Mechanical Waves
2. Longitudinal Waves
Learning outcomes
- I can describe longitudinal waves.
- I can identify compressions and rarefactions.
- I can explain how sound travels through matter.
- I can compare longitudinal and transverse waves.
- I can interpret longitudinal wave diagrams.
Longitudinal Waves
A longitudinal wave is a wave in which the particles of a medium vibrate parallel to the direction in which the wave travels.
Imagine pushing and pulling one end of a stretched spring or slinky. Sections of the spring move backward and forward while the disturbance travels along the length of the spring.
If the wave travels:
→
the particles oscillate:
↔
The particle motion and wave direction are therefore parallel.
This parallel motion is the defining feature of a longitudinal wave.
Waves Transfer Energy
Like all waves, longitudinal waves transfer energy from one location to another.
However, the particles of the medium do not travel along with the wave over large distances.
Instead, they oscillate backward and forward around their equilibrium positions.
For example, when sound travels across a classroom, air molecules do not travel all the way from the speaker to the listener.
Instead:
- one group of air molecules vibrates
- these molecules interact with neighbouring molecules
- the disturbance passes through the air
- energy travels across the room
The matter itself mainly oscillates locally.
Particle Motion in a Longitudinal Wave
The most important feature of a longitudinal wave is the relationship between:
- particle motion
- wave direction
Suppose a longitudinal wave travels to the right.
The particles move:
backward and forward
along the same line.
Therefore:
particle motion ∥ wave direction
The symbol ∥ means parallel to.
A Longitudinal Wave in a Spring
A spring or slinky provides an excellent model of a longitudinal wave.
If several coils are pushed together and released, the compressed region travels along the spring.
The individual coils do not travel from one end of the spring to the other.
Instead, each coil moves backward and forward around its normal position.
The travelling pattern consists of alternating:
- compressions
- rarefactions
Compressions
A compression is a region of a longitudinal wave where particles are closer together than normal.
In a spring:
compression = coils close together
In a sound wave:
compression = air molecules temporarily crowded together
A compression generally corresponds to:
- higher particle density
- higher pressure
than the surrounding undisturbed medium.
Rarefactions
A rarefaction is a region where particles are farther apart than normal.
In a spring:
rarefaction = coils spread farther apart
In air:
rarefaction = air molecules temporarily farther apart
A rarefaction generally corresponds to:
- lower particle density
- lower pressure
than the surrounding undisturbed medium.
Alternating Compressions and Rarefactions
A longitudinal wave consists of a repeating pattern:
compression → rarefaction → compression → rarefaction
As this pattern travels through the medium, individual particles oscillate backward and forward.
The particles themselves do not permanently remain compressed or spread apart.
Instead, different regions become compressed and rarefied as the disturbance passes.
Equilibrium Position
Particles in a medium normally have an equilibrium position.
When a longitudinal wave passes, particles are temporarily displaced from equilibrium.
They then move back toward equilibrium and may continue past it.
This produces an oscillation:
forward → equilibrium → backward → equilibrium
The repeated oscillations allow energy to move through the medium.
How Sound Is Produced
Sound begins with a vibrating source.
Examples include:
- a speaker cone
- vocal cords
- a tuning fork
- a guitar string
- a drum skin
- a vibrating machine
Consider a loudspeaker.
As the speaker cone moves outward, it pushes nearby air molecules closer together.
This produces a compression.
As the speaker cone moves inward, the nearby air molecules spread farther apart.
This produces a rarefaction.
Repeated vibrations produce a continuous series of compressions and rarefactions.
How Sound Travels Through Air
When a vibrating object pushes nearby air particles, those particles collide and interact with neighbouring particles.
The disturbance passes from particle to particle.
A simplified sequence is:
- A source vibrates.
- Nearby air particles are pushed together.
- A compression forms.
- The particles transfer energy to neighbouring particles.
- The source moves backward.
- A rarefaction forms.
- Repeated vibrations create alternating compressions and rarefactions.
- The pattern travels through the air.
This travelling pressure disturbance is a sound wave.
Sound Does Not Carry Air Across the Room
Suppose someone speaks from one side of a classroom.
The air molecules near the speaker's mouth do not travel across the entire room and enter the listener's ear.
Instead, each group of molecules moves a small distance backward and forward.
Energy is transferred from one group of particles to the next.
Therefore:
sound energy travels through the air
while:
air particles oscillate locally
This distinction is fundamental to understanding waves.
Sound Requires a Medium
Sound is a mechanical wave.
Mechanical waves require matter through which to travel.
Sound can travel through:
- gases
- liquids
- solids
Sound cannot travel through a perfect vacuum because there are no particles available to transfer the disturbance.
Sound in Space
Space is often described as silent because most of it is extremely close to a vacuum.
If two astronauts were floating outside a spacecraft with no air between them, shouting would not allow sound to travel directly through the vacuum.
However, they could communicate using radio waves.
Radio waves are electromagnetic waves and do not require a material medium.
Sound Through Solids
Sound can also travel through solids.
If you place your ear against a table while someone taps the other end, vibrations travel through the material.
The particles in the solid vibrate around their equilibrium positions and transfer energy to neighbouring particles.
Sound often travels faster through solids than through gases because the particles are strongly coupled and disturbances can be transmitted efficiently.
Sound Through Liquids
Sound also travels through liquids.
For example:
- whales communicate underwater
- sonar uses sound waves in water
- underwater microphones detect vibrations
- medical ultrasound travels through body tissues and fluids
The particles oscillate and transfer the pressure disturbance through the material.
Wavelength in a Longitudinal Wave
Longitudinal waves have wavelength just as transverse waves do.
The wavelength, λ, is the distance between two corresponding points in consecutive wave cycles.
For a longitudinal wave, wavelength can be measured from:
centre of one compression → centre of next compression
or:
centre of one rarefaction → centre of next rarefaction
Measuring Wavelength
Suppose consecutive compressions are separated by:
0.80 m
Then:
λ = 0.80 m
Suppose the distance from the first compression to the fourth compression is:
2.4 m
There are three complete wavelengths between them.
Therefore:
3λ = 2.4
λ = 2.4/3
λ = 0.80 m
Count the intervals between corresponding regions rather than simply counting the compressions.
Compression to Rarefaction Distance
The distance from the centre of one compression to the centre of the neighbouring rarefaction is:
λ/2
For example, if:
λ = 1.2 m
then:
compression-to-neighbouring-rarefaction distance = 1.2/2
= 0.60 m
A compression and its neighbouring rarefaction are therefore separated by half a wavelength.
Frequency
Frequency is the number of complete wave cycles passing a point each second.
Symbol:
f
Unit:
hertz (Hz)
For example:
f = 200 Hz
means 200 complete cycles occur each second.
For sound, frequency is closely related to our perception of pitch.
Frequency and Pitch
Higher-frequency sound waves are generally heard as higher-pitched sounds.
Lower-frequency waves are heard as lower-pitched sounds.
For example:
- low frequency → low pitch
- high frequency → high pitch
Frequency and loudness are different properties.
A high-pitched sound does not necessarily have to be loud.
Amplitude in Longitudinal Waves
Longitudinal waves also have amplitude, although it is not represented by a physical crest height.
Amplitude relates to the maximum disturbance of the particles or pressure from equilibrium.
For sound:
larger amplitude generally means larger pressure variations.
This is associated with a louder sound.
Therefore, in a simplified comparison:
greater sound-wave amplitude → greater sound intensity
and usually:
greater perceived loudness
Compression Strength and Amplitude
A larger-amplitude longitudinal wave produces:
- more strongly compressed compressions
- more strongly spread rarefactions
- larger pressure variations
A smaller-amplitude wave produces smaller pressure variations.
The spacing between compressions, however, is related to wavelength rather than amplitude.
Wave Speed
The basic wave equation applies to longitudinal waves:
v = fλ
where:
v = wave speed in m/s
f = frequency in Hz
λ = wavelength in m
Worked Example
A sound wave has:
frequency = 250 Hz
wavelength = 1.36 m
Calculate its speed.
Use:
v = fλ
Substitute:
v = 250 × 1.36
v = 340 m/s
The sound wave travels at:
340 m/s
under the conditions represented by the data.
Finding Wavelength
A sound wave travels at:
340 m/s
and has a frequency of:
500 Hz
Use:
v = fλ
Rearrange:
λ = v/f
Substitute:
λ = 340/500
λ = 0.68 m
The wavelength is:
0.68 m
Finding Frequency
A longitudinal wave travels at:
300 m/s
with a wavelength of:
1.5 m
Use:
v = fλ
Rearrange:
f = v/λ
Substitute:
f = 300/1.5
f = 200 Hz
Representing Longitudinal Waves
A longitudinal wave diagram may show particles as dots or vertical lines.
A compression appears as a crowded region:
||||||||
A rarefaction appears as a more widely spaced region:
| | | |
A simplified pattern might look like:
|||||||| | | | |||||||| | | | ||||||||
The crowded regions are compressions.
The spread-out regions are rarefactions.
Reading a Longitudinal Wave Diagram
When examining a longitudinal wave diagram, ask:
- Where are the particles closest together?
- Where are they farthest apart?
- Where is one compression?
- Where is the next compression?
- What distance represents one wavelength?
- Which direction does the wave travel?
- Which direction do the particles oscillate?
The key idea is:
particle motion is parallel to wave direction.
Longitudinal Waves Can Be Drawn as Curves
Sometimes a longitudinal wave is represented using a graph that looks similar to a transverse wave.
This does not mean the particles are moving up and down.
For example, a sound-wave graph might show:
pressure vs position
A peak could represent:
high pressure → compression
A low point could represent:
low pressure → rarefaction
The vertical axis represents a measured quantity such as pressure, not necessarily physical vertical particle displacement.
This distinction is extremely important when interpreting wave diagrams.
Pressure Graphs of Sound
Suppose a graph shows air pressure against position.
A region of maximum pressure corresponds approximately to a:
compression
A region of minimum pressure corresponds approximately to a:
rarefaction
The distance between consecutive pressure maxima represents:
one wavelength
Likewise, the distance between consecutive pressure minima represents one wavelength.
Transverse Waves
In a transverse wave, particle motion is perpendicular to wave travel.
If the wave travels:
→
the particles might move:
↑ ↓
Examples include:
- waves on strings
- electromagnetic waves
- light
- S-waves from earthquakes
Transverse waves are commonly represented using crests and troughs.
Longitudinal vs Transverse Waves
The most important difference is the direction of oscillation.
| Feature | Longitudinal | Transverse |
|---|---|---|
| Particle motion | Parallel to wave direction | Perpendicular to wave direction |
| Main regions/features | Compressions and rarefactions | Crests and troughs |
| Sound in air | Yes | No |
| Wave on a rope | Usually no | Yes |
| Mechanical examples | Sound, spring compression waves | Rope/string waves, S-waves |
| Electromagnetic waves | No | Yes |
| Wavelength | Compression-to-compression or rarefaction-to-rarefaction | Crest-to-crest or trough-to-trough |
What Longitudinal and Transverse Waves Have in Common
Despite their different particle motions, both types of waves:
- transfer energy
- can have wavelength
- can have frequency
- can have amplitude
- can have a period
- travel at a particular speed
- can be described using v = fλ
- involve oscillations
The main difference is the orientation of the oscillation.
Comparing Particle Motion
Suppose both waves travel to the right.
For a transverse wave:
wave →
particles ↑↓
For a longitudinal wave:
wave →
particles ↔
Therefore:
transverse → perpendicular
longitudinal → parallel
This is the most reliable way to classify the two types.
Sound from a Tuning Fork
A vibrating tuning fork demonstrates how vibrations produce sound.
As a prong moves outward, it pushes nearby air particles together.
A compression forms.
As the prong moves inward, a lower-pressure region forms.
A rarefaction develops.
Repeated vibration creates alternating compressions and rarefactions that move outward through the air.
From Source to Ear
When someone speaks:
- Vocal cords vibrate.
- The vibrations produce pressure changes in the air.
- Compressions and rarefactions travel outward.
- The sound wave reaches the ear.
- The pressure variations cause the eardrum to vibrate.
- The ear converts these vibrations into signals that can be interpreted by the brain.
Matter does not travel from the speaker's mouth all the way to the listener's ear.
The energy and disturbance travel.
Longitudinal Waves in Everyday Life
Longitudinal waves are involved in:
- speech
- music
- headphones and speakers
- microphones
- sonar
- ultrasound
- medical imaging
- seismic investigations
- machinery vibrations
- acoustic measurements
Understanding longitudinal waves therefore helps explain many familiar technologies.
Ultrasound
Ultrasound uses sound frequencies above the normal range of human hearing.
Ultrasound waves can travel through body tissues.
Reflections occur where the waves encounter boundaries between different materials or tissues.
The returning signals can be used to construct images.
The underlying wave travelling through the tissue is a mechanical pressure disturbance.
Sonar
Sonar uses sound waves to detect objects or determine distances underwater.
A sound pulse is sent through water.
It reflects from an object and returns.
By measuring the travel time, the distance can be estimated.
If the sound travels to an object and back:
distance to object = speed × total time / 2
The division by 2 is required because the sound travels the distance twice.
Seismic P-Waves
Earthquakes can produce P-waves.
P-waves are primarily longitudinal seismic waves.
Material oscillates approximately parallel to the direction the wave travels.
P-waves can travel through:
- solids
- liquids
They generally arrive at seismic stations before slower S-waves, which is why they are called primary waves.
Interpreting Spacing in Diagrams
Suppose a longitudinal-wave diagram contains compressions at:
x = 1 m
x = 4 m
x = 7 m
The spacing between consecutive compressions is:
4 − 1 = 3 m
and:
7 − 4 = 3 m
Therefore:
λ = 3 m
The diagram shows two complete wavelengths from the first compression to the third compression.
A Complete Example
A longitudinal wave travels through a spring.
Consecutive compressions are:
0.75 m apart
The frequency is:
6 Hz
Therefore:
λ = 0.75 m
Use:
v = fλ
v = 6 × 0.75
v = 4.5 m/s
If the wave travels horizontally along the spring, the coils oscillate horizontally backward and forward.
The wave is longitudinal because:
particle motion ∥ wave direction
Interpreting an Unfamiliar Diagram
When presented with an unfamiliar wave diagram, do not classify it only by its visual shape.
Instead:
- Determine what the diagram represents.
- Identify the direction of wave propagation.
- Identify the direction of particle oscillation.
- Locate high-density regions.
- Locate low-density regions.
- Identify corresponding points.
- Measure wavelength between matching regions.
- Check any axis labels and units.
A curved graph can represent a longitudinal wave if the vertical axis represents pressure, density, or displacement rather than the physical shape of the medium.
A Reliable Strategy for Identifying Longitudinal Waves
Ask these questions:
What is oscillating?
Particles of a medium.
Which direction do they oscillate?
Backward and forward.
Which direction does the wave travel?
Along the medium.
Are these directions parallel?
If yes, the wave is longitudinal.
Then identify:
crowded region → compression
spread-out region → rarefaction
Common Mistakes
Thinking particles travel through the medium with the sound
Particles oscillate locally while the disturbance transfers energy.
Thinking longitudinal particles move perpendicular to the wave
They move parallel to the wave direction.
Calling a compression a crest
Compressions belong to longitudinal waves. Crests describe transverse-wave displacement.
Calling a rarefaction a trough
A rarefaction is a low-density or low-pressure region, not a physical trough.
Measuring wavelength from compression to neighbouring rarefaction
That distance is approximately λ/2, not λ.
Counting compressions instead of intervals
Three consecutive compressions contain two complete wavelengths between the first and third.
Thinking sound can travel through empty space
Mechanical sound requires matter.
Assuming a curved wave graph must represent transverse motion
A curved graph might represent pressure or density in a longitudinal wave.
Confusing pitch and loudness
Frequency is associated with pitch.
Amplitude/intensity is associated with loudness.
Thinking frequency determines wave speed by itself
Wave speed depends strongly on the medium and its conditions. Frequency and wavelength are connected to that speed through v = fλ.
Key Terms
Longitudinal wave — wave in which particles oscillate parallel to the direction of wave travel.
Mechanical wave — wave that requires a material medium.
Medium — material through which a mechanical wave travels.
Particle — small constituent of matter that participates in the local oscillation of a mechanical wave.
Oscillation — repeated motion around an equilibrium position.
Equilibrium position — normal position around which particles oscillate.
Compression — region where particles are closer together than normal.
Rarefaction — region where particles are farther apart than normal.
Pressure — force per unit area; sound waves produce changing pressure in a medium.
Density — amount of mass per unit volume; compressions and rarefactions involve small changes in density.
Sound wave — mechanical wave produced by vibrations and transmitted through matter.
Wavelength (λ) — distance between corresponding points of consecutive wave cycles.
Frequency (f) — number of complete cycles per second, measured in hertz.
Period (T) — time required for one complete cycle.
Amplitude — maximum size of a disturbance from equilibrium.
Wave speed (v) — speed at which a disturbance propagates through a medium.
Pitch — perception associated mainly with sound frequency.
P-wave — primarily longitudinal seismic wave produced by an earthquake.
Ultrasound — sound with frequency above the normal human hearing range.
Sonar — technology using sound and its reflections to detect objects or determine distances.
Key Takeaways
- In a longitudinal wave, particle motion is parallel to the direction of wave travel.
- Particles oscillate around equilibrium rather than travelling with the wave.
- Longitudinal waves transfer energy through a medium.
- Compressions are regions where particles are closer together.
- Rarefactions are regions where particles are farther apart.
- A longitudinal wave contains alternating compressions and rarefactions.
- Sound in air is a longitudinal mechanical wave.
- A vibrating source produces changing pressure in the surrounding medium.
- Sound can travel through gases, liquids, and solids.
- Sound cannot travel through a perfect vacuum because there are no particles to transmit the mechanical disturbance.
- Wavelength can be measured from compression to compression or rarefaction to rarefaction.
- The distance from a compression to its neighbouring rarefaction is approximately half a wavelength.
- Frequency is the number of cycles per second.
- Higher sound frequency generally corresponds to higher pitch.
- Greater sound-wave amplitude is associated with larger pressure variations and generally greater intensity.
- Longitudinal waves obey v = fλ.
- A curved representation of a sound wave does not necessarily show particles moving vertically.
- Graphs may represent pressure, density, or displacement.
- Transverse waves have perpendicular oscillations; longitudinal waves have parallel oscillations.
- P-waves are important examples of longitudinal seismic waves.
- The most reliable way to identify a longitudinal wave is to compare the direction of particle oscillation with the direction of wave travel.
Check Your Understanding
- Define a longitudinal wave in terms of particle motion and wave direction.
- Explain what happens to particles of a medium as a longitudinal wave passes.
- Define compression and rarefaction.
- A spring contains alternating crowded and spread-out regions. Explain which regions represent compressions and which represent rarefactions.
- Explain how the vibration of a speaker produces a sound wave in air.
- Explain why sound cannot travel through a perfect vacuum.
- Consecutive compressions are 1.5 m apart. Determine the wavelength.
- The distance from the first compression to the fifth compression is 8 m. Determine the wavelength.
- A longitudinal wave has a frequency of 12 Hz and wavelength of 2.5 m. Calculate its speed.
- A sound wave travels at 340 m/s and has a frequency of 680 Hz. Calculate its wavelength.
- Compare transverse and longitudinal waves in terms of particle motion, wave direction, and characteristic features.
- A student sees a curved graph of a sound wave and concludes that the air particles must move vertically up and down. Explain why this conclusion may be incorrect.
- A longitudinal-wave diagram shows compressions at x = 2 m, 5 m, 8 m, and 11 m. Determine the wavelength and the number of complete wavelengths between the first and fourth compressions.
- Explain how sound can travel from a speaker to a listener even though individual air molecules do not travel across the room.
- Sketch a longitudinal wave containing at least three compressions and three rarefactions. Identify the compressions, rarefactions, one wavelength, the direction of wave travel, and the direction of particle motion.