A rope wave and a sound wave side by side: watch what really moves

How to use this simulation

1. Press play and watch both lanes at once. The rope on top and the air column below are driven by the same source, so they share one frequency, one amplitude and one wave speed. Only the direction of the particle motion differs.
2. Find the orange particle in each lane. Its dashed line marks its rest position. Watch it for a few seconds: on the rope it bobs up and down, in the air it slides left and right, and in both cases it keeps returning to the dashed line. It never follows the wave to the right.
3. Read the small trace on the right of each lane. It plots the orange particle's displacement against time for the last 4 seconds, and it is a sine curve with a period of T = 1/f.
4. Drag the frequency slider up. The waves squeeze closer together, the wavelength λ in the readings falls, and the product f × λ stays equal to the wave speed. The wave restarts from the source each time you change f or v, so you can see the front travelling out.
5. Raise the wave speed with the frequency fixed. The wavelength stretches and the pattern spreads out. Then change only the amplitude: the waves get taller on the rope and the compressions get tighter in the air, but λ does not change.
6. Tick the graph box. The third lane plots how far each air particle has moved from its rest position against where it sits. The red and green dots show that a compression is where this curve crosses zero going down, and a rarefaction is where it crosses zero going up.

A stadium wave, a rope and a speaker

Picture a crowd doing the wave in a stadium. A ripple races around the stands, yet nobody in the crowd runs anywhere. Each person just stands up and sits down in their own seat, and the pattern of standing people travels. That one picture holds the most important idea in this whole chapter: a wave moves, but the things that make it up do not move along with it.

Now tie one end of a long rope to a door handle and flick the other end up and down. A hump runs along the rope towards the door. Look at a knot tied on the rope: it goes up, comes down, and stays in the same place along the rope. The hump travels metres, the knot travels centimetres and returns home.

Sound works the same way, with one twist. When a speaker cone pushes out, it shoves the air next to it, squeezing it. That squeezed patch pushes the next layer, which pushes the next, and a pulse of squeezed air runs through the room. Yet the air near your ear does not travel from the speaker to you; each little bit of air only rocks forwards and backwards a tiny distance. The twist is the direction of the rocking: along the line the sound travels, not across it.

That difference in the direction of the rocking is exactly what separates the two families of waves in this page. The simulation draws both at once, from one source, so you can compare them directly.

First, what is a wave?

A wave is a disturbance that travels through a medium and carries energy from one place to another without carrying the medium itself. Water, rope, air and a metal spring are all examples of a medium: a material made of particles that can push and pull on their neighbours.

The recipe is always the same. Something, called the source, shakes one particle. That particle drags or pushes the next one through the forces between neighbours. The next one then does the same to its neighbour, and so on. Every particle ends up repeating the source's motion, just a little later than the one before it. That small delay, passed down the line, is what we see as the wave moving.

This is why waves are so useful. You can send energy and information across a room, an ocean or a planet without moving a single bucket of material from here to there. A phone call, a guitar string, an earthquake and the voice of your teacher all work this way.

Waves that need a material medium, like the two in this simulation, are called mechanical waves. Light is different, since it needs no medium, but this page is about mechanical waves only.

Transverse waves: the particles shake sideways

In a transverse wave the particles of the medium vibrate at right angles (perpendicular) to the direction in which the wave travels. In the top lane of the simulation the wave runs left to right while each bead of the rope moves up and down.

A transverse wave has a very clear shape. The highest points are called crests, the lowest points are called troughs, and the line the rope would lie along at rest is the equilibrium line. The size of the largest sideways displacement from that line is the amplitude. In the simulation you can see crests and troughs travel, but the orange bead only rises and falls.

A medium can carry a transverse wave only if it pushes back when you try to slide one layer sideways over the next. That kind of springy resistance to sliding is called shear stiffness, and solids have it. Liquids and gases do not, which is why a transverse wave cannot travel through the body of a liquid or a gas. A rope, a guitar string and a drum skin all work because they are held tight.

Light and radio waves are also transverse, but that is a different story: they do not need a medium, and what oscillates is an electric and a magnetic field, not particles. For this chapter keep to the rope picture.

Longitudinal waves: the particles shuttle along the line

In a longitudinal wave the particles vibrate back and forth along the same line the wave travels. In the second lane, each dot only moves left and right, yet the pattern of bunching and spreading marches to the right.

Where the particles are crowded together, the pressure and density are higher than normal. That region is a compression. Where they are spread apart, the pressure and density are lower than normal. That region is a rarefaction. A longitudinal wave is a train of compressions and rarefactions, one after another.

You can make one with a slinky laid on a table: push a few coils together and let go, and a bunch of squeezed coils runs down the spring, followed by a stretched-out patch. Sound in air is the same thing, with air molecules in place of coils. A loudspeaker or a tuning fork makes the squeezed and stretched regions.

Because a longitudinal wave works by compressing and stretching, it can travel in solids, liquids and gases. Every material resists being squeezed. That is why you can hear a sound through a wall, underwater and in the open air.

The key idea: the wave travels, the particle does not

Almost everyone, at first, imagines that the orange particle must be carried to the right along with the crest or the compression. The simulation exists to cure that picture.

Watch the dashed line through the orange particle. It marks where the particle sits when nothing disturbs it. The particle always comes back to that line. On the rope it goes up to the amplitude, down to minus the amplitude and up again. In the air it goes forward, backward and forward again. In neither case does it drift to the right.

What moves to the right is the pattern: the position of the crest, the position of the compression. Because the pattern moves with the wave speed v, a crest can cross the whole screen while each particle moves only a few centimetres from home.

This matters for everyday life. When a sound reaches you from the other side of a field, no air has travelled from the speaker to your ear. A candle flame near the speaker flickers slightly but is not blown out. The wave has delivered energy to your eardrum, and the air that carried it is still roughly where it started.

Words you need, in plain language

Here are the terms the question papers use, with the symbol and the unit.

Measuring one wavelength

The wavelength is the length of one complete wave. On a transverse wave you can measure it from one crest to the next crest, or from one trough to the next trough. On a longitudinal wave you measure from the centre of one compression to the centre of the next compression, or from one rarefaction to the next.

The rule is that the two points you pick must be matching, meaning the particles at both points are doing exactly the same thing at the same moment. A crest and the next trough are not matching, and neither are a compression and the next rarefaction. They are half a wavelength apart. In the simulation the arrows labelled λ always join matching points.

With the default settings the wavelength is 3 m, so a compression and the next rarefaction sit 1.5 m apart, which is half a wavelength.

Frequency, period and the formula v = fλ

Frequency is how many complete vibrations each particle makes in one second. It is measured in hertz (Hz). A frequency of 1 Hz means one full up-and-down (or back-and-forth) vibration per second. The period is the time for one vibration, so the two are reciprocals of each other.

Now think about one full vibration of the source. In that time T, the first crest leaves the source and travels one full wavelength before the next crest is born. So the wave covers a distance λ in a time T. Speed is distance divided by time, which gives the wave equation.

Why a sound wave is drawn as a sine curve

A longitudinal wave does not look like a sine curve. It looks like a row of dots with crowded and sparse patches. Yet in textbooks and oscilloscopes, sound is always drawn as a smooth wiggle. Why?

Because the wiggle is a graph, not a picture. For each particle, plot how far it has moved from its rest position (up for a shift to the right, down for a shift to the left) against where the particle sits at rest. Because every particle repeats the source's smooth back-and-forth motion a little later than its neighbour, the graph comes out as a sine curve.

Tick the graph box in the simulation and compare the lanes. Where the curve crosses zero going downwards, the particles to the left have moved right and the particles to the right have moved left, so they are squeezed towards that point: a compression (red dot). Where the curve crosses zero going upwards, the neighbours move away from the point: a rarefaction (green dot). The crests and troughs of the curve are the points of greatest displacement, where the density is normal.

That is why a compression is not at a crest of the displacement graph. It sits at a zero of the displacement, where the slope is steepest. Mixing this up is a classic exam trap.

Try it in the simulation

Guess what will happen first, then run it and check.

• Leave the defaults (f = 1 Hz, v = 3 m/s, A = 6 cm). Wait for the wave to reach the orange particle. The readings show λ = 3 m and T = 1 s. Check that two neighbouring crests are really 3 m apart.
• Pause at any moment and look at the dashed line through the orange particle in each lane. Step through a few frames by pressing play and pause again. The particle is never far from its dashed line.
• Set the speed to 0.25× and watch the trace on the right. The orange dot slides along a sine curve. One full bump takes exactly T seconds.
• Raise the frequency to 2 Hz. Before you press anything, predict the new wavelength. Check with the readout: it should halve.
• Raise the wave speed to 4 m/s at 0.5 Hz. Predict λ, then read it. Then lower the speed to 1.5 m/s at 2 Hz and see the crowded waves.
• Change only the amplitude. Watch the rope's height and the strength of the compressions change while λ and T stay put. Amplitude does not affect the wavelength or the speed.
• Turn on the graph and find one compression and one rarefaction in both the dot lane and the graph lane. Confirm the zero-crossing rule from the section above.

Solved problems

Each solution lists what is known, then the formula, then the numbers. Writing it in this order earns full marks in an exam.

Transverse and longitudinal side by side

Use this table to settle any "which one is it?" question.

Mistakes almost everyone makes

Check these before an exam and you will save marks.

• Thinking the particles travel with the wave. They only oscillate about a fixed rest position. The pattern and the energy travel, the matter does not.
• Measuring wavelength between a crest and a trough, or between a compression and a rarefaction. That is only half a wavelength. Always use matching points.
• Saying a higher frequency makes a faster wave. In a given medium the speed is fixed. A higher frequency gives a shorter wavelength, not a higher speed.
• Drawing sound as a transverse wave in the picture of the air. The curve is a graph of displacement; the air itself moves along the direction of travel.
• Putting the compression at the crest of the displacement graph. A compression sits where the displacement is zero and the curve slopes downward.
• Confusing amplitude with wavelength. Amplitude is how big the vibration is, wavelength is how long one wave is. They are independent.
• Forgetting that frequency does not change when a wave enters a new medium; the speed and wavelength do.
• Writing the unit of frequency as seconds. Frequency is in hertz, and period is in seconds.

Where you meet these waves in real life

Both kinds of wave are in the world around you.

• Speech and music: your voice makes compressions and rarefactions in the air, a longitudinal wave that your eardrum turns into a vibration.
• Earthquakes: the fast P-wave is longitudinal and arrives first. The slower S-wave is transverse and arrives second. The gap between them tells seismologists how far away the quake was.
• Musical instruments: a plucked guitar string carries transverse waves, but the sound it makes in the air is longitudinal.
• Ultrasound scans: sound at a frequency far above hearing travels through the body, and the echoes build the picture of a baby or an organ.
• Sonar: ships send a sound pulse down and time the echo to map the sea floor, using the speed of sound in water.
• Stadium waves and flag ripples: transverse patterns where each person or each bit of cloth simply moves up and down.

Exam corner: SSC, HSC and general

Questions on this topic usually ask for definitions (transverse, longitudinal, wavelength), a distinction table, the wave equation v = fλ with a calculation, and the explanation of why sound cannot be a transverse wave in air. Draw a labelled diagram whenever you can: crest, trough, wavelength and amplitude on one sketch, and compression and rarefaction on another.

Revision: last-minute summary

If you only have five minutes, read this list.

• A wave carries energy and the pattern forward. The particles of the medium only oscillate about their rest positions.
• Transverse: particles move at right angles to the wave. Parts: crest, trough. Example: rope, S-wave.
• Longitudinal: particles move along the wave. Parts: compression, rarefaction. Example: sound, P-wave.
• Wavelength λ: distance between two matching points. Frequency f in Hz. Period T = 1/f in seconds.
• Wave equation: v = fλ. In a given medium v is fixed, so a bigger f means a smaller λ.
• When a wave enters another medium, f stays the same while v and λ change together.
• A displacement–position graph makes any longitudinal wave look like a sine curve; a compression sits at a zero crossing, not at a peak.

Frequently asked questions