Water Waves and Depth: Aboriginal Observations and Physics

A stone drops into a still pool. Circular ripples spread out from the splash, travelling across the water's surface. Now imagine that part of the pool has a submerged rock shelf. The water becomes shallower above it. Would the ripples travel at the same speed, or would their behaviour change?

The answer provides a useful way to investigate something you cannot easily see: the depth of the water beneath a moving wave.

Aboriginal Peoples' observations of water waves provide an important context for this investigation. By connecting wave velocity and wavelength with changes in water depth, we can explore how careful observations of the natural environment can reveal information about what lies beneath its surface.

[IMAGE: Aerial oblique view of a calm natural waterhole with a stone splash creating expanding circular ripples. A submerged rock shelf is visible beneath the clearer shallow water, with wave crests approaching it and changing their spacing. Show the deeper and shallower sections of the waterhole as recognisable physical features.]

## Reading water through waves

### What is documented?

The relationship between Aboriginal Knowledge and water is not limited to finding or collecting water. Understanding waterways, their changing conditions, and the features beneath their surfaces has also been important to communities living with and caring for Country.

A useful community-specific example comes from Torres Strait Islander Peoples. In her research on Torres Strait seascapes, anthropologist Maureen Fuary describes how Islanders interpret different aspects of the sea, including water depth, water clarity, surface patterns, waves, currents, and tides.

Her account includes observations associated with Mabuiag in western Torres Strait, where the movement and collision of waves around the island feature in local knowledge and song. It also describes how surface behaviour and knowledge of particular locations help identify submerged features such as reefs, rocks, and sandbanks.

This is an example of detailed, place-based environmental knowledge. It should not be presented as a practice belonging to every Aboriginal community, or as evidence that all communities use an identical technique.

For further educational context, the Ngarrngga resource on the Torres Strait Islander seascape draws on Fuary's research.

### Connecting these observations to physics

The NSW Physics 11-12 Syllabus (2025) includes the use of water-wave velocity and wavelength to approximate water depth in its study of wave properties. Throwing stones into still water provides an example of how observable wave patterns can be produced and studied.

There are two things worth keeping separate:

- **Documented Cultural Knowledge:** Aboriginal Peoples have developed detailed understandings of their local waterways and environmental conditions. Specific practices and knowledge must be attributed to the relevant communities.
- **Modern physical interpretation:** Physics explains how the speed and wavelength of surface waves depend on water depth. We can use this relationship to construct a mathematical estimate of depth.

The physics model helps explain the connection. It does not mean that every community used the same measurements, terminology, or mathematical procedures.

The community-specific account above is discussed in Maureen Fuary’s 2009 paper, Reading and Riding the Waves: The Sea as a Known Universe in Torres Strait. The Ngarrngga Torres Strait Islander seascape case study provides an additional educational resource.

Aboriginal Peoples and Torres Strait Islander Peoples should not be treated as one interchangeable cultural group. The Torres Strait example is a specific case of First Nations seascape knowledge, not evidence that every Aboriginal community shares the same practices. NESA also emphasises respect for Indigenous Cultural and Intellectual Property and the importance of working with local knowledge holders when teaching cultural material. See the NESA guidance on the Physics syllabus.

01What actually moves when a wave travels?

Look at the ripples produced by a stone. The circular rings move outwards, but the water itself doesn’t travel outwards with them.

A floating leaf near the splash might rise and fall as the waves pass. It may move slightly forwards and backwards, but it doesn’t travel all the way to the edge of the pool with each crest.

This is similar to a stadium wave. The pattern travels around the stadium, even though each spectator mostly stays in their seat. The analogy describes the movement of the pattern, although water particles actually follow small oscillating paths.

The properties we need to measure

There are four important quantities.

• Wavelength (\(\lambda\)): the horizontal distance between two successive crests, measured in metres (m).
• Period (\(T\)): the time taken for one complete wave cycle, measured in seconds (s).
• Frequency (\(f\)): the number of complete wave cycles passing a fixed point each second, measured in hertz (Hz).
• Wave speed (\(v\)): the speed at which a particular part of the wave, such as a crest, travels, measured in metres per second (m/s).

Frequency and period are inversely related:

\[ f=\frac{1}{T} \]

For example, if a crest passes a fixed point every 4 seconds, the period is 4 seconds and the frequency is:

\[ f=\frac{1}{4}=0.25\text{ Hz} \]

Wave speed connects frequency and wavelength:

\[ \boxed{v=f\lambda} \]

This means that a wave travelling at 2 m/s with a wavelength of 8 m has a frequency of:

\[ f=\frac{v}{\lambda} =\frac{2}{8} =0.25\text{ Hz} \]

You can also calculate wave speed directly by measuring how far a crest travels over a known time:

\[ v=\frac{\text{distance travelled}}{\text{time taken}} \]

The Bureau of Meteorology’s guide to waves and swell provides further explanations of wavelength and period.

02Why water depth changes wave speed

Imagine two sections of the same waterway.

One section is relatively deep. The other passes over a submerged rock shelf, making it much shallower.

Would a wave behave identically in both sections?

No. The seabed affects how the water moves beneath the wave.

The effect of the seabed

As a surface gravity wave travels, water particles beneath the surface oscillate. Their motion extends down into the water column.

In sufficiently deep water, the seabed has little influence on the surface wave’s motion.

As the water becomes shallower, the wave motion increasingly interacts with the bottom. The particle paths become flatter, and the seabed constrains their movement.

The wave travels more slowly.

This gives us an important relationship:

For shallow-water gravity waves, a decrease in water depth produces a decrease in wave speed.

But something else changes as well: wavelength.

Why wavelength decreases too

Consider a regular wave train produced by a source operating at a constant frequency.

As the waves enter a shallower region, their speed decreases. However, their frequency remains constant because the source is still producing waves at the same rate.

Remember:

\[ v=f\lambda \]

If \(f\) stays constant while \(v\) decreases, \(\lambda\) must also decrease.

The crests become closer together.

This is the key observation connecting waves to water depth:

• Deeper water can support faster-moving waves.
• Shallower water slows the waves.
• For the same source frequency, slower waves have shorter wavelengths.

If waves approach a submerged feature at an angle, the part of a wavefront entering shallow water first slows before the rest. The wavefront bends. This behaviour is called refraction.

It means that the shape and movement of wave patterns can provide information about underwater features, even when those features cannot be seen directly.

03Using wave speed to estimate water depth

We now have a way to connect an observable wave property to something hidden beneath the surface.

For sufficiently shallow water, the speed of a surface gravity wave can be approximated using:

\[ \boxed{v\approx\sqrt{gh}} \]

Here:

• \(v\) is the wave speed relative to the water, in m/s.
• \(g\) is gravitational acceleration, approximately \(9.8\text{ m/s}^2\).
• \(h\) is the water depth, in metres.

Rearranging gives:

\[ \boxed{h\approx\frac{v^2}{g}} \]

This is the relationship that allows us to estimate water depth from wave speed.

Notice that the speed is squared. If wave speed doubles, the estimated depth becomes four times as large, provided the approximation remains valid.

When is this equation valid?

There is an important condition.

The equation \(v\approx\sqrt{gh}\) applies to shallow-water waves, where the wavelength is much greater than the water depth.

A useful approximate criterion is:

\[ h<\frac{\lambda}{20} \]

In other words, the water depth should be less than one-twentieth of the wavelength.

For example, if a wave has a wavelength of 10 m, the shallow-water approximation is appropriate when the depth is less than approximately 0.5 m.

The wave amplitude must also be small compared with the water depth.

The approximation works best when:

• The wavelength is much greater than the water depth.
• The wave amplitude is small compared with the depth.
• The waves are surface gravity waves rather than tiny surface-tension-dominated ripples.
• Strong currents and significant wave reflections are absent.
• The water depth changes gradually over the region being measured.

The Institute of Physics explains the shallow-water approximation and its assumptions in its resource, The speed of water waves.

Important syllabus distinction

The relationship \(v\approx\sqrt{gh}\) is a useful explanatory extension for understanding how wave speed can reveal water depth. It is not one of the listed mandatory wave equations in the 2025 NSW Physics syllabus. The syllabus explicitly includes \(f=1/T\) and \(v=f\lambda\).

You should understand the physical reasoning behind the depth estimate, including when the approximation is valid.

Why wavelength matters

You might notice that the depth equation only contains wave speed. So why does the syllabus also emphasise wavelength?

There are two reasons.

First, wavelength allows us to calculate wave speed when frequency is known:

\[ v=f\lambda \]

Second, wavelength allows us to check whether the shallow-water approximation is appropriate:

\[ h<\frac{\lambda}{20} \]

Without this check, you might calculate a depth that looks reasonable but is based on an invalid model.

If the water is not sufficiently shallow compared with the wavelength, wave speed depends on both depth and wavelength. The simple equation \(v\approx\sqrt{gh}\) is no longer reliable.

This is why we must not apply a deep-water wave relationship, or a shallow-water approximation, indiscriminately to every water wave.

04Reconstructing the observation as a measurement

Let’s turn the stone-in-water example into a measurement process.

The aim is to use a visible wave pattern to estimate an otherwise hidden water depth. The following is a modern physics reconstruction of the syllabus example, not a claim that every Aboriginal community followed this exact experimental procedure.

Setting up the observation

Imagine a calm body of water with a relatively flat surface.

A stone is dropped at a known point, creating circular ripples. Along one direction from the impact, two fixed reference markers are placed a known distance apart.

A ruler or scale is used to measure the distance between successive wave crests.

The measurement follows a logical sequence:

1. Create observable waves.

Drop a stone into the water to generate circular wavefronts.

2. Measure the wave’s movement.

Track an identifiable crest as it moves between the two reference markers. Record the distance travelled and the time taken.

3. Measure the wavelength.

Determine the horizontal distance between successive crests in the same region.

4. Calculate wave speed.

Use the measured distance and time, or calculate the speed using \(v=f\lambda\) if the frequency is known.

5. Estimate water depth.

If the shallow-water conditions are satisfied, use:

\[ h\approx\frac{v^2}{g} \]

What should the observation show?

Suppose the waves move from a deeper region towards a submerged shelf.

The prediction is that the waves will slow as they enter the shallower water. If the source frequency remains constant, the wavelength will decrease too.

The resulting change in crest spacing and movement provides information about the changing water depth.

This is the physical connection behind using water-wave observations to investigate submerged features.

Why the first measurement might fail

A single stone drop does not necessarily produce a perfectly regular wave train.

It creates a short wave packet containing different wavelengths and frequencies. Some crests may be difficult to distinguish, especially close to the splash.

Very short ripples can also be affected by surface tension, so they should not automatically be treated as the long gravity waves described by the shallow-water model.

Other problems include:

• Wind disturbing the water surface.
• Currents carrying the wave pattern along.
• Reflections from nearby banks interfering with incoming waves.
• Measuring the motion of a floating object rather than the speed of a crest.
• Using a wavelength and a speed measurement that refer to different parts of the wave packet.

Improving the measurement

A more controlled modern investigation would use repeated disturbances at a consistent interval, or a wave generator, to produce a more regular wave train.

The observer could then measure several successive crests, track the same wave component over a known distance, and repeat the measurements.

This improves the reliability of the frequency, wavelength, and speed estimates.

It also makes it easier to identify changes caused by water depth rather than wind, currents, or irregular wave generation.

The important conclusion is that wave observations provide useful information, but the interpretation depends on the physical conditions of the waterway.

05Worked examples

06Questions and solutions

Question 1

A regular wave train travels across a calm waterhole. Its frequency is 0.40 Hz and its wavelength is 3.0 m.

Calculate:

a) The wave speed.

b) The approximate water depth, assuming shallow-water conditions.

c) Whether the shallow-water approximation is valid.

Solution 1

The wave speed is 1.2 m/s, and the estimated water depth is approximately 0.15 m. The shallow-water condition is satisfied, although only just.

Step 1: Calculate wave speed.

\[ v=f\lambda \]

Substitute:

\[ v=0.40\times3.0 \]

\[ \boxed{v=1.2\text{ m/s}} \]

Step 2: Calculate the approximate depth.

\[ h\approx\frac{v^2}{g} \]

\[ \begin{aligned} h&\approx\frac{1.2^2}{9.8}\\\ &=\frac{1.44}{9.8}\\\ &=0.147\text{ m} \end{aligned} \]

Step 3: Check the shallow-water condition.

\[ \frac{\lambda}{20}=\frac{3.0}{20} =0.15\text{ m} \]

Since:

\[ 0.147<0.15 \]

the condition is satisfied.

Final answer:

• Wave speed: \(1.2\text{ m/s}\).
• Approximate water depth: \(0.15\text{ m}\).

The estimate is relatively close to the limit of the approximation, so small measurement errors could affect its reliability.

Question 2

A regular wave train is generated by a source with a period of 3.0 seconds.

It travels across two regions of a stationary waterway:

• Region A has a wavelength of 3.0 m.
• Region B has a wavelength of 1.0 m.

Assume both regions satisfy the shallow-water conditions.

Calculate:

a) The frequency of the wave.

b) The wave speed in each region.

c) The approximate depth of each region.

d) The ratio of the depth of region B to region A.

Explain why the frequency does not change between the regions.

Solution 2

Region B is approximately one-ninth the depth of region A. The waves slow down and their wavelengths decrease as they enter the shallower region, while their frequency remains constant.

Step 1: Calculate frequency.

\[ f=\frac{1}{T} \]

\[ f=\frac{1}{3.0} \]

\[ \boxed{f=0.333\text{ Hz}} \]

Step 2: Calculate wave speed in region A.

\[ v_A=f\lambda_A \]

\[ v_A=0.333\times3.0 \]

\[ \boxed{v_A=1.0\text{ m/s}} \]

Step 3: Calculate wave speed in region B.

\[ v_B=f\lambda_B \]

\[ v_B=0.333\times1.0 \]

\[ \boxed{v_B=0.333\text{ m/s}} \]

Step 4: Calculate the depth of region A.

\[ h_A\approx\frac{v_A^2}{g} \]

\[ h_A\approx\frac{1.0^2}{9.8} \]

\[ \boxed{h_A=0.102\text{ m}} \]

Step 5: Calculate the depth of region B.

\[ h_B\approx\frac{v_B^2}{g} \]

\[ h_B\approx\frac{0.333^2}{9.8} \]

\[ \boxed{h_B=0.0113\text{ m}} \]

Step 6: Calculate the depth ratio.

\[ \frac{h_B}{h_A} =\frac{0.0113}{0.102} \]

\[ \boxed{\frac{h_B}{h_A}\approx0.111} \]

Therefore, region B is approximately one-ninth the depth of region A.

Step 7: Explain the frequency.

The wave source determines the frequency. As the waves travel between regions of different depths, their speed and wavelength change, but the source continues producing waves at the same rate.

This is why the frequency remains constant.

Question 3

Two different regular wave sources produce waves in the same calm waterhole.

Their measurements are:

A student argues that Wave Y must be travelling through shallower water because its wavelength is shorter.

Evaluate this claim. Calculate the wave speed and approximate water depth for both waves, assuming the shallow-water approximation is valid.

Solution 3

The student’s conclusion is incorrect. Both waves have the same speed and therefore give the same estimated water depth, approximately 0.10 m.

The important difference is their frequency, not the depth of the water.

Step 1: Calculate the speed of Wave X.

\[ v_X=f_X\lambda_X \]

\[ v_X=0.20\times5.0 \]

\[ \boxed{v_X=1.0\text{ m/s}} \]

Step 2: Calculate the speed of Wave Y.

\[ v_Y=f_Y\lambda_Y \]

\[ v_Y=0.40\times2.5 \]

\[ \boxed{v_Y=1.0\text{ m/s}} \]

Both waves travel at the same speed.

Step 3: Calculate the approximate depth.

For either wave:

\[ h\approx\frac{1.0^2}{9.8} \]

\[ \boxed{h\approx0.102\text{ m}} \]

Step 4: Check both wavelengths.

For Wave X:

\[ \frac{h}{\lambda_X} =\frac{0.102}{5.0} =0.0204 \]

For Wave Y:

\[ \frac{h}{\lambda_Y} =\frac{0.102}{2.5} =0.0408 \]

Both ratios are below 0.05, so both satisfy the shallow-water condition.

The shorter wavelength of Wave Y is explained by its higher frequency.

This question exposes a common mistake: assuming that wavelength alone determines water depth.

You must consider frequency as well. Two waves can have different wavelengths while travelling at the same speed through water of the same depth.

Question 4

A student drops a single stone into a pond and observes a ripple pattern.

They measure:

• Wavelength: 0.40 m.
• Distance travelled by an identifiable crest: 1.2 m.
• Time taken: 2.0 seconds.

The student calculates the water depth using:

\[ h\approx\frac{v^2}{g} \]

They conclude that the pond is approximately 0.037 m deep.

Evaluate whether this conclusion is justified.

Identify two limitations of the investigation and explain how the measurement process could be improved.

Solution 4

The estimated depth of 0.037 m is not justified using the shallow-water approximation. The calculated depth fails the required wavelength condition, and the one-off stone impact does not produce a reliably regular wave train.

Step 1: Calculate the measured wave speed.

\[ v=\frac{\text{distance}}{\text{time}} \]

\[ v=\frac{1.2}{2.0} \]

\[ \boxed{v=0.60\text{ m/s}} \]

Step 2: Calculate the student’s estimated depth.

\[ h\approx\frac{v^2}{g} \]

\[ h\approx\frac{0.60^2}{9.8} \]

\[ \boxed{h\approx0.0367\text{ m}} \]

The arithmetic is correct. However, the model still needs to be checked.

Step 3: Test the shallow-water condition.

\[ \frac{\lambda}{20} =\frac{0.40}{20} =0.020\text{ m} \]

The calculated depth is:

\[ 0.0367\text{ m} \]

This is greater than 0.020 m.

Therefore, the shallow-water approximation is not satisfied.

The calculated depth cannot be accepted as a reliable estimate.

Step 4: Identify the experimental limitations.

There are two important problems.

First, a single stone impact produces a transient wave packet containing different wave components. The spacing between two neighbouring crests may not represent a single regular wavelength throughout the packet. The measured crest speed and wavelength must refer to the same identifiable wave component.

Second, the water may not satisfy the assumptions of the shallow-water model. The wavelength is not sufficiently large compared with the estimated depth, so the speed may depend on both wavelength and depth.

Step 5: Improve the investigation.

A better experimental design would:

• Use repeated disturbances at a fixed interval to produce a regular wave train.
• Track the same identifiable crest between fixed reference points.
• Measure several successive wavelengths and average the results.
• Use longer-wavelength gravity waves rather than very short ripples.
• Minimise wind, currents, and reflections.
• Check the shallow-water condition before calculating depth.

The central lesson is that correct substitution does not guarantee a physically valid answer. You must check whether the model applies to the situation.

Question 5

A researcher investigates wave behaviour in two parallel water channels using the same shore-mounted wave generator.

The generator produces regular waves at a frequency of 0.25 Hz.

The channels have the following conditions:

Both channels contain low-amplitude gravity waves, and the current in Channel A is uniform and flows in the direction of wave propagation.

A student applies the shallow-water equation directly to the measured crest speeds:

\[ h\approx\frac{v^2}{g} \]

They conclude that Channel A is deeper than Channel B.

Determine whether this conclusion is correct.

Calculate the actual approximate water depth in each channel, accounting for the current. Check the shallow-water condition for both channels.

Solution 5

Both channels have the same approximate water depth of 0.10 m. The apparent difference in wave speed is caused by the current in Channel A, not a difference in water depth.

The student’s mistake is using the speed measured relative to the shore instead of the wave speed relative to the water.

Step 1: Identify the effect of the current.

In Channel A, the current moves in the same direction as the wave.

Therefore, the wave’s speed relative to the shore is greater than its speed relative to the water.

The relationship is:

\[ v_{\text{shore}}=v_{\text{water}}+u \]

where \(u\) is the water current speed.

Step 2: Calculate the actual wave speed relative to the water in Channel A.

Rearrange:

\[ v_{\text{water}}=v_{\text{shore}}-u \]

Substitute:

\[ v_{\text{water,A}}=1.50-0.50 \]

\[ \boxed{v_{\text{water,A}}=1.00\text{ m/s}} \]

Step 3: Determine the wave speed relative to the water in Channel B.

There is no current, so:

\[ \boxed{v_{\text{water,B}}=1.00\text{ m/s}} \]

The two waves have identical speeds relative to the water.

Step 4: Calculate the depth of Channel A.

Use the corrected wave speed:

\[ h_A\approx\frac{v_{\text{water,A}}^2}{g} \]

\[ h_A\approx\frac{1.00^2}{9.8} \]

\[ \boxed{h_A\approx0.102\text{ m}} \]

Step 5: Calculate the depth of Channel B.

\[ h_B\approx\frac{v_{\text{water,B}}^2}{g} \]

\[ h_B\approx\frac{1.00^2}{9.8} \]

\[ \boxed{h_B\approx0.102\text{ m}} \]

Step 6: Check the shallow-water conditions.

For Channel A:

\[ \frac{h_A}{\lambda_A} =\frac{0.102}{6.0} =0.017 \]

For Channel B:

\[ \frac{h_B}{\lambda_B} =\frac{0.102}{4.0} =0.0255 \]

Both values are less than 0.05.

Therefore, the shallow-water approximation is valid in both channels.

Step 7: Explain the misleading measurements.

The measured speeds are also consistent with the wave relationship in the shore reference frame:

For Channel A:

\[ v=f\lambda =0.25\times6.0 =1.50\text{ m/s} \]

For Channel B:

\[ v=f\lambda =0.25\times4.0 =1.00\text{ m/s} \]

The generator produces the same frequency in both channels, but the current changes the speed and wavelength observed from the shore.

This is the crucial distinction: the shallow-water depth relationship uses the wave speed relative to the water, not the speed relative to the shore.

A larger measured wave speed does not automatically mean deeper water.

07Sources and further reading

The following resources provide the cultural, educational, and scientific context for this topic.

• NESA: Physics 11-12 Syllabus (2025) – official syllabus content.
• NESA: Physics syllabus overview and implementation – syllabus implementation dates and protocols for working with Aboriginal and Torres Strait Islander Communities.
• Maureen Fuary (2009): Reading and Riding the Waves: The Sea as a Known Universe in Torres Strait – research documenting Torres Strait Islander knowledge of the seascape.
• Ngarrngga: The Torres Strait Islander Seascape – an educational case study exploring community-specific knowledge.
• Institute of Physics: The Speed of Water Waves – explains the physics of water-wave speed and the shallow-water approximation.
• Bureau of Meteorology: Waves and Swell – explains wave properties, wavelength, period, and coastal wave behaviour.

Syllabus timing: The 2025 Physics 11-12 syllabus begins implementation for Year 11 in 2027, with Year 12 implementation commencing in Term 4 of 2027 and the first HSC examination under the new syllabus in 2028. Students completing Year 12 in 2026 continue with the 2017 syllabus.

The next useful concept is wave refraction. Once you understand why waves slow down over shallow water, you can explain why wavefronts bend as they approach a coastline or travel across an uneven seabed. That connects the measurement of wave speed and wavelength to the changing direction of waves around underwater features.