Waves transfer energy by local disturbance

A wave is a disturbance that transfers energy from one place to another without requiring a net transfer of matter. A medium is a material substance whose particles or fields can be disturbed so that a wave can pass through it. In a water ripple, an air sound wave or a wave on a stretched string, neighbouring particles pass the disturbance along. After the wave has passed, each particle is back where it started on average.

A travelling wave is a wave whose pattern moves through space, carrying energy away from its source. Keep this sentence in your head: the wave travels, but the particles of the medium only oscillate about equilibrium positions. Energy is transferred even though the medium has no resultant displacement.

Transverse waves

A transverse wave is a travelling wave in which the disturbance of the medium is perpendicular to the direction of energy transfer. Stretch a slinky horizontally and shake one end up and down: the pulse travels along the slinky, while each coil moves up and down. A crest is a point of maximum positive displacement on a transverse wave, and a trough is a point of maximum negative displacement.

For a transverse mechanical wave, a displacement–distance graph often looks like the actual shape of the wave at that instant. Careful, though: the graph models particle displacement against position. It isn’t a picture of the path taken by the energy.

Longitudinal waves

A longitudinal wave is a travelling wave in which the disturbance of the medium is parallel to the direction of energy transfer. If a slinky is pushed and pulled along its length, the pulse travels along the slinky while each coil moves backwards and forwards along the same line.

A compression is a region in a longitudinal wave where particles of the medium are closer together than at equilibrium. A rarefaction is a region in a longitudinal wave where particles are farther apart than at equilibrium. On a displacement–distance graph for a longitudinal wave, maximum compression and maximum rarefaction occur where the particle displacement is zero, not at the peaks and troughs of the displacement graph. That surprises students every year, so pause on it: bunching depends on how neighbouring particles are displaced relative to one another.

Describing particle motion as the wave passes

IB questions often ask what a named particle is doing at a particular instant. Don’t say the particle is “moving along with the wave”. For a transverse wave travelling to the right, a particle may be moving up, down or momentarily at rest depending on where it is on the waveform. For a longitudinal wave travelling to the right, a particle may be moving right, left or momentarily at rest.

A displacement–distance graph is a graph that shows the displacement of all particles along a wave at one instant. It gives spatial information, such as wavelength and amplitude. A displacement–time graph is a graph that shows the displacement of one particle as time passes. It gives temporal information, such as period and amplitude.

How to read displacement–distance and displacement–time graphs for the same travelling wave.

A sequence of displacement–distance graphs drawn at equal time intervals shows the wave pattern moving while labelled particles oscillate about fixed equilibrium positions. This is the link between the wave model and simple harmonic motion: each point in the medium can oscillate while the wave pattern itself travels through the medium.

The quantities used to describe a travelling wave

Wavelength is the shortest distance between two points on a wave that are in the same state of oscillation, such as crest to crest, trough to trough, compression to compression, or rarefaction to rarefaction.

Frequency is the number of complete oscillations made per unit time by a particle of the medium. You can also think of it as the number of complete wave cycles passing a fixed point per unit time.

Time period is the time taken for one complete oscillation of a particle of the medium, or the time taken for one wavelength to pass a fixed point.

Amplitude is the maximum displacement of a disturbed quantity from its equilibrium value. In a mechanical wave, this might be a particle displacement. In an electromagnetic wave, it might mean the size of the electric or magnetic field variation. So when you see “displacement” in waves, read it in context.

Wave speed is the speed at which a fixed phase point of the wave pattern, such as a crest or compression, travels through space.

For wave motion,

The hertz is just $\text{s}^{-1}$. A wave with frequency 50 Hz makes 50 complete oscillations each second, and 50 wavelengths pass a fixed point each second.

Why the wave equation is sensible

During one period, the wave pattern moves forward by one wavelength. Speed is distance divided by time, so $v = \frac{\lambda}{T}$. Since frequency is the reciprocal of period, we can write the same relationship as $v = f\lambda$. This equation is not a special new kind of speed; it is ordinary kinematics applied to a repeating pattern.

What happens when frequency changes depends on the wave speed. In a given medium, the speed of a mechanical wave is usually fixed mainly by the properties of that medium, so a higher source frequency gives a shorter wavelength. If the wave enters a different medium, its speed can change. The frequency is set by the source and stays the same across the boundary, so the wavelength changes when the speed changes.

Reading graphs correctly

On a displacement–distance graph, read the amplitude from the equilibrium line to a maximum displacement. Read the wavelength between matching points in phase. This graph does not directly give the period or frequency unless you also know the wave speed or have information about how the pattern changes with time.

On a displacement–time graph for one particle, the amplitude is again measured from equilibrium to maximum displacement. The period is the time for one complete cycle. The graph does not directly give the wavelength, because there is no spatial axis.

For a finite wave train, if a source produces $n$ cycles, the length of the wave train is

Equivalently, if the source operates for a duration $\Delta t$, then the wave train length is

Here is the promised link to kinematics: a wavefront travelling at speed $v$ for a time $\Delta t$ covers a distance $v\Delta t$.

Phase language, used lightly

Two points are in phase if they have the same displacement and the same direction of motion at the same time. Points one wavelength apart are in phase. Points half a wavelength apart are in antiphase: when one is at a crest, the other is at a trough. This language helps when comparing two particles on a wave, or two microphone traces in a sound experiment.

When waves overlap, their disturbances combine at each point. This is superposition, a principle stating that the resultant displacement at a point is the vector sum of the individual displacements there. You meet the full machinery of superposition in C.3, but the idea already fits naturally inside the wave model.

Sound as a longitudinal mechanical wave

A sound wave is a longitudinal mechanical wave made up of pressure and density variations travelling through a medium. In air, a vibrating source, for example a loudspeaker cone, pushes nearby air molecules together and then lets them spread apart again. These compressions and rarefactions then move away from the source.

Sound travels through gases, liquids and solids because their particles interact and pass on a pressure disturbance. Gases and liquids cannot maintain a sideways displacement through their bulk, so sound in these media is longitudinal. In solids, particles are held in fixed arrangements by bonding forces, which allows both longitudinal and transverse mechanical waves.

Particle displacement and pressure variation do not reach their maxima at the same positions in a sound wave. At the centre of a compression or rarefaction, particles are momentarily at their equilibrium positions, while the pressure difference from normal is greatest. Where particle displacement is maximum, the pressure is approximately normal. In phase terms, the pressure variation and particle displacement variation are a quarter of a cycle out of step.

Hearing and energy transfer

The ear responds to pressure variations. When a sound wave reaches the eardrum, it makes the eardrum vibrate, and that vibration is converted into nerve signals. Energy has been transferred from the source to the ear; the air itself has not travelled from the loudspeaker to the listener.

As sound spreads out from a source, its amplitude usually falls. Part of the decrease comes from geometric spreading, where the same energy is shared over a larger area. Part also comes from dissipation, where wave energy is transferred into internal energy of the air and surroundings.

Measuring the speed of sound

One simple direct method uses a sharp sound with a visible cue, such as a clapperboard. Measure a large distance, record the time delay between seeing the clap and hearing it, and then use $\text{speed} = \text{distance} / \text{time}$. A large distance helps because human reaction time is not small compared with the sound travel time over a few metres. Echo methods use the round-trip distance to a large flat wall, so the sound distance is twice the wall distance.

A more precise school-lab method uses two microphones and an oscilloscope. A signal generator drives a loudspeaker at known frequency. Keep one microphone fixed and move the other until the two oscilloscope traces are in phase, then keep moving it until they are in phase again. The distance moved is one wavelength, so $v = f\lambda$. Measuring several wavelengths and averaging reduces percentage uncertainty.

Smartphones can also work as acoustic timers, provided you think carefully about sampling rate and timing resolution. If the time interval you want to measure is only a few milliseconds, a low sampling rate gives too few samples to locate the arrival time reliably. Good experimental design here is not about having expensive kit; it is about making the travel time large enough and the timing uncertainty small enough.

Electromagnetic waves as transverse field waves

An electromagnetic wave is a travelling wave made of oscillating electric and magnetic fields, at right angles to each other, that can transfer energy through a vacuum or through a medium. We model it as transverse because the fields vary perpendicular to the direction in which the wave travels.

In a vacuum, all electromagnetic waves travel at the same speed: $3.00 \times 10^8\ \mathrm{m\ s^{-1}}$. The family includes radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. The wave equation still applies, so when the speed is fixed, a long wavelength goes with a low frequency, while a short wavelength goes with a high frequency.

Light, then, is not separate from electromagnetic radiation. Visible light is simply the narrow part of the electromagnetic spectrum that the human eye detects. A prism can disperse white light because different visible wavelengths travel differently through glass, so they leave the prism in different directions.

Production and propagation

Electromagnetic radiation is produced when charged particles accelerate or when charged particles change energy state. A helpful field picture is to imagine a charge moving: information about the change in its electric field cannot show up everywhere at once. The changing field pattern moves outward at the speed of light, and the changing electric and magnetic fields keep one another going as the wave travels.

That is why electromagnetic waves do not need a material medium. In the nineteenth century, physicists suggested an invisible “aether” to carry light waves, but modern physics has no need for it. The field itself varies and carries the energy.

When electromagnetic radiation enters a different medium, its frequency is still fixed by the source, but its speed and wavelength may change. The same wave model is used for radio communication, light optics, medical imaging and astronomical observation; the details change because matter interacts differently with different wavelengths.

The electromagnetic spectrum

The electromagnetic spectrum is continuous, so the named regions do not have sharp boundaries. The Physics data booklet gives approximate wavelength ranges, and you should know the order from longest to shortest wavelength: radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays.

Electromagnetic spectrum ordered from longest to shortest wavelength, with approximate regions and typical interactions or uses.

Radio waves have long wavelengths, which makes them useful for communication because they can diffract around large obstacles and may reflect from regions of the upper atmosphere. Microwaves are used in communications, radar and heating when molecules in food absorb their energy. Infrared radiation is strongly linked with thermal emission from warm objects. Visible light passes through much of Earth’s atmosphere, which is fortunate for us. Ultraviolet can excite or ionise atoms and is partly absorbed by the atmosphere. X-rays and gamma rays have very short wavelengths and can penetrate matter, although they may also ionise atoms and damage living tissue.

X-rays give a lovely nature-of-science example. Wilhelm Röntgen noticed unexpected fluorescence near a gas discharge tube and investigated it rather than dismissing it. The new radiation passed through materials opaque to visible light and was quickly used for medical imaging. The lesson is not “luck is enough”; careful attention to an anomaly can open up a new area of physics.

The wave model also helped push physics towards quantum mechanics. Light behaves as a wave in interference and diffraction, but as photons in other observations. Later, matter itself was found to show wave-like behaviour. Schrödinger’s wave mechanics is a major example of a familiar model being extended into a new domain, with all the caution that such extension requires.

Two wave families, one wave model

A mechanical wave needs a material medium. Its disturbance is passed along by the motion or deformation of particles in that medium. Sound in air, waves on a string and seismic waves are mechanical waves.

An electromagnetic wave is different: it is carried by oscillating fields and can travel through a vacuum. Both types transfer energy. Both can be described using wavelength, frequency, period, amplitude and speed. They can also reflect, refract, diffract and superpose, although those behaviours are treated more fully in later wave topics.

Comparison of mechanical and electromagnetic waves.

The key difference is what oscillates. In a mechanical wave, particles of the medium oscillate about equilibrium positions. In an electromagnetic wave, electric and magnetic field strengths vary with time and position. That is why a sound wave cannot cross empty space, while sunlight can travel from the Sun to Earth through a vacuum.

Intensity and spreading from a source

Intensity is the power transferred by a wave per unit area perpendicular to the direction of energy transfer. For a point source radiating uniformly in three dimensions,

So $I \propto \frac{1}{r^2}$. The reason is straightforward: as distance increases, the same power spreads over the surface area of a larger sphere. Double the distance and the area becomes four times larger, so the intensity falls to one quarter of its original value. This inverse square idea is especially useful for electromagnetic radiation from stars, lamps and transmitters, provided absorption and beaming are not dominating the situation.

Waves in technology and society

Wave technology is useful because waves can carry energy and information without transporting matter from source to receiver. Electromagnetic waves are used in radio, mobile communication, radar, thermal imaging, optical fibres, X-ray imaging and radiotherapy. Mechanical waves are used in sonar, ultrasound scans, non-destructive testing and seismology.

There is a good NOS point here: the same wave model supports many different technologies, but each application depends on how waves interact with matter. Microwaves are useful in communication and cooking for different reasons; X-rays are useful in imaging because they penetrate soft tissue more than bone, but the same ionising ability makes radiation protection essential. Physics models are powerful because they generalise — and useful because we also know where the details matter.