Standing Waves

A standing wave is a wave pattern that appears to remain fixed in position rather than traveling through a medium. Standing waves are produced when two waves with the same frequency, wavelength, and approximately equal amplitude travel in opposite directions and interfere.

Standing waves are important in many vibrating systems, including:

• guitar, violin, and piano strings
• organ pipes and wind instruments
• air columns
• microwave cavities
• antennas
• bridges and buildings
• laboratory wave experiments

Standing waves also provide a clear connection between interference, resonance, wavelength, and frequency.

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How Standing Waves Form

Imagine a wave traveling along a stretched string.

When the wave reaches a fixed end, it reflects and travels back along the string. The incoming wave and reflected wave now travel in opposite directions.

The two waves overlap and interfere.

At certain frequencies, this interference creates a stable pattern in which some points remain stationary while other points oscillate strongly.

This pattern is a standing wave.

The individual waves are still traveling. It is the resulting interference pattern that appears stationary.

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Conditions for a Stable Standing Wave

A clear standing-wave pattern is produced when the two interfering waves have:

• the same frequency
• the same wavelength
• similar or equal amplitudes
• opposite directions of travel

The waves continuously undergo both constructive and destructive interference.

At particular locations, the interference is always destructive.

At other locations, the interference alternates to produce maximum oscillation.

These locations are called nodes and antinodes.

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Nodes

A node is a point on a standing wave where the displacement is always zero.

At a node, the two traveling waves interfere destructively.

If one wave attempts to displace the medium upward, the other produces an equal downward displacement.

The resulting displacement is:

zero

Nodes therefore appear stationary.

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Antinodes

An antinode is a point where the oscillation has maximum amplitude.

At an antinode, the interfering waves produce maximum constructive interference during the oscillation.

The particles of the medium move through the largest range.

Therefore:

• nodes → minimum displacement
• antinodes → maximum displacement

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Nodes and Antinodes Alternate

Along a standing wave, nodes and antinodes alternate.

A typical pattern is:

Node — Antinode — Node — Antinode — Node

The distance relationships are important.

Distance between adjacent nodes:

λ/2

Distance between adjacent antinodes:

λ/2

Distance from a node to the nearest antinode:

λ/4

where λ represents wavelength.

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Determining Wavelength from a Standing Wave

Suppose adjacent nodes are separated by:

0.40 m

Since adjacent nodes are separated by λ/2:

λ/2 = 0.40 m

Therefore:

λ = 0.80 m

If the wave speed is 120 m/s:

v = fλ

120 = f(0.80)

f = 150 Hz

Therefore:

f = 150 Hz

Standing-wave patterns can therefore be used to determine wavelength and frequency experimentally.

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Waves on a String

Consider a string fixed at both ends.

The ends cannot move, so both ends must be:

nodes

Only certain wavelengths can fit between these fixed endpoints.

This means the string cannot form a stable standing wave at every possible frequency.

Instead, it has a set of allowed frequencies called its natural frequencies.

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Fundamental Frequency

The simplest standing-wave pattern on a string is called the fundamental mode or first harmonic.

For a string fixed at both ends, the fundamental pattern contains:

• a node at each end
• one antinode in the middle
• half of one wavelength along the string

Therefore:

L = λ₁/2

so:

λ₁ = 2L

where:

• L = length of the string
• λ₁ = wavelength of the fundamental

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Fundamental Frequency of a String

Using:

v = fλ

and:

λ₁ = 2L

we obtain:

f₁ = v/(2L)

Therefore:

f₁ = v/(2L)

The fundamental frequency is the lowest natural frequency at which the string can form a standing wave.

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Example: Fundamental Frequency

A string is:

0.80 m

long.

Wave speed along the string is:

240 m/s

Fundamental wavelength:

λ₁ = 2L

λ₁ = 2(0.80)

λ₁ = 1.60 m

Now:

f₁ = v/λ₁

f₁ = 240/1.60

f₁ = 150 Hz

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Harmonics

Higher-frequency standing-wave patterns are called harmonics.

For a string fixed at both ends:

first harmonic = fundamental

second harmonic = 2 × fundamental frequency

third harmonic = 3 × fundamental frequency

and so on.

In general:

fₙ = nf₁

where n is the harmonic number.

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Harmonics on a String

For a string fixed at both ends:

The general wavelength relationship is:

λₙ = 2L/n

Therefore:

fₙ = nv/(2L)

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Understanding the Harmonic Number

For a string fixed at both ends, the harmonic number tells us how many half-wavelengths fit along the string.

First harmonic:

L = λ/2

Second harmonic:

L = 2(λ/2)

Third harmonic:

L = 3(λ/2)

In general:

L = nλ/2

This leads directly to:

λₙ = 2L/n

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Nodes and Antinodes in String Harmonics

For a string fixed at both ends:

First harmonic:

• 2 nodes
• 1 antinode

Second harmonic:

• 3 nodes
• 2 antinodes

Third harmonic:

• 4 nodes
• 3 antinodes

In general, for the nth harmonic:

• number of antinodes = n
• number of nodes = n + 1

including the fixed endpoints.

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Worked Example: Third Harmonic

A string has length:

L = 1.20 m

Find the wavelength of its third harmonic.

Use:

λₙ = 2L/n

λ₃ = 2(1.20)/3

λ₃ = 2.40/3

λ₃ = 0.80 m

If the wave speed is:

v = 200 m/s

then:

f₃ = v/λ₃

f₃ = 200/0.80

f₃ = 250 Hz

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Finding the Fundamental from a Higher Harmonic

Suppose the fourth harmonic of a string is:

480 Hz

Since:

f₄ = 4f₁

480 = 4f₁

Therefore:

f₁ = 120 Hz

The other harmonics would include:

f₂ = 240 Hz

f₃ = 360 Hz

f₄ = 480 Hz

f₅ = 600 Hz

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Standing Waves in Air Columns

Standing waves can also form in columns of air.

This occurs in musical instruments such as:

• flutes
• clarinets
• recorders
• trumpets
• trombones
• organ pipes

The boundary conditions depend on whether the end of the air column is:

• open
• closed

This changes the allowed standing-wave patterns.

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Open Ends and Closed Ends

For the common displacement representation of air-column standing waves:

At an open end, air particles can move freely.

Therefore, an open end is a:

displacement antinode

At a closed end, air particles cannot move freely.

Therefore, a closed end is a:

displacement node

This distinction determines which wavelengths fit inside the air column.

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Pressure Nodes and Antinodes

Sound waves can also be described using variations in pressure.

For pressure standing waves:

• an open end is approximately a pressure node
• a closed end is a pressure antinode

Therefore, it is important to know whether a diagram represents:

particle displacement

or:

pressure variation

The node and antinode locations are reversed between these two descriptions.

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Open-Open Air Columns

An air column open at both ends has displacement antinodes at both ends.

For the fundamental mode:

L = λ₁/2

Therefore:

λ₁ = 2L

and:

f₁ = v/(2L)

This is the same mathematical relationship as a string fixed at both ends, even though the boundary conditions are physically different.

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Harmonics in an Open-Open Pipe

An ideal open-open pipe supports:

• first harmonic
• second harmonic
• third harmonic
• fourth harmonic
• and so on

The relationships are:

λₙ = 2L/n

and:

fₙ = nv/(2L)

Therefore:

fₙ = nf₁

All integer harmonics are possible in the ideal model.

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Example: Open Pipe

An open pipe has length:

L = 0.85 m

Take the speed of sound as:

v = 340 m/s

Fundamental wavelength:

λ₁ = 2L

λ₁ = 1.70 m

Fundamental frequency:

f₁ = 340/1.70

f₁ = 200 Hz

Second harmonic:

f₂ = 400 Hz

Third harmonic:

f₃ = 600 Hz

Fourth harmonic:

f₄ = 800 Hz

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Closed-Open Air Columns

Now consider an air column that is:

• closed at one end
• open at the other

For displacement:

• closed end = node
• open end = antinode

The shortest standing-wave pattern that satisfies these conditions contains:

one quarter of a wavelength

Therefore:

L = λ₁/4

so:

λ₁ = 4L

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Fundamental Frequency of a Closed-Open Pipe

Using:

v = fλ

and:

λ₁ = 4L

we obtain:

f₁ = v/(4L)

This is half the fundamental frequency of an open-open pipe of the same length, assuming the same wave speed.

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Harmonics in a Closed-Open Pipe

A closed-open pipe supports only the odd harmonics in the ideal model.

Its frequencies are:

f₁, 3f₁, 5f₁, 7f₁, ...

The allowed frequencies can be written:

fₙ = nv/(4L)

where:

n = 1, 3, 5, 7, ...

The allowed wavelengths are:

λₙ = 4L/n

for odd n.

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Why Even Harmonics Are Missing

A closed-open pipe must always have:

• a displacement node at the closed end
• a displacement antinode at the open end

Patterns corresponding to the even harmonics of the fundamental cannot satisfy both boundary conditions simultaneously.

Therefore, the allowed modes correspond to:

1st harmonic

3rd harmonic

5th harmonic

7th harmonic

and so on.

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Important Harmonic Numbering

The second allowed mode of a closed-open pipe is actually the:

third harmonic

The third allowed mode is the:

fifth harmonic

This distinction is important.

For example:

First mode → f₁

Second mode → 3f₁

Third mode → 5f₁

Do not automatically assume that the second mode means the second harmonic.

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Example: Closed Pipe

A closed-open pipe has length:

L = 0.50 m

Take:

v = 340 m/s

Fundamental frequency:

f₁ = v/(4L)

f₁ = 340/[4(0.50)]

f₁ = 340/2

f₁ = 170 Hz

The next allowed frequency is the third harmonic:

f₃ = 3(170)

f₃ = 510 Hz

The next is:

f₅ = 5(170)

f₅ = 850 Hz

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Comparing Standing-Wave Systems

This table is one of the most useful summaries for solving standing-wave problems.

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Resonance

Every vibrating system has one or more natural frequencies.

If an external periodic force drives the system at or near one of these natural frequencies, the system can oscillate with a much larger amplitude.

This phenomenon is called:

resonance

Examples include:

• pushing a playground swing at the correct rhythm
• vibrating a guitar string
• blowing across an air column
• tuning a radio circuit
• vibration of mechanical structures

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Natural Frequency

A natural frequency is a frequency at which a system tends to oscillate when disturbed.

For example, if a stretched string is plucked, it naturally vibrates using combinations of its allowed frequencies.

These frequencies are determined by physical properties such as:

• length
• tension
• mass
• shape
• material
• boundary conditions

Different systems therefore have different natural frequencies.

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Driving Frequency

A driving frequency is the frequency of an external force acting repeatedly on a system.

Suppose a swing has a natural frequency.

If someone pushes it randomly, the motion may remain relatively small.

If the pushes occur at an appropriate frequency and phase, energy is transferred efficiently to the swing.

The amplitude increases.

This is resonance.

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Maximum Amplitude at Resonance

When the driving frequency is close to the natural frequency:

f_drive ≈ f_natural

the amplitude becomes large.

In an idealized system:

maximum response occurs at resonance

Real systems experience damping, so the exact frequency of maximum amplitude can differ slightly from the undamped natural frequency.

For introductory problems, it is normally treated as occurring at the natural frequency.

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Energy Transfer During Resonance

Resonance occurs because energy is transferred efficiently from the driving source to the oscillating system.

If the driving force repeatedly acts in a way that reinforces the motion, the system gains energy.

The amplitude increases until energy supplied by the driving force is balanced by energy losses.

These losses may result from:

• friction
• air resistance
• internal deformation
• sound production
• thermal energy transfer

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Damping

Damping is the loss of mechanical energy from an oscillating system.

Greater damping generally:

• reduces the maximum resonant amplitude
• makes the resonance less sharp
• causes oscillations to die away more quickly

A lightly damped system can produce a much larger resonant response than a heavily damped system.

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Resonance Curve

If amplitude is plotted against driving frequency, the graph typically shows a peak near the natural frequency.

Far from resonance:

amplitude is relatively small

Near resonance:

amplitude increases

At resonance:

amplitude reaches its maximum

The width and height of the peak depend strongly on damping.

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Resonance and Musical Instruments

Musical instruments depend heavily on resonance.

When a guitar string is plucked, the string vibrates at its natural frequencies.

The body of the guitar also vibrates and transfers energy to the surrounding air.

In wind instruments, the air column resonates at particular frequencies determined partly by:

• tube length
• whether the ends are open or closed
• air temperature
• instrument geometry

This produces the instrument's characteristic notes.

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Harmonics and Musical Tone

Two instruments can play the same fundamental frequency but sound different.

This occurs partly because they produce different combinations and strengths of harmonics.

The fundamental largely determines the perceived pitch.

The mixture of harmonics contributes to the instrument's:

timbre

or tone quality.

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Changing the Length of a String

For a string:

f₁ = v/(2L)

If wave speed remains constant, decreasing L increases the fundamental frequency.

Therefore:

shorter string → higher frequency

longer string → lower frequency

This is why pressing a guitar string against a fret changes its pitch.

The vibrating portion of the string becomes shorter.

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Changing Wave Speed on a String

Wave speed on a stretched string depends on tension and linear mass density.

A commonly used relationship is:

v = √(T/μ)

where:

• T = tension
• μ = mass per unit length

Increasing tension increases wave speed.

Since:

f₁ = v/(2L)

increasing the tension generally increases the resonant frequencies.

This is why tightening a musical string raises its pitch.

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Changing the Length of an Air Column

For an open-open pipe:

f₁ = v/(2L)

For a closed-open pipe:

f₁ = v/(4L)

In both cases:

longer air column → lower fundamental frequency

shorter air column → higher fundamental frequency

This principle is used in wind instruments.

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Speed of Sound and Temperature

The speed of sound in air depends on temperature.

At typical room conditions, it is approximately:

340–343 m/s

As air temperature increases, the speed of sound increases.

Since:

f = v/λ

changes in sound speed can slightly change the resonant frequencies of air columns.

This is one reason temperature can affect the tuning of some instruments.

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Worked Problem: String Harmonics

A string is:

L = 1.50 m

long and has wave speed:

v = 180 m/s

Find the fundamental frequency.

f₁ = v/(2L)

f₁ = 180/[2(1.50)]

f₁ = 180/3

f₁ = 60 Hz

Third harmonic:

f₃ = 3f₁

f₃ = 3(60)

f₃ = 180 Hz

Third-harmonic wavelength:

λ₃ = v/f₃

λ₃ = 180/180

λ₃ = 1.0 m

Check:

λ₃ = 2L/3

λ₃ = 3/3

λ₃ = 1.0 m

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Worked Problem: Open Air Column

An open-open pipe is:

L = 0.75 m

long.

Take:

v = 330 m/s

Fundamental frequency:

f₁ = 330/[2(0.75)]

f₁ = 330/1.50

f₁ = 220 Hz

Third harmonic:

f₃ = 3(220)

f₃ = 660 Hz

Third-harmonic wavelength:

λ₃ = 2L/3

λ₃ = 1.50/3

λ₃ = 0.50 m

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Worked Problem: Closed-Open Air Column

A pipe closed at one end has:

L = 0.40 m

and:

v = 320 m/s

Fundamental frequency:

f₁ = v/(4L)

f₁ = 320/[4(0.40)]

f₁ = 320/1.60

f₁ = 200 Hz

The next allowed harmonic is the third:

f₃ = 3(200)

f₃ = 600 Hz

The next is the fifth:

f₅ = 5(200)

f₅ = 1000 Hz

There is no second or fourth harmonic in the ideal closed-open model.

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Worked Problem: Identifying a Harmonic from Nodes

A string fixed at both ends shows:

5 antinodes

Since the nth harmonic has n antinodes:

n = 5

Therefore, the string is vibrating in its:

fifth harmonic

It will also contain:

6 nodes

including the two endpoints.

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Worked Problem: Finding Wavelength from Nodes

Adjacent nodes in a standing wave are:

0.25 m apart

Since:

node-to-node distance = λ/2

then:

λ/2 = 0.25

λ = 0.50 m

If:

v = 150 m/s

then:

f = v/λ

f = 150/0.50

f = 300 Hz

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Standing Waves Do Not Transport Energy Like Traveling Waves

A traveling wave carries energy from one region to another.

In an ideal standing wave, there is no net energy transfer along the medium averaged over time.

Energy is instead exchanged locally between different forms as parts of the medium oscillate.

This is another important distinction between traveling and standing waves.

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Standing Waves in Structures

Standing-wave and resonance effects are important in engineering.

Structures such as:

• bridges
• towers
• aircraft
• machines
• buildings

have natural frequencies.

If repeated forces occur near one of those frequencies, large oscillations may develop.

Engineers therefore analyze natural frequencies and resonance when designing structures.

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Preventing Unwanted Resonance

Engineers can reduce dangerous resonance by:

• increasing damping
• changing the mass of the system
• changing stiffness
• changing dimensions
• altering natural frequencies
• isolating vibrations

The goal is often to prevent external driving frequencies from matching important natural frequencies.

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Useful Problem-Solving Relationships

For all waves:

v = fλ

For a string fixed at both ends:

λₙ = 2L/n

fₙ = nv/(2L)

For an open-open pipe:

λₙ = 2L/n

fₙ = nv/(2L)

For a closed-open pipe:

λₙ = 4L/n

fₙ = nv/(4L)

where n = 1, 3, 5, 7...

Distance between adjacent nodes:

λ/2

Distance between adjacent antinodes:

λ/2

Distance from node to nearest antinode:

λ/4

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A Reliable Strategy for Standing-Wave Problems

First identify the system:

• string fixed at both ends
• open-open pipe
• closed-open pipe

Then determine the boundary conditions.

Ask:

• Where must nodes occur?
• Where must antinodes occur?
• Which harmonics are allowed?

Next determine how much of a wavelength fits into the length L.

Then:

1. calculate λ
2. use v = fλ
3. calculate f
4. check that the harmonic number is allowed
5. include correct units

Drawing the standing-wave pattern can make the problem much easier.

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Common Mistakes

Thinking a standing wave is not made from traveling waves

Standing waves result from interference between waves traveling in opposite directions.

Confusing nodes and antinodes

Nodes have zero displacement. Antinodes have maximum displacement.

Thinking adjacent nodes are one wavelength apart

Adjacent nodes are separated by:

λ/2

Thinking a node and neighboring antinode are λ/2 apart

They are separated by:

λ/4

Assuming L = λ for the fundamental of a fixed string

For the first harmonic:

L = λ/2

so:

λ = 2L

Using open-open formulas for a closed-open pipe

Closed-open fundamental:

λ = 4L

not 2L.

Including even harmonics in an ideal closed-open pipe

Only odd harmonics occur:

1, 3, 5, 7...

Calling the second allowed mode of a closed-open pipe the second harmonic

It is the:

third harmonic

Confusing displacement and pressure nodes in air columns

At an open end there is approximately a displacement antinode but a pressure node.

Assuming resonance means frequency increases

Resonance describes a large response when the driving frequency matches or approaches a natural frequency.

Ignoring units

Length should normally be converted to metres when using wave speed in m/s.

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Key Terms

Standing wave — stationary interference pattern produced by waves of the same frequency traveling in opposite directions.

Interference — combination of overlapping waves.

Constructive interference — interference that produces increased displacement.

Destructive interference — interference that reduces or cancels displacement.

Node — point in a standing wave where displacement remains zero.

Antinode — point where the oscillation has maximum amplitude.

Fundamental frequency — lowest natural frequency of a vibrating system.

First harmonic — fundamental mode of vibration.

Harmonic — allowed frequency related to the fundamental frequency.

Natural frequency — frequency at which a system naturally tends to oscillate.

Resonance — large-amplitude response produced when a system is driven near one of its natural frequencies.

Driving frequency — frequency of an external periodic force.

Damping — reduction of oscillation caused by energy loss.

Air column — column of air capable of supporting sound-wave standing patterns.

Open-open pipe — air column open at both ends.

Closed-open pipe — air column closed at one end and open at the other.

Boundary condition — physical restriction determining whether a node or antinode must occur at a boundary.

Displacement node — location where particle displacement is zero.

Pressure node — location where sound-pressure variation is minimal.

Mode — particular allowed standing-wave pattern of a system.

Timbre — characteristic quality of a sound influenced by its harmonic content.

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Key Takeaways

• Standing waves form when waves of the same frequency and wavelength travel in opposite directions and interfere.
• Nodes are locations of zero displacement.
• Antinodes are locations of maximum displacement.
• Adjacent nodes are separated by λ/2.
• Adjacent antinodes are separated by λ/2.
• A node and its nearest antinode are separated by λ/4.
• Strings fixed at both ends have nodes at both ends.
• The fundamental wavelength of a fixed string is λ₁ = 2L.
• The fundamental frequency of a fixed string is f₁ = v/(2L).
• Fixed strings support all integer harmonics.
• For strings, fₙ = nf₁ and λₙ = 2L/n.
• An ideal open-open air column also supports all integer harmonics.
• An open end is a displacement antinode and approximately a pressure node.
• A closed end is a displacement node and approximately a pressure antinode.
• A closed-open pipe has fundamental wavelength λ₁ = 4L.
• A closed-open pipe has fundamental frequency f₁ = v/(4L).
• Ideal closed-open pipes support only odd harmonics.
• Resonance occurs when a system is driven near one of its natural frequencies.
• Resonance produces a large amplitude because energy is transferred efficiently to the oscillating system.
• Damping limits resonant amplitude.
• Standing waves explain important aspects of musical instruments, acoustics, engineering, and mechanical vibration.

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Check Your Understanding

1. Explain how two traveling waves can combine to produce a standing wave.
2. Describe the difference between a node and an antinode.
3. The distance between two adjacent nodes is 0.60 m. Determine the wavelength.
4. A node and its nearest antinode are 0.20 m apart. Determine the wavelength.
5. A string fixed at both ends is 1.2 m long. Determine the wavelength of its fundamental mode.
6. A 0.80 m string has a wave speed of 240 m/s. Calculate its fundamental frequency.
7. The fundamental frequency of a string is 120 Hz. Determine the frequencies of its second, third, and fifth harmonics.
8. A string fixed at both ends displays four antinodes. Which harmonic is present? How many nodes are there?
9. A string is 1.5 m long and supports waves traveling at 300 m/s. Calculate the wavelength and frequency of its third harmonic.
10. An open-open pipe is 0.50 m long. Using a sound speed of 340 m/s, calculate its fundamental frequency and second harmonic frequency.
11. A closed-open pipe is 0.50 m long. Using a sound speed of 340 m/s, calculate its fundamental frequency and the next two allowed harmonic frequencies.
12. Explain why an ideal closed-open pipe does not support the second harmonic.
13. Explain the difference between displacement nodes and pressure nodes at the open and closed ends of an air column.
14. Describe resonance in terms of natural frequency, driving frequency, energy transfer, and amplitude.
15. A vibrating system has a natural frequency of 250 Hz. Predict what happens to its amplitude as a driving frequency is gradually increased from 100 Hz through 250 Hz and then to 400 Hz. Explain how damping would affect the maximum response.