Relation v = λf: surface waves
Verify the fundamental wave relationship v = λ × f by varying the frequency and measuring the wavelength with the FizziQ Web Waves on a Lake simulation.
Activity Summary
The student uses the FizziQ Web Waves on a Lake simulation to observe circular wave propagation. By fixing the propagation speed and varying the frequency, they measure the wavelength and verify that the product λ × f equals the wave speed.
Introduction
Throw a pebble into a lake and watch the circular ripples spread across the surface. Does their propagation speed depend on the frequency? On the amplitude? The Waves on a Lake simulation in FizziQ Web lets you answer these questions by systematically varying the frequency while keeping the speed fixed, and measuring the wavelength from the pattern of concentric circles. You will discover the universal relationship that connects three fundamental wave quantities.
Learning Objectives
- Measure the wavelength from the simulation
- Experimentally verify the relationship v = λ × f
- Understand that propagation speed depends on the medium, not on frequency
- Establish that wavelength and frequency are inversely proportional at constant speed
Scientific Concepts
Instruments and sensors
Scientific instruments
- FizziQ Web Waves on a Lake simulation
FizziQ Features
- Experiment notebook — Used to record the frequency and wavelength values in a table and plot λ versus 1/f to read off the slope (wave speed).
Required Materials
- - Computer, tablet, or smartphone with FizziQ Web - Note: the protocol remains adaptable to any comparable simulation or data-plotting tool.
Experimental Protocol
Open the Waves on a Lake simulation in FizziQ Web (Experiment → Simulations → Waves on a Lake).
Fix the propagation speed to a given value (for example 2 m/s) and the amplitude to 0.5 m. These parameters will not change during the experiment.
Set the frequency to 0.5 Hz. Start the simulation (START). Observe the circular waves propagating on the surface.
Observe the distance between two successive crests: this is the wavelength λ. Measure it or note it from the data display.
Stop the simulation. Change the frequency to 1.0 Hz, then restart. Has the distance between crests changed? In which direction?
Repeat for frequencies 1.5 Hz, 2.0 Hz, 2.5 Hz, and 3.0 Hz. For each frequency, note the wavelength λ in a table.
Create a table with three columns: Frequency f (Hz), Wavelength λ (m), and Product λ × f (m/s).
Calculate the product λ × f for each row. What do you notice? Is this product constant?
Plot the graph of λ versus 1/f. If the relationship v = λ × f holds, you should get a straight line through the origin with slope v.
Verify that the slope matches the propagation speed you set at the beginning. Conclusion: v = λ × f.
Expected Results
For v = 2 m/s: λ = 4.0 m (f = 0.5 Hz), λ = 2.0 m (f = 1.0 Hz), λ = 1.33 m (f = 1.5 Hz), λ = 1.0 m (f = 2.0 Hz), λ = 0.8 m (f = 2.5 Hz), λ = 0.67 m (f = 3.0 Hz). The product λ × f is constant and equals 2 m/s (the propagation speed). The graph λ(1/f) is a straight line through the origin with slope 2 m/s.
Scientific Questions
- Does the wave amplitude affect the propagation speed or the wavelength?
- Why do waves on deep water travel faster than waves on shallow water?
- Could you determine the propagation speed using only a float and a stopwatch?
- What is the difference between the speed of a wave and the speed of a float?
- Does the v = λf relationship also apply to electromagnetic waves like light?
Scientific Background
A **mechanical wave** is a disturbance that propagates through a material medium without transporting matter. On a lake, a float bobs up and down but does not travel with the wave.
Three quantities characterize a wave. The **frequency** f (in Hz) is the number of oscillations per second. The **wavelength** λ (in m) is the distance between two consecutive crests. The **speed** v (in m/s) is the distance traveled by the wavefront per second.
These three quantities are linked by the **fundamental relationship**: **v = λ × f**. This relationship is universal: it applies to water waves, sound waves, light waves, and seismic waves.
The propagation speed depends on the **medium** and not on the frequency. On a lake, it depends on the water depth. In air, sound speed depends on temperature. When you change the frequency, only the wavelength adjusts (λ = v/f).
The **amplitude** is the maximum height of the wave (the crest). It is independent of frequency and wavelength. A high-frequency wave and a low-frequency wave can have the same amplitude.
Extensions
- Observe the float motion and verify that their oscillation period matches 1/f
- Measure the time delay between two floats at different distances to determine the propagation speed independently
- Compare the wave pattern at high frequency (many close crests) with that at low frequency (few distant crests)
- Investigate whether the amplitude affects the propagation speed
Frequently Asked Questions
How do I measure the wavelength in the simulation?
The wavelength is the distance between two successive crests visible on the screen. You can also calculate it: λ = v/f if you know the speed and frequency.
The wavelength seems to change during the simulation.
Make sure the frequency and speed settings are not changing. The visual appearance may vary as the wavefront spreads, but the actual wavelength remains constant.
Does changing the amplitude affect the wavelength?
No. Amplitude and wavelength are independent. Changing the amplitude makes the waves taller or shorter but does not change the spacing between crests.
Why is the product λ × f not perfectly constant in my data?
Small discrepancies are due to measurement precision. If you are reading wavelength from the visual display, there is inherent imprecision. The theoretical product should be exactly equal to v.
Detailed Description
The student uses the FizziQ Web Waves on a Lake simulation to observe circular wave propagation. By fixing the propagation speed and varying the frequency, they measure the distance between consecutive crests (wavelength) and verify that the product λ × f equals the wave speed. FizziQ Web provides this ready-to-use simulation with adjustable frequency, propagation speed, and amplitude, and its experiment notebook lets students build the frequency/wavelength table and plot the λ versus 1/f graph directly in the app.
This activity is part of our sound resources. To dig deeper into measuring and analyzing sound (waveform, spectrum, spectrogram, decibels) and find all twelve experiments, read our complete guide to measuring and analyzing sound with a smartphone or computer.
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