point x, at time t, is doing what the end did at time t − x/v.
The same wave can be written using the shape of the string at t = 0, g(x):
So and are travelling waves, but is not.
A pulse on a long string (x in cm, t in s) is y = (3 mm) exp[−(t/2 − x/4)²]. Find (a) its speed, (b) how the point x = 0 moves, (c) the string’s shape at t = 0, and (d) where the peak is at t = 3 s.
- 1A wave transports energy without transporting the medium; mechanical waves need a medium, light does not.
- 2A pulse on a uniform string travels at constant speed without changing shape; repeated shaking gives a wave train.
- 3Travelling wave: or toward +x; replace − by + for −x.
Check yourself
Keep exploring
Hand-picked videos, simulations and readings that explain this idea from a different angle.