Representing Waves

Waves are constantly changing with both position and time, so physicists use graphs to represent them clearly. Two of the most important representations are displacement-distance graphs and displacement-time graphs.

These graphs may look very similar, but they show different information. Understanding which graph you are looking at is essential when finding quantities such as wavelength, period, frequency, and amplitude.

Learning Targets

• I can interpret displacement-distance graphs.
• I can interpret displacement-time graphs.
• I can distinguish between spatial and temporal wave graphs.
• I can determine wavelength and period from graphs.
• I can sketch simple wave diagrams.

---

Why Do We Use Wave Graphs?

A travelling wave involves many particles or points behaving differently at different positions and times.

Trying to describe all of this using words would be difficult.

Graphs allow us to represent important wave properties such as:

• Displacement
• Amplitude
• Wavelength
• Period
• Frequency

However, we need to pay close attention to the horizontal axis.

A wave graph with distance on the horizontal axis represents something different from a wave graph with time on the horizontal axis.

---

Displacement

Displacement describes how far a point in the medium has moved from its equilibrium position.

The equilibrium position is the position the point would occupy if no wave were passing through it.

Displacement can be:

• Positive
• Zero
• Negative

For a transverse wave:

• A crest represents maximum positive displacement.
• A trough represents maximum negative displacement.
• The centre line represents zero displacement.

---

Amplitude

The amplitude of a wave is the maximum displacement from the equilibrium position.

It is measured from:

Equilibrium → Crest

or:

Equilibrium → Trough

It is not measured from the trough to the crest.

If the vertical distance from a crest to a trough is 8 cm:

Amplitude = 8 / 2

Amplitude = 4 cm

---

Displacement-Distance Graphs

A displacement-distance graph shows the shape of a wave at a particular instant in time.

The vertical axis represents:

Displacement

The horizontal axis represents:

Distance

This is sometimes called a spatial wave graph because it shows how displacement varies with position in space.

Think of it as taking a photograph of the wave at one instant.

---

Reading a Displacement-Distance Graph

A displacement-distance graph can tell us:

• The displacement at different positions
• The amplitude
• The wavelength
• The shape of the wave at that instant

Suppose the graph shows crests at:

x = 2 m

and:

x = 8 m

The wavelength is the distance between the two neighbouring crests:

λ = 8 − 2

λ = 6 m

Therefore:

Wavelength = 6 m

---

Finding Wavelength

Wavelength is the distance between two consecutive points that are at the same stage of a wave cycle.

The symbol is:

λ

Wavelength can be measured:

Crest → next crest

Trough → next trough

or between any other two consecutive equivalent points.

Wavelength is a distance, so its SI unit is:

metre (m)

---

Worked Example — Finding Wavelength

A displacement-distance graph has one crest at:

x = 1.5 m

and the next crest at:

x = 5.5 m

Calculate the wavelength.

λ = 5.5 − 1.5

λ = 4.0 m

Therefore:

Wavelength = 4.0 m

Notice that no time information was required.

The horizontal axis represents distance.

---

Displacement-Time Graphs

A displacement-time graph shows how the displacement of one particular point changes as time passes.

The vertical axis represents:

Displacement

The horizontal axis represents:

Time

This is sometimes called a temporal wave graph.

Instead of showing the entire wave at one moment, this graph follows one position over time.

---

Imagine a Floating Cork

Imagine a cork floating on water while waves pass underneath it.

The cork moves:

Up → equilibrium → down → equilibrium → up

repeatedly.

If we recorded the vertical displacement of the cork over time, we could draw a displacement-time graph.

The graph describes the oscillation of the cork at one location.

It does not directly show the physical shape of the entire water wave.

---

Period

The period is the time required for one complete oscillation or wave cycle.

The symbol is:

T

The SI unit is:

second (s)

On a displacement-time graph, the period can be measured from:

One crest → next crest

or:

One trough → next trough

or between any two consecutive equivalent points.

---

Worked Example — Finding Period

Suppose one crest occurs at:

t = 1 s

and the next crest occurs at:

t = 5 s

Then:

T = 5 − 1

T = 4 s

Therefore:

Period = 4 s

This means one complete oscillation takes four seconds.

---

Period and Frequency

Period and frequency are closely related.

Frequency tells us how many oscillations occur each second.

Period tells us how long one oscillation takes.

Therefore:

f = 1 / T

and:

T = 1 / f

For example, if:

T = 0.25 s

then:

f = 1 / 0.25

f = 4 Hz

Four complete oscillations occur every second.

---

Worked Example — Finding Frequency from a Graph

A displacement-time graph shows that one complete oscillation takes:

0.20 s

Therefore:

T = 0.20 s

Use:

f = 1 / T

f = 1 / 0.20

f = 5 Hz

Therefore:

Frequency = 5 Hz

---

Spatial vs Temporal Wave Graphs

This distinction is extremely important.

The graphs may have very similar shapes.

Always check the horizontal axis before interpreting the graph.

---

A Useful Way to Remember

Think of the two graphs this way:

Displacement-Distance

A photograph

You see the wave at one instant across many positions.

Displacement-Time

A video of one point

You watch how one location moves as time passes.

This distinction prevents many common errors.

---

Same Wave, Different Graphs

A single travelling wave can be represented using both types of graph.

Suppose a wave has:

Wavelength = 4 m

Frequency = 2 Hz

Its period is:

T = 1/f

T = 1/2

T = 0.5 s

A displacement-distance graph would repeat every:

4 m

A displacement-time graph would repeat every:

0.5 s

These represent the same wave but describe it in different ways.

---

Connecting the Two Graphs

The wave equation connects the spatial and temporal descriptions:

v = λf

Because:

f = 1/T

we can also write:

v = λ/T

This makes physical sense.

During one period, a travelling wave moves forward by one wavelength.

Therefore:

Wave speed = One wavelength / One period

---

Worked Example — Using Both Graphs

A displacement-distance graph gives:

λ = 3 m

A displacement-time graph for the same wave gives:

T = 0.50 s

First calculate frequency:

f = 1/T

f = 1/0.50

f = 2 Hz

Now calculate wave speed:

v = λf

v = 3 × 2

v = 6 m/s

Therefore:

Wave speed = 6 m/s

---

Determining Amplitude from Either Graph

Amplitude can be determined from either a displacement-distance graph or a displacement-time graph.

Suppose the vertical axis shows:

Maximum displacement = +0.30 m

Minimum displacement = −0.30 m

The amplitude is:

0.30 m

Do not calculate:

0.30 − (−0.30) = 0.60 m

That is the total crest-to-trough displacement, not the amplitude.

---

Reading Points on a Displacement-Distance Graph

Suppose a displacement-distance graph shows a point at:

x = 3 m

with displacement:

+0.20 m

This means:

At the instant represented by the graph, the point in the medium located 3 m from the origin is displaced 0.20 m in the positive direction.

It does not mean that the point travelled 3 m.

The horizontal coordinate describes its position in the medium.

---

Reading Points on a Displacement-Time Graph

Suppose a displacement-time graph shows:

At:

t = 3 s

the displacement is:

−0.10 m

This means that after 3 seconds, the particular point being observed is displaced 0.10 m in the negative direction from equilibrium.

The horizontal coordinate is now time, not position.

---

Sketching a Simple Wave

To sketch a wave accurately:

• Draw and label the horizontal axis.
• Draw and label the vertical displacement axis.
• Mark the equilibrium position.
• Draw smooth crests and troughs.
• Show a consistent wavelength or period if the wave is periodic.
• Label important quantities.

A wave should normally be drawn as a smooth curve rather than as a series of sharp triangles.

---

Sketching a Displacement-Distance Graph

Suppose you are told:

Amplitude = 2 cm

Wavelength = 4 m

Your graph should:

• Have displacement on the vertical axis.
• Have distance on the horizontal axis.
• Reach +2 cm at the crests.
• Reach −2 cm at the troughs.
• Repeat every 4 m.

The distance between neighbouring crests should therefore be:

4 m

---

Sketching a Displacement-Time Graph

Suppose:

Amplitude = 3 cm

Period = 2 s

Your graph should:

• Have displacement on the vertical axis.
• Have time on the horizontal axis.
• Reach +3 cm at maximum positive displacement.
• Reach −3 cm at maximum negative displacement.
• Repeat every 2 s.

The time between neighbouring crests should therefore be:

2 s

---

Wave Direction

A single displacement-distance graph shows the shape of the wave at one instant, but it does not necessarily tell us which direction the wave is travelling.

The same wave shape could potentially be moving:

Left

or:

Right

Additional information is required, such as:

• A second graph taken slightly later.
• An arrow showing the direction of propagation.
• Information about how a particular point is moving.

This is an important limitation of a single spatial graph.

---

Successive Wave Snapshots

Imagine taking two photographs of a travelling wave separated by a short time.

In the first image, a crest is at:

x = 2 m

A short time later, the crest is at:

x = 3 m

The wave pattern has shifted to the right.

Therefore:

Wave direction = right

If we know the time between the snapshots, we can also calculate the wave speed.

---

Worked Example — Wave Speed from Snapshots

A crest moves from:

x = 2.0 m

to:

x = 3.5 m

in:

0.50 s

Distance travelled:

3.5 − 2.0 = 1.5 m

Wave speed:

v = distance / time

v = 1.5 / 0.50

v = 3.0 m/s

Therefore:

Wave speed = 3.0 m/s

---

Representing Longitudinal Waves

Longitudinal waves can also be represented graphically.

A sound wave, for example, consists of alternating:

• Compressions
• Rarefactions

Although longitudinal waves do not physically look like a transverse sine curve, quantities such as displacement, pressure, or density can still be plotted against distance or time.

Therefore, a sinusoidal graph does not necessarily mean that the particles physically move up and down.

---

Wave Graphs and Sound

Suppose a microphone records a sound wave.

A graph might show the signal changing with time.

The horizontal axis is therefore:

Time

From this graph, we can determine:

• Period
• Frequency
• Relative amplitude

A higher-frequency sound produces more oscillations in the same time interval.

This type of representation is widely used in:

• Music recording
• Audio editing
• Speech analysis
• Telecommunications

---

Comparing Two Displacement-Time Graphs

Suppose Wave A completes:

2 cycles in 1 second

and Wave B completes:

5 cycles in 1 second

Then:

Wave A frequency = 2 Hz

Wave B frequency = 5 Hz

Wave B has the greater frequency.

Its period is also shorter.

For Wave A:

T = 1/2 = 0.50 s

For Wave B:

T = 1/5 = 0.20 s

Therefore:

Higher frequency → shorter period

---

Worked Example — Counting Cycles

A displacement-time graph shows:

6 complete cycles

during:

3 seconds

Frequency is:

f = number of cycles / time

f = 6 / 3

f = 2 Hz

Period is:

T = 1/f

T = 1/2

T = 0.5 s

Therefore:

Frequency = 2 Hz

Period = 0.5 s

---

Common Mistakes

Mistake: Finding wavelength from a displacement-time graph

A displacement-time graph gives period, not wavelength directly.

Mistake: Finding period from a displacement-distance graph

A displacement-distance graph gives wavelength, not period directly.

Mistake: Ignoring the horizontal axis

Two graphs may look almost identical while representing completely different information.

Always check whether the horizontal axis is:

Distance or time?

Mistake: Measuring wavelength from crest to neighbouring trough

Crest to trough represents half a wavelength.

A complete wavelength can be measured from crest to the next crest.

Mistake: Measuring period from crest to neighbouring trough

Crest to trough represents half a period.

A complete period is measured between equivalent points on consecutive cycles.

Mistake: Measuring amplitude from crest to trough

Amplitude is measured from the equilibrium position to a crest or trough.

Mistake: Assuming the graph shows the physical path of a particle

A wave graph represents displacement. A particle in the medium does not necessarily travel along the shape drawn on the graph.

Mistake: Assuming a displacement-distance graph automatically shows wave direction

A single snapshot generally does not provide enough information to determine the direction of propagation.

---

Key Terms

Displacement – distance and direction from the equilibrium position.

Equilibrium position – the undisturbed position of a point in the medium.

Amplitude – maximum displacement from equilibrium.

Displacement-distance graph – a graph showing displacement against position at one instant.

Displacement-time graph – a graph showing displacement of one point as time passes.

Spatial graph – a representation showing variation with position.

Temporal graph – a representation showing variation with time.

Wavelength – distance between consecutive equivalent points on a wave.

Period – time required for one complete oscillation.

Frequency – number of complete oscillations per second.

Wave profile – the shape of a wave at a particular instant.

---

Check Your Understanding

1. What does a displacement-distance graph represent?

2. What does a displacement-time graph represent?

3. Which type of graph should you use to determine wavelength directly?

4. Which type of graph should you use to determine period directly?

5. Consecutive crests on a displacement-distance graph occur at 3 m and 9 m. Calculate the wavelength.

6. Consecutive crests on a displacement-time graph occur at 1.5 s and 4.0 s. Calculate the period.

7. Calculate the frequency of the wave in Question 6.

8. A graph has a maximum displacement of +4 cm and a minimum displacement of −4 cm. What is the amplitude?

9. Explain why measuring from one crest to the next trough does not give one complete wavelength.

10. A wave has a wavelength of 8 m and a period of 2 s. Calculate its frequency and wave speed.

11. Explain the difference between a spatial wave graph and a temporal wave graph.

12. Sketch a displacement-distance graph for a wave with an amplitude of 2 m and wavelength of 6 m.

13. Sketch a displacement-time graph for an oscillation with an amplitude of 3 cm and period of 4 s.

14. Explain why a single displacement-distance graph usually cannot tell you whether a wave is travelling left or right.

---

Key Takeaways

• Wave graphs allow us to represent displacement, amplitude, wavelength, period, and frequency.
• A displacement-distance graph shows displacement at different positions at one instant.
• A displacement-distance graph is a spatial representation of a wave.
• Wavelength can be determined directly from a displacement-distance graph.
• A displacement-time graph shows how one point changes displacement over time.
• A displacement-time graph is a temporal representation.
• Period can be determined directly from a displacement-time graph.
• Frequency and period are related by f = 1/T.
• The wave equation connects the spatial and temporal descriptions through v = λf.
• During one period, a travelling wave moves forward by one wavelength, so v = λ/T.
• Always inspect the horizontal axis before interpreting a wave graph.
• Amplitude is measured from the equilibrium position to the maximum displacement.
• A single spatial snapshot usually does not reveal the direction in which the wave is travelling.
• Clear wave sketches should include labelled axes, equilibrium position, amplitude, and either wavelength or period, depending on the graph.