Propagation of Mechanical Waves

Lessons

Senior 5 · Physics

Propagation of Mechanical Waves

4 lessons

Lesson 1 of 4

Types and Characteristics of Waves

25 min

A shake that travels

At 07:34 GMT on 3 February 2008 an earthquake of magnitude 5.9 struck under Lake Kivu, about 20 km north of Bukavu, and was felt across Rwanda's Western Province and beyond. No rock travelled from Bukavu to Kigali, yet the shaking did. A wave is exactly that: a disturbance that carries energy from place to place without carrying the material along with it. Seismometers recorded two distinct arrivals, a small fast shake followed by a larger slow one, and from the time between them seismologists could tell how far away the earthquake was. By the end of this lesson you can do the same calculation.

What a wave is

Drop a stone into Lake Kivu and ripples spread outwards; a leaf on the water bobs up and down as they pass but is not carried away. Each bit of water oscillates about its own equilibrium position, and passes the motion to its neighbour. A mechanical wave needs a material medium whose particles are linked by forces (elastic forces in a solid, pressure in a fluid, tension in a string). Light and radio waves are electromagnetic: they need no medium and travel through a vacuum. This unit is about mechanical waves.

Transverse and longitudinal waves

	Transverse	Longitudinal
particle motion	perpendicular to the direction of travel	parallel to the direction of travel
features	crests and troughs	compressions and rarefactions
media	solids, strings, the surface of a liquid; not through the bulk of a fluid, which cannot resist a sideways shear	solids, liquids and gases
examples	waves on a guitar string or rope; seismic S waves	sound; a push along a slinky; seismic P waves

Definition

A progressive (travelling) wave is a disturbance that moves through a medium, transferring energy without a net transfer of matter. In a transverse wave the particles oscillate perpendicular to the direction of propagation; in a longitudinal wave they oscillate parallel to it. Example: a sound wave from a drum is longitudinal; a wave sent along a skipping rope is transverse.

Describing a wave

Quantity	Symbol, unit	Meaning
amplitude	$A$ , m	greatest displacement of a particle from equilibrium
wavelength	$\lambda$ , m	distance between two neighbouring particles in the same phase (crest to crest, or compression to compression)
period	$T$ , s	time for one particle to make one complete oscillation
frequency	$f = \frac{1}{T}$ , Hz	number of oscillations of a particle per second
wave speed	$v$ , $\mathrm{m\,s^{-1}}$	the speed at which the disturbance (a crest, a compression) moves through the medium

The wave equation $v = f\lambda$

In one period $T$ , each particle completes one oscillation, and the wave pattern moves forward by exactly one wavelength. So

v = \frac{\lambda}{T} = f\lambda

The frequency is set by the source; the speed is set by the medium. When a wave passes from one medium into another, its frequency stays the same and its wavelength changes to fit the new speed.

What sets the speed?

For every mechanical wave, the speed depends on a balance between how strongly the medium pulls displaced particles back (an elastic property) and how much inertia they have (a density):

v = \sqrt{\frac{\text{elastic property}}{\text{inertia property}}}

Wave	Speed	Symbols
transverse wave on a stretched string	$v = \sqrt{\frac{T}{\mu}}$	$T$ tension (N), $μ \mu$ mass per unit length ( $\mathrm{kg\,m^{-1}}$ )
longitudinal wave in a solid rod	$v = \sqrt{\frac{E}{\rho}}$	$E$ Young modulus (Pa), $\rho$ density
longitudinal wave in a liquid	$v = \sqrt{\frac{K}{\rho}}$	$K$ bulk modulus (Pa)
sound in a gas	$v = \sqrt{\frac{\gamma p}{\rho}}$	$\gamma$ ratio of heat capacities (1.40 for air), $p$ pressure

You can check each formula with units: for the string, $\sqrt{\frac{\mathrm{N}}{\mathrm{kg\,m^{-1}}}} = \sqrt{\frac{\mathrm{kg\,m\,s^{-2}}}{\mathrm{kg\,m^{-1}}}} = \mathrm{m\,s^{-1}}$ . The string formula is derived in Lesson 4.

For a gas, $\frac{p}{\rho} = \frac{RT}{M}$ for a fixed temperature, so the speed of sound does not depend on the pressure. In Kigali the air pressure is lower than at sea level, but so is the density, in the same proportion, so sound travels at the same speed at the same temperature. It does rise with temperature: $v \propto \sqrt{T}$ .

Worked example 1: speeds in three media

Calculate the speed of (a) sound in air with $p = 1.01 \times 10^5\,\mathrm{Pa}$ , $\rho = 1.20\,\mathrm{kg\,m^{-3}}$ , $\gamma = 1.40$ ; (b) a longitudinal wave in steel with $E = 2.0 \times 10^{11}\,\mathrm{Pa}$ , $\rho = 7800\,\mathrm{kg\,m^{-3}}$ ; (c) a transverse wave on a string of tension 80 N and mass per unit length $5.0\,\mathrm{g\,m^{-1}}$ .

Solution

(a) $v = \sqrt{\frac{1.40 \times 1.01 \times 10^5}{1.20}} = \sqrt{1.18 \times 10^5} = 343\,\mathrm{m\,s^{-1}}$ .

(b) $v = \sqrt{\frac{2.0 \times 10^{11}}{7800}} = \sqrt{2.56 \times 10^7} = 5.1 \times 10^3\,\mathrm{m\,s^{-1}}$ , about 15 times faster than in air.

(c) $\mu = 5.0 \times 10^{-3}\,\mathrm{kg\,m^{-1}}$ : $v = \sqrt{\frac{80}{5.0 \times 10^{-3}}} = \sqrt{1.6 \times 10^4} = 126\,\mathrm{m\,s^{-1}}$ .

Seismic waves

An earthquake sends out both kinds of wave through the Earth. P waves (primary) are longitudinal and travel faster, typically about $6\,\mathrm{km\,s^{-1}}$ in the upper crust. S waves (secondary) are transverse and slower, typically about $3.5\,\mathrm{km\,s^{-1}}$ . S waves cannot pass through the liquid outer core, which is how seismologists first learned that part of the core is liquid. A seismometer close to an earthquake records the P wave first and the S wave a few seconds later; the further away the earthquake, the longer the gap.

Worked example 2: how far away was the earthquake?

A seismometer records the S wave 10 s after the P wave. Taking typical crustal speeds of $6.0\,\mathrm{km\,s^{-1}}$ for P waves and $3.5\,\mathrm{km\,s^{-1}}$ for S waves, find the distance to the earthquake.

Solution

Both waves travel the same distance $d$ : $\frac{d}{3.5} - \frac{d}{6.0} = 10$ . So $d\left(\frac{1}{3.5} - \frac{1}{6.0}\right) = 10$ , $d \times 0.119 = 10$ , $d = 84\,\mathrm{km}$ . Check: the P wave takes $\frac{84}{6.0} = 14\,\mathrm{s}$ and the S wave $\frac{84}{3.5} = 24\,\mathrm{s}$ ; the difference is 10 s. With readings from three stations, the three distances fix the epicentre.

Warning

Common mistakes. (1) Thinking a wave carries the medium with it: particles oscillate about fixed positions; only energy travels. (2) Confusing particle speed with wave speed: a rope particle may move slowly up and down while the wave races along the rope. (3) Thinking a higher frequency travels faster: in a given medium (without dispersion), $v$ is fixed, so a higher $f$ means a shorter $\lambda$ . (4) Saying sound is faster at high altitude because the air is thinner: $\frac{p}{\rho}$ is unchanged; only temperature matters.

Tip

When a wave crosses into a new medium, ask "what stays the same?" The answer is always the frequency, because it is set by the source and each layer of the new medium is driven by the one before it.

P and S waves on a slinky (Cambridge Volcano Seismology). Watch the push-pull P wave race ahead of the side-to-side S wave along the same spring, just as in the Earth.

✦

Practice

Distinguish between transverse and longitudinal waves, giving one example of each.
A wave on a rope has frequency 2.5 Hz and wavelength 1.6 m. Find its speed, and the time a crest takes to travel 10 m.
A tuning fork of frequency 256 Hz is struck in air where sound travels at $343\,\mathrm{m\,s^{-1}}$ . Find the wavelength. The sound passes into water, where its speed is $1480\,\mathrm{m\,s^{-1}}$ . Find the new frequency and wavelength.
Find the speed of a transverse wave on a wire of tension 120 N and mass per unit length $3.0\,\mathrm{g\,m^{-1}}$ .
Explain why S waves cannot travel through a liquid.
Show that the speed of sound in an ideal gas does not depend on its pressure at constant temperature, and explain how the speed changes from 20 °C to 30 °C.
Two seismometers, A and B, record S–P intervals of 6.0 s and 15 s for the same earthquake. Using the speeds of Worked example 2, find the distance of each from the earthquake.

Show answersHide answers

Transverse: particles oscillate perpendicular to the direction of travel (waves on a string). Longitudinal: particles oscillate parallel to it (sound).
$v = f\lambda = 2.5 \times 1.6 = 4.0\,\mathrm{m\,s^{-1}}$ ; $t = \frac{10}{4.0} = 2.5\,\mathrm{s}$ .
$\lambda = \frac{343}{256} = 1.34\,\mathrm{m}$ . In water $f = 256\,\mathrm{Hz}$ (unchanged), $\lambda = \frac{1480}{256} = 5.78\,\mathrm{m}$ .
$v = \sqrt{\frac{120}{3.0 \times 10^{-3}}} = \sqrt{4.0 \times 10^4} = 200\,\mathrm{m\,s^{-1}}$ .
An S wave is transverse: it needs the medium to resist a sideways (shear) deformation. A liquid flows under shear and has no restoring force against it.
$v = \sqrt{\frac{\gamma p}{\rho}}$ and $\frac{p}{\rho} = \frac{RT}{M}$ , so $v = \sqrt{\frac{\gamma RT}{M}}$ , with no pressure term. $v \propto \sqrt{T}$ : $\frac{v_{30}}{v_{20}} = \sqrt{\frac{303}{293}} = 1.017$ , so the speed rises by about 1.7%.
$d = \frac{\Delta t}{0.119}$ : A: $\frac{6.0}{0.119} = 50\,\mathrm{km}$ ; B: $\frac{15}{0.119} = 126\,\mathrm{km}$ .

Key points

A mechanical wave transfers energy through a medium without transferring matter.
Transverse: oscillation perpendicular to travel; longitudinal: parallel (compressions and rarefactions).
$v = f\lambda$ ; the source fixes $f$ , the medium fixes $v$ .
$v = \sqrt{\frac{T}{\mu}}$ (string), $\sqrt{\frac{E}{\rho}}$ (solid rod), $\sqrt{\frac{\gamma p}{\rho}}$ (gas).

Sources and credits

16.1 Traveling Waves and 16.3 Wave Speed on a Stretched String, from University Physics Volume 1 by William Moebs, Samuel J. Ling and Jeff Sanny, OpenStax, licensed under CC BY 4.0. Adapted by Ganzaa.
Facts on the earthquake of 3 February 2008: 2008 Lake Kivu earthquake, Wikipedia, licensed under CC BY-SA 4.0; facts only, summarised by Ganzaa.
Video: P and S waves on a slinky, Cambridge Volcano Seismology, embedded from YouTube.

The three diagrams, the worked examples and the practice exercises are written by Ganzaa.

1A shake that travels

2What a wave is

3Transverse and longitudinal waves

4Describing a wave

The wave equation v=fλ v = f\lambda v=fλ

5What sets the speed?

6Seismic waves