Wave properties — amplitude frequency wavelength speed

Something travels, but nobody moves

Picture a full football stadium. A few dozen people stand up, raise their arms and sit down again. The people next to them do the same, a moment later. Then the next people, and the next. A wave sweeps around the whole stadium.

In 2002, three physicists studied videos of these crowd waves in stadiums holding more than 50 000 people. They found that a typical wave moves at about 12 metres per second, which is about 20 seats every second (Farkas, Helbing and Vicsek, Nature, 2002).

Now ask a strange question. What exactly is moving at 12 m/s? No person travels around the stadium. Every person stands up and sits down in their own seat. Yet something clearly goes around, and you can measure its speed.

What travels is the pattern: the place where people are standing. This is the idea at the centre of every wave in physics. Sound, light, ripples on a pond and earthquake waves all carry something from place to place, while the material they pass through stays where it is. This lecture gives you the words to describe any wave, and the one equation that links them.

The material a wave travels through is called the medium. For sound in a room, the medium is the air. For the stadium wave, the medium is the crowd. Light is the exception: it needs no medium at all, and it can cross empty space.

Two ways for the particles to move

In the stadium, each person moves up and down, while the wave moves sideways around the stadium. The two directions are at right angles. Waves like this are called transverse.

A wave on a rope works the same way. Shake one end up and down, and each piece of the rope moves up and down while the wave travels along it. Light and all other electromagnetic waves are transverse too.

Now push and pull one end of a long, loose spring, along its length. Each coil moves backwards and forwards, in the same direction that the wave travels. Waves like this are called longitudinal. Sound in air is longitudinal. A loudspeaker pushes the air forwards and pulls it back, again and again.

In a longitudinal wave, the particles crowd together in some places and spread apart in others. A region where they are crowded is a compression. A region where they are spread out is a rarefaction.

Earthquakes show both kinds at once. An earthquake sends out longitudinal waves, called P waves, and transverse waves, called S waves. P waves travel faster through rock, so a distant measuring station feels them first. Try it yourself question 4 uses the gap between them.

The words for describing a wave

To describe a wave precisely, we need a picture of it. There are two different pictures, and this is where students most often lose marks. So read this part slowly.

Picture 1 is a photograph. Freeze the rope at one instant. Plot how far each piece of rope is from its rest position, against its distance along the rope. This is a displacement–distance graph. It shows the whole wave at one moment.

Picture 2 is a recording of one piece. Watch one single piece of the rope over time. Plot its displacement against time. This is a displacement–time graph. It shows one particle at every moment.

Both graphs have the same wavy shape, so they are easy to confuse. But they measure different things.

Here are the five quantities, each with its picture.

Amplitude, $A$, is the largest displacement of a particle from its rest position, in metres. It is the height of a crest above the middle line, not the height from a trough to a crest. You can read it from either graph.

Wavelength, $\lambda$ (the Greek letter lambda), is the length of one complete wave, in metres. Measure it from crest to next crest, from trough to next trough, or between any two neighbouring points doing exactly the same thing. You read it from the displacement–distance graph only.

Period, $T$, is the time for one complete oscillation of a particle, in seconds. You read it from the displacement–time graph only.

Frequency, $f$, is the number of complete oscillations each second. Its unit is the hertz (Hz), and 1 Hz means one oscillation per second.

Period and frequency describe the same thing in two ways. If a particle makes 5 oscillations each second, each oscillation takes $\frac{1}{5}$ of a second. So:

Wave speed, $v$, is the speed of the pattern through the medium, in m/s. It is the 12 m/s of the stadium wave. It is not the speed of any particle.

Why speed equals frequency times wavelength

Now the central idea. Watch one point on the rope. In one period $T$, the particle there makes one complete oscillation. During that same time, the wave pattern moves forward by exactly one wavelength. This is because, after one period, the next crest has arrived where the last crest was.

So the wave travels a distance $\lambda$ in a time $T$. Speed is distance divided by time:

That is it. That is the whole idea. You can also see it by counting. If 5 waves pass you every second, and each wave is 0.8 m long, then 5 × 0.8 = 4 m of wave passes you every second. The wave's speed is 4 m/s.

Here is the part most students get backwards. What decides the speed? For a rope, the speed depends on the rope: how tight it is and how heavy it is for its length. For sound, it depends on the air. In air at 20 °C, sound travels at about 343 m/s, whatever note is played. So in a given medium, $v$ is fixed, and if you raise the frequency, the wavelength must get shorter to keep $f\lambda$ the same.

Explore this below. Choose a medium, change the frequency, and watch what the wavelength does. Then switch to the spring to see the same wave as a longitudinal one.

What should you have seen? The dark particle stays where it is and only oscillates. The triangle on the crest moves at the speed of the rope, whatever the frequency. When you raise the frequency in the same medium, the wavelength gets shorter. When you change to the faster medium at the same frequency, the wavelength doubles. The simulation runs at half speed, so that you can follow each particle.

Phase: which points move together

Look at points P and Q on the displacement–distance graph. Both are at a crest. A moment later, both will be moving down together. Points that are doing exactly the same thing at the same moment are in phase. Any two points one whole wavelength apart are in phase.

Now look at P and R. When P is at a crest, R is at a trough. When P moves down, R moves up. They always do opposite things. Points like this are in antiphase. Any two points half a wavelength apart are in antiphase.

We measure how far out of step two points are as a phase difference, in degrees. One whole wavelength is 360°, like one full turn of a circle. So for two points a distance $d$ apart:

P and R are 0.4 m apart on a wave with $\lambda$ = 0.8 m, so their phase difference is $\frac{0.4}{0.8}\times 360^{\circ }=180^{\circ }$. At Pre-University you will also meet phase difference in radians, where 360° is $2\pi$ rad.

Amplitude and energy

A wave carries energy, and the amplitude tells you how much. Here is why, using the idea of kinetic energy, $E_{\text{k}}=\frac{1}{2}m{v}^2$.

In one period, a particle moves from its rest position up to a crest, down to a trough, and back. That is a distance of $4A$. If you double the amplitude at the same frequency, the particle must cover twice the distance in the same time. So it moves twice as fast. Kinetic energy depends on speed squared, so its energy becomes $2^2=4$ times as large.

So, for a wave of the same frequency in the same medium, the energy it carries is proportional to the square of its amplitude. Double the amplitude, and the wave carries four times the energy. This is why a loud sound, with a large amplitude, carries far more energy than a quiet one at the same pitch.

Worked examples

Conversions like the one in Worked Example 2 cause many lost marks. Find the slip in this student's working.

A check that reuses your own numbers only catches arithmetic slips. A check against a known size, such as the length of an aerial, can catch a wrong idea.

Three classic mistakes