IB DP Physics · Theme C · Wave behavior

Standing waves and resonance.

C.4 · Nodes and antinodes, strings and pipes, driven oscillations and damping
Standard level and higher level

SL + HL · 4 hours C.3 Wave phenomena ↗

The journey

Three parts.

Part 01

Standing waves

How they form, nodes and antinodes, amplitude and phase

Part 02

Strings and pipes

Boundary conditions and the harmonics they allow

Part 03

Resonance and damping

Natural and driving frequencies, and the effect of damping

C.3 added two waves. Here the two waves are the same wave, going both ways.

C.4 STANDING WAVES AND RESONANCE · SL + HL

Part 01

Standing waves.

A wave that oscillates without going anywhere

C.4 STANDING WAVES AND RESONANCE · SL + HL

Superposition again

How a standing wave forms.

Two identical waves traveling in opposite directions
Usually a wave and its own reflection
Their sum has fixed points that never move

Same frequency, same wavelength, same amplitude.

TWO TRAVELING WAVES AND THEIR SUM

C.4 STANDING WAVES AND RESONANCE · SL + HL

Watch one dot

At each black dot on the bottom line, the two traveling waves are always exactly opposite: whenever one has a displacement upwards there, the other has an equal displacement downwards. Their sum is zero at every moment, so that point never moves. Half way between two dots the waves are always in step with each other, and the sum swings to twice the amplitude of either.

Where the second wave comes from

Pluck a guitar string and a wave runs along it, reflects at the fixed end and runs back through the wave that is still arriving. The outgoing and returning waves are automatically identical, apart from direction. Sound reflecting from the end of a pipe, and microwaves reflecting from the metal wall of an oven, do the same.

Why “standing”

The crests of a traveling wave move along. Here the pattern stays where it is: the positions of greatest and of zero movement are fixed. It is also called a stationary wave.

A definition worth learning: a standing wave is formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions.

The fixed points and the busiest points

Nodes and antinodes.

Node: the displacement is always zero
Antinode: the amplitude is greatest
Node to next node: \(\dfrac{\lambda}{2}\)
Node to next antinode: \(\dfrac{\lambda}{4}\)

THREE LOOPS

C.4 STANDING WAVES AND RESONANCE · SL + HL

Finding a wavelength

One loop of a standing wave is half a wavelength, not a whole one: a full wavelength of a sine curve has a crest and a trough, and those are two neighboring loops. If the nodes on a string are 0.40 m apart, the wavelength is 0.80 m. This is the most direct way there is of measuring a wavelength, since the nodes stay still to be measured.

A microwave oven

With the turntable removed, a bar of chocolate heats in patches. The melted spots are antinodes of the standing wave inside the oven, and they are about 6 cm apart. So \(\lambda = 12\) cm, and with the frequency printed on the back of the oven, 2.45 GHz, \(v = f \lambda = 2.45 \times 10^{9} \times 0.12 = 2.9 \times 10^{8}\) m s⁻¹: the speed of light, measured in a kitchen.

Taking the distance between neighboring nodes as the wavelength. It is half the wavelength.

How the points of a standing wave move

Amplitude and phase.

Every point oscillates with the same frequency
The amplitude varies: zero at a node, greatest at an antinode
Between two nodes: all in phase
Either side of a node: in antiphase

ONE INSTANT

C.4 STANDING WAVES AND RESONANCE · SL + HL

Compare with a traveling wave

In a traveling wave every particle has the same amplitude, and each is slightly behind its neighbor in phase, so the phase changes steadily along the wave. A standing wave is the reverse on both counts. The amplitude depends on where you are. The phase has only two possible values: every point in a loop reaches its greatest displacement at the same instant as all the others in that loop, and exactly half a cycle away from the points in the loops next door.

Twice a cycle the string is flat

All the points pass through their equilibrium positions at the same moment. At that instant the string is straight, but every part of it is moving, fastest at the antinodes. All the energy is kinetic. A quarter of a cycle later the string is at its greatest displacement and momentarily at rest: all the energy is potential.

Phase difference between two points on a standing wave: 0 if there is an even number of nodes between them (including none), \(\pi\) if there is an odd number.

Compare

Standing and traveling waves.

	Traveling wave	Standing wave
Energy	Transferred along the wave	Not transferred: stored between the nodes
Amplitude	The same for every point	Varies from zero at nodes to greatest at antinodes
Phase	Changes steadily along the wave	The same within a loop; opposite in the next
Pattern	Moves at the wave speed	Stays in place
Wavelength	Crest to crest	Twice the node-to-node distance

C.4 STANDING WAVES AND RESONANCE · SL + HL

Quick check

Standing waves.

01What two waves make a standing wave?

Identical waves in opposite directions

02What is a node?

A point of zero displacement at all times

03Neighboring nodes are 0.30 m apart. Wavelength?

0.60 m

04Two points in the same loop. Phase difference?

Zero

05Two points in neighboring loops. Phase difference?

\(\pi \,\mathrm{rad}\)

06Does a standing wave transfer energy along the string?

No

PRESS → TO REVEAL EACH ANSWER

Part 02

Strings and pipes.

The ends decide which standing waves fit

C.4 STANDING WAVES AND RESONANCE · SL + HL

The story · Leipzig, 1787

Background story · not assessed

Chladni makes sound visible.

Ernst Chladni, drawn in 1809

Four brass plates of different shapes on wooden stands, each covered with a pattern of curved lines of sand — **Chladni figures** on brass plates (a modern exhibit)

Chladni, a physicist and musician, sprinkled sand on a metal plate and drew a violin bow across its edge
The sand jumped off the moving parts and gathered along the nodal lines: a standing wave in two dimensions
In Paris in 1809 Napoleon watched, and set a prize for the mathematics of the patterns

Portrait: Gilles-Louis Chrétien, 1809, public domain. Photograph: Rodrigo Tetsuo Argenton; exhibit by Estes Objethos Atelier, CC BY-SA 4.0.

C.4 STANDING WAVES AND RESONANCE · SL + HL

What the ends allow

Boundary conditions.

A fixed end of a string · a closed end of a pipe

Node

The string is held, or the air is up against a wall: nothing can move there

A free end of a string · an open end of a pipe

Antinode

Nothing holds it back, so the movement is greatest there

Only wavelengths that put the right thing at each end can form a standing wave.

C.4 STANDING WAVES AND RESONANCE · SL + HL

Three cases, for strings and for pipes

Two nodes at the ends: a string fixed at both ends, or a pipe closed at both ends. Two antinodes at the ends: a string free at both ends, or a pipe open at both ends. A node at one end and an antinode at the other: a string fixed at one end only, or a pipe closed at one end. The first two cases give the same list of wavelengths. The third gives a different one.

What moves in a pipe

The air. Its molecules oscillate to and fro along the length of the pipe: a longitudinal standing wave. The diagrams draw the size of that movement as a curve across the pipe, but the air is not moving sideways. In this course the nodes and antinodes of a pipe always refer to displacement.

Why an open end is an antinode

At the mouth of a pipe the air is free to move in and out into the room, with nothing to stop it, so its movement is greatest there.

Pipes are described as open or closed. The lowest mode is the first harmonic; the words “fundamental” and “overtone” are not used in this course.

A node at each end

A string fixed at both ends.

Whole half wavelengths fit \(L = n \dfrac{\lambda}{2}\) so \(\lambda_{n} = \dfrac{2L}{n}\) \(f_{n} = \dfrac{v}{\lambda_{n}} = \dfrac{n v}{2L} = n f_{1}\)

\(n = 1, 2, 3 \dots\): the harmonics. All are possible.

THE FIRST THREE HARMONICS

C.4 STANDING WAVES AND RESONANCE · SL + HL

Why only these

A wave on the string reflects at each end and overlaps with itself again and again. For most wavelengths the many reflections are out of step and cancel. Only when a whole number of half wavelengths fits exactly between the two fixed ends do they all reinforce, and a large, steady standing wave builds up.

A real string does several at once

A plucked string vibrates in a mixture of harmonics. The first sets the pitch you hear. The others, at twice, three times and more of that frequency, give the note its character, which is why a guitar and a violin playing the same note sound different.

Changing the note

\(f_{1} = \frac{v}{2L}\). Shorten the string by pressing it against a fret and the note rises. Tighten it and the wave speed increases, so the note rises. A heavier string has a lower wave speed and a lower note.

The same results hold for a pipe closed at both ends, and (with antinodes for nodes) for a string free at both ends.

Worked example

A guitar string.

\(L = 0.65 \,\mathrm{m}\) · wave speed \(v = 286 \,\mathrm{m} \,\mathrm{s}^{-1}\) \(\lambda_{1} = 2L = 2 \times 0.65 = 1.30 \,\mathrm{m}\) \(f_{1} = \dfrac{v}{\lambda_{1}} = \dfrac{286}{1.30}\) \(f_{1} = \class{res}{220 \,\mathrm{Hz}}\)

The third harmonic \(\lambda_{3} = \dfrac{2L}{3} = \dfrac{1.30}{3} = 0.433 \,\mathrm{m}\) \(f_{3} = 3 f_{1} = 3 \times 220 = \class{res}{660 \,\mathrm{Hz}}\) Nodes every 0.217 m along the string

C.4 STANDING WAVES AND RESONANCE · SL + HL

The method for any harmonic

Sketch the pattern. Count the loops and write the length of the string as that many half wavelengths. Find the wavelength, then use \(f = \frac{v}{\lambda}\). The sketch does the remembering: there is no need to learn a formula for each case.

Two wavelengths, two speeds

1.30 m is the wavelength of the wave on the string, which travels at 286 m s⁻¹. The sound it makes in the air has the same frequency, 220 Hz, but travels at 340 m s⁻¹, so its wavelength in the air is \(\frac{340}{220} = 1.5\) m. The string forces the air to vibrate at its own frequency; the wavelength then depends on the medium.

Touch a vibrating string lightly at its midpoint and you force a node there. The odd harmonics die, and the second harmonic sings out an octave higher.

An antinode at each end

A pipe open at both ends.

Whole half wavelengths again \(\lambda_{n} = \dfrac{2L}{n}\) \(f_{n} = \dfrac{n v}{2L} = n f_{1}\)

The same wavelengths as the string. All harmonics.

DISPLACEMENT OF THE AIRANTINODES AT BOTH ENDS

C.4 STANDING WAVES AND RESONANCE · SL + HL

Reading the picture

The curves show how far the air moves at each place along the pipe, to and fro along its length. Where the two curves are far apart the air is surging in and out: an antinode. Where they cross the air stays still: a node. In the first harmonic the air at the two ends moves in opposite directions, rushing out of both ends together and then in at both ends together, while the air in the middle does not move at all.

Why the same list as a string

From one antinode to the next is half a wavelength, just as from one node to the next. A pipe with an antinode at each end therefore fits a whole number of half wavelengths, like a string with a node at each end.

How sound reflects from an open end

There is no wall, but there is a sudden change: inside, the air is confined; outside, it is not. Part of the wave is reflected at any such change, and that reflected wave makes the standing wave.

A flute and an open organ pipe behave like this. Opening a hole in the side of a flute shortens the vibrating column of air and raises the note.

A node at one end, an antinode at the other

A pipe closed at one end.

Odd quarter wavelengths fit \(L = n \dfrac{\lambda}{4}\) so \(\lambda_{n} = \dfrac{4L}{n}\) \(f_{n} = \dfrac{n v}{4L} = n f_{1}\)

\(n = 1, 3, 5 \dots\): odd harmonics only.

DISPLACEMENT OF THE AIRCLOSED AT THE LEFT

C.4 STANDING WAVES AND RESONANCE · SL + HL

Why a quarter

The shortest piece of a standing wave that has a node at one end and an antinode at the other is a quarter of a wavelength. So the first harmonic has \(\lambda = 4L\): twice the wavelength, and half the frequency, of an open pipe of the same length. A closed pipe sounds an octave lower than an open one.

Why the even harmonics are missing

To get from a node back to an antinode you must add whole half wavelengths: \(\frac{1}{4}, \frac{3}{4}, \frac{5}{4} \dots\) of a wavelength in the pipe. Two quarters or four quarters would put a node, or an antinode, at both ends, which the pipe does not allow. The second pattern in the diagram is therefore the third harmonic, at three times the frequency of the first.

The same holds for a string fixed at one end and free at the other.

Calling the second pattern of a closed pipe the “second harmonic”. Its frequency is \(3 f_{1}\), so it is the third harmonic. There is no second.

Worked example

Two pipes of the same length.

\(L = 0.50 \,\mathrm{m}\) · closed at one end · \(v = 340 \,\mathrm{m} \,\mathrm{s}^{-1}\) \(\lambda_{1} = 4L = 2.0 \,\mathrm{m}\) \(f_{1} = \dfrac{v}{\lambda_{1}} = \dfrac{340}{2.0} = \class{res}{170 \,\mathrm{Hz}}\) Next: \(f_{3} = 3 f_{1} = \class{res}{510 \,\mathrm{Hz}}\)

The same pipe, open at both ends \(\lambda_{1} = 2L = 1.0 \,\mathrm{m}\) \(f_{1} = \dfrac{v}{\lambda_{1}} = \dfrac{340}{1.0} = \class{res}{340 \,\mathrm{Hz}}\) Next: \(f_{2} = 2 f_{1} = \class{res}{680 \,\mathrm{Hz}}\)

C.4 STANDING WAVES AND RESONANCE · SL + HL

Measuring the speed of sound

Hold a tuning fork of known frequency over a tube that stands in water, and raise the tube slowly. The sound swells when the air column is the right length for a standing wave. The first loud position is at a quarter of a wavelength and the next at three quarters, so the distance between the two is exactly half a wavelength. With a 512 Hz fork and positions 33.2 cm apart, \(\lambda = 0.664\) m and \(v = 512 \times 0.664 = 340\) m s⁻¹.

Why use the difference

The antinode at an open end actually sits slightly beyond the end of the tube. Taking the difference between two positions cancels that small shift. The course ignores it elsewhere: no end corrections are needed in calculations.

A narrow tube closed at the bottom, such as a test tube, is a closed pipe. Add water and the air column shortens, so the note rises.

Quick check

Strings and pipes.

01What is found at the closed end of a pipe?

A displacement node

02A 1.2 m string fixed at both ends, first harmonic. Wavelength?

2.4 m

03Its first harmonic is 100 Hz. Frequency of the fourth?

400 Hz

04Which harmonics does a pipe closed at one end have?

Odd ones only

05A pipe closed at one end is 0.85 m long. First harmonic? (340 m s⁻¹)

\(\frac{340}{4 \times 0.85} = 100 \,\mathrm{Hz}\)

06Its closed end is opened. The first harmonic becomes?

200 Hz: double

PRESS → TO REVEAL EACH ANSWER

Part 03

Resonance and damping.

What happens when an oscillator is pushed, and when it is held back

C.4 STANDING WAVES AND RESONANCE · SL + HL

Free and forced oscillations

Natural and driving frequency.

Left to itself

Natural frequency

The frequency at which a system oscillates freely after being displaced and released. Set by the system: \(m\) and \(k\), or \(l\) and \(g\)

Pushed from outside

Driving frequency

The frequency of a periodic force applied to the system. Set by whatever is doing the pushing

A driven system ends up oscillating at the driving frequency. How much it oscillates depends on how close that is to its natural frequency.

C.4 STANDING WAVES AND RESONANCE · SL + HL

A child on a swing

Let go and the swing moves back and forth at its natural frequency, set by its length as for any pendulum (C.1). Now push it. Push at random and little happens. Push once every swing, at the natural frequency, and each push adds to the motion already there: the amplitude grows and grows.

Everything has natural frequencies

A wine glass rings at one when tapped. A string or a column of air has a whole set: the harmonics of Part 2. A building, a bridge and a car on its springs each have theirs. A molecule of carbon dioxide has natural frequencies in the infrared, which is why it absorbs infrared (B.2).

Three frequencies to keep apart: the natural frequency (of the system), the driving frequency (of the force), and the frequency of the resulting oscillation, which equals the driving frequency.

Driving at the natural frequency

Resonance.

The amplitude is greatest when driving frequency = natural frequency
Energy is transferred from the driver most efficiently
Far from that frequency the response is small

THE RESPONSE OF A DRIVEN OSCILLATOR

C.4 STANDING WAVES AND RESONANCE · SL + HL

Why the peak

At resonance the driving force is always pushing in the direction the system is already moving, so it does positive work during every part of every cycle. At any other frequency the force and the motion fall out of step, and for part of each cycle the force acts against the motion and takes energy back out.

What limits the amplitude

The amplitude does not grow for ever. As it grows, more energy is lost in each cycle to friction and air resistance. It settles at the size for which the energy supplied by the driver in each cycle equals the energy lost. Less resistance means a larger amplitude before the two balance.

The two ends of the graph

Driven very slowly, the system simply follows the driver, moving with about the driver’s own amplitude. Driven very fast, its inertia stops it responding in time, and it hardly moves.

Barton’s pendulums show this in the laboratory: several light pendulums of different lengths are driven by one heavy one, and the one of matching length swings most.

Losing energy

Damping.

Damping: a resistive force takes energy from the oscillation
Light: many oscillations, slowly shrinking
Critical: back to rest in the shortest time, with no oscillation
Heavy: no oscillation, a slow return

RELEASED FROM REST

C.4 STANDING WAVES AND RESONANCE · SL + HL

Light damping

A pendulum swinging in air. Each swing is slightly smaller than the last, and the amplitude falls by the same fraction in every cycle, so the peaks lie on a smooth decaying curve. The period hardly changes: it is very slightly longer than with no damping at all.

Critical damping

A car’s suspension is designed to be close to this. After a bump the body should return to its level smoothly and quickly, without bouncing. The same is wanted of a door closer, and of the needle of a measuring instrument, which should settle on its reading at once.

Heavy damping

Picture the pendulum hanging in thick oil. Pulled aside and released, it oozes back towards the vertical and takes a long time to arrive. More damping than critical does not mean a quicker return: it means a slower one.

Thinking that the heaviest damping stops the motion fastest. Critical damping is the quickest way back to equilibrium.

Qualitative

Damping and resonance.

More damping: a lower peak
A broader peak
At a slightly lower frequency

Light damping: a tall, sharp resonance, very close to the natural frequency.

THREE AMOUNTS OF DAMPING

C.4 STANDING WAVES AND RESONANCE · SL + HL

Lower

A damped system loses more energy in each cycle, so the balance between energy supplied and energy lost is reached at a smaller amplitude.

Broader

A lightly damped system responds strongly only to a narrow range of driving frequencies and ignores the rest. A heavily damped one responds modestly to a wide range. Engineers choose between the two. A radio tuner needs a sharp resonance, to pick one station out of hundreds. A loudspeaker cone needs a broad one, so that it reproduces all the notes of the music about equally and does not boom at one of them.

Shifted

With heavier damping the greatest amplitude occurs at a driving frequency a little below the natural frequency. For light damping the shift is too small to notice.

A sketch of the effect of extra damping needs all three: a lower peak, a wider peak, and the top moved slightly to the left. The curves should never cross.

Where it matters

Resonance: useful and destructive.

Useful

Musical instruments

Strings and air columns resonate at their harmonics; the body of a violin strengthens the sound

Useful

Tuning a radio

A circuit is set to resonate at the frequency of one station

Useful

MRI scanners

Nuclei in a strong magnet absorb radio waves of one frequency

Destructive

Buildings in earthquakes

Ground motion near a building’s natural frequency can bring it down

Destructive

Bridges and footsteps

Soldiers break step on a bridge; walkways are fitted with dampers

Destructive

Machinery

An unbalanced wheel or engine shakes a car violently at one particular speed

C.4 STANDING WAVES AND RESONANCE · SL + HL

Two ways to prevent damage

Move the natural frequency away from the driving frequency, by changing the mass or the stiffness of the structure. Or add damping, so that the resonance peak is low even if it is driven. Tall buildings in earthquake zones use both: flexible foundations, and in some a huge suspended mass near the top that swings against the motion of the building and absorbs its energy.

The wine glass

A glass has very little damping, so its resonance is sharp and tall. A loud note at exactly its natural frequency can drive the rim to an amplitude large enough to break it. A note that is slightly off does nothing: the peak is too narrow.

Have two useful and two destructive examples ready, and be able to say what is driving and what is resonating in each.

The story · Washington State, 7 November 1940

Background story · not assessed

Galloping Gertie.

A frame of black and white film showing the roadway of a suspension bridge breaking and falling into the water between its towers — The **Tacoma Narrows Bridge** as its deck fell, filmed by Barney Elliott

Opened on 1 July 1940. Its 853 m span moved so much in wind that the builders gave it a nickname
Four months later, in a wind of about 64 km h⁻¹, the deck began to twist and tore itself apart
Textbooks long called it resonance. It was not

The cause was aeroelastic flutter: the twisting deck changed the airflow, and the airflow fed the twisting. A steady wind, not a periodic push.

Film still: Barney Elliott, 1940, public domain.

C.4 STANDING WAVES AND RESONANCE · SL + HL

Quick check

Resonance and damping.

01At what frequency does a driven system oscillate?

The driving frequency

02When does resonance occur?

Driving frequency = natural frequency

03Which damping returns a system to rest fastest?

Critical damping

04Damping is increased. What happens to the height of the resonance peak?

It falls

05And to the frequency of the peak?

Slightly lower

06Give one useful example of resonance.

Tuning a radio; a musical instrument

PRESS → TO REVEAL EACH ANSWER

Summary

Standing waves and resonance on one slide.

Standing wave

Two identical waves, opposite ways

No energy transferred
Node to node: \(\frac{\lambda}{2}\)

Along the wave

Amplitude varies; phase 0 or \(\pi\)

In phase within a loop, antiphase in the next

Same ends

\(\lambda_{n} = \frac{2L}{n}\)

String fixed at both ends; pipe open at both ends
All harmonics

Different ends

\(\lambda_{n} = \frac{4L}{n}\), \(n\) odd

Pipe closed at one end
Odd harmonics only

Resonance

Driving = natural frequency

Greatest amplitude and energy transfer

Damping

Light · critical · heavy

More damping: a lower, broader peak at a slightly lower frequency

C.4 STANDING WAVES AND RESONANCE · SL + HL

Summary

Key words.

Standing waveA pattern that oscillates in place

NodeZero displacement at all times

AntinodeGreatest amplitude

HarmonicOne of the allowed standing wave patterns

First harmonicThe pattern of lowest frequency

Natural frequencyThe frequency of free oscillation

Driving frequencyThe frequency of the applied force

ResonanceGreatest amplitude: driving = natural frequency

Critical dampingThe quickest return with no oscillation

C.4 STANDING WAVES AND RESONANCE · SL + HL