Travelling waves that do not travel on

A standing wave is a wave pattern where the positions of maximum displacement and zero displacement stay fixed, because two identical travelling waves superpose as they move in opposite directions. That last phrase matters. In this course, standing waves are formed from two waves, not from three or more.

A travelling wave is a disturbance that transfers energy through a medium or field as its shape moves through space. A standing wave behaves differently. The pattern doesn’t travel along the string or pipe; each point in the medium oscillates with its own amplitude. Energy is not carried steadily along the standing-wave pattern as it is in a single travelling wave.

The usual setup is straightforward: a wave reaches a boundary, reflects, and overlaps with the incoming wave. If the incident and reflected waves have the same frequency, wavelength and amplitude, their superposition can produce a fixed pattern of cancellation and reinforcement. The wave equation is

In a real experiment, the reflected wave is often slightly weaker because the support absorbs some energy. The “zero” points may not be perfectly still, but their positions are still where the model predicts. That’s why a school string vibrator can give a convincing standing wave even though the laboratory is not a perfect physics diagram.

Reflection and boundary behaviour

A boundary condition is a constraint at the end of a wave-bearing system that determines what displacement is allowed there. For a string fixed to a wall, the end cannot move, so the displacement at the fixed end is always zero. The reflected pulse is inverted: it returns with opposite displacement, a phase change of $\pi$ rad.

At a free end, the end of the string can move. The reflected pulse is not inverted, so the end becomes a place of large displacement. In both cases, the reflected wave travels back through the incident wave and the two superpose.

Seeing the pattern in the lab

A standard way to see standing waves is to attach a string to a vibration generator, pass it over or attach it to a fixed support, and then vary the generator frequency until a clear pattern appears. Sketch each clear pattern rather than just admiring it. Successive patterns show the allowed harmonics set by the boundary conditions.

A stroboscope set close to the driving frequency can make the string look almost frozen, which helps you identify the fixed positions and the positions of maximum displacement. In this practical, useful observations include the number of loops, the positions of nodes, whether the end is fixed or free, and how sharply the pattern appears. Random errors usually come from judging amplitude and node position by eye; systematic errors can come from an uncertain string length, uneven tension, or assuming the vibrator end is a perfect node when it is actually moving slightly.

Fixed points in a moving pattern

A node is a point on a standing wave where the displacement is always zero, because the two contributing waves always cancel there. An antinode is a point on a standing wave where the displacement amplitude is maximum, because the two contributing waves reinforce most strongly there.

Adjacent nodes are separated by $\lambda/2$. Adjacent antinodes are also separated by $\lambda/2$. The distance from a node to the nearest antinode is $\lambda/4$. In a standing-wave diagram, these distances are often the quickest route to the wavelength.

Relative amplitude

The amplitude is not the same everywhere along a standing wave. At a node it is zero; at an antinode it is largest; between them it changes smoothly. This is a big difference from an ideal travelling wave, where every point of the medium has the same amplitude if there is no energy loss.

So, when you compare two points on a standing wave, don’t just write “same wave, same amplitude”. Check where the points are. A point closer to an antinode has a larger amplitude than a point closer to a node.

Phase along a standing wave

Points between the same pair of adjacent nodes oscillate in phase. They reach maximum positive displacement together, pass through equilibrium together, and reach maximum negative displacement together. Their amplitudes may be different, but the timing is the same.

Points in neighbouring loops, separated by a node, oscillate in antiphase. Their phase difference is $\pi$ rad, so when one loop is displaced upward, the next is displaced downward. In the IB style of diagram, the solid and dashed curves usually show the two extreme positions half a cycle apart, not two different waves.

Here’s the contrast with travelling waves. In a travelling wave, neighbouring points usually have a continuously changing phase difference along the wave. In a standing wave, whole regions between nodes move together, and adjacent regions are $\pi$ rad out of phase.

Interpreting diagrams carefully

For a transverse wave on a string, the drawn curve can be read as the actual shape of the string at an instant. For a sound wave in a pipe, the drawn curve is not the shape of the air. It is a graph of displacement amplitude of air molecules along the pipe axis. This warning saves many wrong answers later.

Harmonics and allowed wavelengths

A harmonic is an allowed standing-wave mode with a frequency that is an integer multiple of the lowest allowed frequency for that system. That lowest-frequency mode is the first harmonic. In this course, use “first harmonic”, not the alternative terms often used in music or older textbooks.

The method is usually the same: apply the boundary conditions, work out how much of a wavelength fits into length $L$, then use $v = f\lambda$.

Strings: two fixed boundaries

A fixed end is a node. So, for a string fixed at both ends, both ends are nodes. The first harmonic places half a wavelength on the string:

In general,

$$L = \frac{n\lambda_n}{2}$$

so

$$\lambda_n = \frac{2L}{n}$$

The frequency is

$$f_n = \frac{nv}{2L},$$

where the symbols have the meanings already given above. All integer harmonics are possible: $n = 1, 2, 3, \dots$

Strings: one fixed and one free boundary

A fixed end is a node, while a free end is an antinode. With one fixed and one free boundary, the first harmonic fits one quarter of a wavelength into the length:

$$L = \frac{\lambda_1}{4}$$

The next allowed pattern still has to keep a node at one end and an antinode at the other. That allows only odd harmonics.

For this case,

$$L = \frac{n\lambda_n}{4}, \quad \lambda_n = \frac{4L}{n},$$

and

$$f_n = \frac{nv}{4L}$$

for $n = 1, 3, 5, \dots$

This is why the second visible pattern is the third harmonic, not the second harmonic. The name comes from the frequency ratio to the first harmonic, not from the order in which you happen to draw the patterns.

Strings: two free boundaries

A free end is an antinode, so a string or flexible rod with two free ends has antinodes at both ends. Its frequency sequence matches the case with two fixed ends:

$$f_n = \frac{nv}{2L}$$

for $n = 1, 2, 3, \dots$

The shapes look different because the nodes and antinodes swap positions. The allowed wavelengths and frequencies, however, follow the same integer pattern. In practice such a system has to be supported at a node; otherwise the support changes the boundary condition.

Pipes: displacement nodes and antinodes only

For standing waves in air, describe vibration modes using displacement nodes and displacement antinodes. Pressure nodes and pressure antinodes are not required here. Open-pipe end corrections are not used in this course either, so take the pipe length exactly as the length shown or given.

A closed end is a displacement node because the wall prevents air molecules from moving along the pipe axis. An open end is a displacement antinode because air molecules at the open end are free to move. Pipes are described simply as open or closed.

Pipe formulae from boundary conditions

For a pipe open at both ends, both ends are displacement antinodes. The allowed sequence is the same as for two free ends:

$$f_n = \frac{nv}{2L}$$

for $n = 1, 2, 3, \dots$

For a pipe closed at both ends, both ends are displacement nodes. The allowed sequence is the same as for two fixed ends:

$$f_n = \frac{nv}{2L}$$

for $n = 1, 2, 3, \dots$

For a pipe closed at one end and open at the other, one end is a displacement node and the other is a displacement antinode:

$$f_n = \frac{nv}{4L}$$

for $n = 1, 3, 5, \dots$

A pipe closed at one end has a first harmonic at half the frequency of an open-open pipe of the same length, assuming the same wave speed. That is why instruments with different end conditions jump to different next notes when driven harder: an open-open pipe can move to the second harmonic, while a closed-open pipe has no even harmonics and jumps to the third.

Determining wavelength and frequency

To find an unknown wavelength or frequency, begin with the boundary sketch. Count how many quarter-wavelengths or half-wavelengths fit into $L$. Then use $v = f\lambda$. For example, if a closed-open pipe is in its third harmonic, $L = \frac{3\lambda_3}{4}$, so $\lambda_3 = \frac{4L}{3}$ and $f_3 = \frac{3v}{4L}$.

When the gas in a pipe changes, or the temperature changes, the wave speed changes. For the same pipe and the same harmonic, the pipe length and boundary conditions fix the allowed wavelength, so the frequency changes in proportion to the wave speed.

Natural and forced oscillations

A natural frequency is a frequency at which a system oscillates freely after being disturbed and then left alone. A free vibration is an oscillation of a system at its own natural frequency without a continuing external periodic driver. Many systems have more than one natural frequency, although a single mass on a spring gives the clearest starting model.

For a mass–spring oscillator,

A stiffer spring raises the natural frequency; a larger mass lowers it.

For a simple pendulum at small amplitude,

Notice the shared pattern: a stronger “restoring effect” gives a higher frequency, while greater inertia gives a lower frequency.

A forced vibration is an oscillation in which an external periodic force makes a system oscillate. The driving frequency is the frequency of that external periodic force. In a motor-driven mass–spring system, for example, the motor sets the driving frequency, while the mass and spring still have their own natural frequency.

Resonance as efficient energy transfer

Resonance is a phenomenon in which a driven oscillator reaches a large amplitude because the driving frequency is close to one of its natural frequencies. If the driving frequency is far from the natural frequency, the driver pushes at badly timed points in successive cycles, so the amplitude stays small.

At resonance, the timing works. The driver transfers energy to the oscillator efficiently, rather like pushing a swing at the right moment each cycle. In steady state, the amplitude becomes constant when the energy supplied per cycle by the driver equals the energy dissipated per cycle by damping.

Resonance still obeys conservation of energy. A large amplitude does not mean energy has appeared from nowhere; energy has been fed into the oscillator over many cycles and stored as kinetic and potential energy, while the driver continuously replaces losses.

Resonance and simple harmonic motion

Simple harmonic motion is the ideal model: the restoring force is proportional to displacement, and there is no energy loss. Real resonant systems usually have damping, so their motion is not perfectly simple harmonic. Damping adds a dissipative effect that depends on motion, often on speed, and slightly changes the natural frequency when it is not negligible.

The SHM formulae still help because they identify the natural frequency around which resonance occurs. The actual resonant frequency and maximum amplitude, though, depend on both the oscillator and the energy losses in the system.

Demonstrating resonance

Barton’s pendulums make a lovely classroom demonstration. A heavy driver pendulum shakes a support, and several lighter pendulums of different lengths respond. The pendulum with the same length as the driver has the same natural frequency and develops the largest amplitude. Shorter and longer pendulums respond weakly because their natural frequencies do not match the driver.

In a quantitative investigation, a clamped hacksaw blade can be driven by an electromagnet connected to a variable-frequency signal generator. The free-end amplitude is measured against driving frequency, then repeated for different blade lengths or different damping. Good data collection means choosing enough frequencies near the peak, not just evenly spaced values far away from the interesting part.

Resonance of molecules and the greenhouse effect

The greenhouse effect can be modelled using resonance ideas. Gas molecules have natural modes of vibration and rotation. Infrared radiation is absorbed strongly when its frequency matches an allowed molecular mode, so energy transfers to the molecule. Greenhouse gases are effective because they absorb in infrared regions emitted by Earth’s surface.

As a Nature of Science point, this model links several areas of physics and chemistry: waves, energy transfer, molecular structure and thermal radiation. The resonance picture explains selective absorption, but a full climate model also needs emission, convection, atmospheric composition and feedbacks.

What damping does to resonance

Damping is a dissipative process in which mechanical energy is transferred from an oscillating system to the surroundings, usually by friction, air resistance, turbulence or internal heating. It matters at resonance because the driver is trying to build up energy while damping takes it away.

With little damping, the resonance peak is tall and narrow. The system can reach a large maximum amplitude, but only across a small range of driving frequencies. Increase the damping, and the peak drops and spreads out. The oscillator becomes less selective: it responds over a wider range, but the amplitude is never as dramatic.

The resonant frequency is the driving frequency at which the amplitude of the driven oscillator is maximum. With zero damping, the simple model places this at the natural frequency. With light or moderate damping, the frequency for maximum amplitude is slightly less than the undamped natural frequency. As damping increases, the peak moves to a lower frequency and becomes harder to pick out. With heavy damping, there may be no clear resonance peak at all.

Only a qualitative treatment is required here. You should be able to read the family of resonance curves: more damping gives a smaller maximum amplitude, a broader response, and a resonant frequency shifted downward when amplitude is used to define resonance.

Phase and energy input

The phase between the driver and the driven oscillator changes with driving frequency. At low driving frequency, the driven system tends to move nearly in phase with the driver. Near resonance, the driver leads by about a quarter of a cycle, so it supplies energy at a very effective part of the motion. At high driving frequency, the driven system is nearly half a cycle out of phase with the driver.

Damping changes how sharp this phase change is. Light damping gives a rapid transition around resonance; heavier damping spreads the transition over a wider frequency range.

Useful and destructive resonance

Resonance is not “good” or “bad”; it is powerful. Useful resonance appears in musical instruments, microwave heating, optical cavities in lasers, and medical magnetic resonance imaging. In each case, the useful feature is selective energy transfer at particular frequencies.

Destructive resonance happens when unwanted vibrations build up. Bridges, vehicle parts, washing machines and rotating machinery can all suffer large oscillations if a driving frequency matches a natural frequency. Engineers reduce the risk by changing natural frequencies, adding damping, or avoiding operating speeds that drive the system strongly.

Three damping regimes

Light damping is damping where an oscillator keeps oscillating, but its amplitude steadily falls. On a displacement–time graph, the curve still crosses equilibrium again and again; the peaks just get smaller. In many lightly damped systems, the amplitude envelope is approximately exponential, so equal time intervals reduce the amplitude by the same fraction.

With light damping, the period changes only slightly compared with the undamped oscillator. Energy is being removed, but not so strongly that the oscillation stops.

Critical damping is damping where an oscillator returns to equilibrium in the shortest possible time without oscillating. It’s the Goldilocks case in many engineering designs: no overshoot, no slow drift back. Car suspension and door-closing mechanisms are designed with this idea in mind, even if the real device is more complicated than the textbook curve.

Heavy damping is damping where an oscillator returns to equilibrium without oscillating, but more slowly than in critical damping. It is sometimes called overdamping. The resistance is so strong that the system creeps back instead of snapping back.

Reading damping graphs

On a displacement–time graph, light damping gives repeated oscillations with decreasing amplitude. Critical damping reaches equilibrium fastest without crossing it. Heavy damping also stays on one side of equilibrium, but takes longer to settle.

On a resonance curve, light damping produces a tall, sharp maximum. Critical and heavy damping remove the dramatic peak; the response becomes smaller and less frequency-selective. Keep the two graph types separate: one shows displacement against time after release, while the other shows steady-state amplitude against driving frequency.