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Standing Waves Guide

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Standing Waves Guide

Nathan and Keira are testing a stretched string with a vibration driver. Use the information in each question. Show working for calculations and include units.

Model and measurements

For a string fixed at both ends, the allowed wavelengths are λₙ = 2L/n, where L is the vibrating length and n = 1, 2, 3, … . Wave speed is v = fλ. For a stretched string, v = √(T/μ), where T is tension and μ is mass per unit length.

1.Choose the best explanation for how a standing wave forms on the string.

A single wave travels along the string and gradually stops.
Two waves of the same frequency and amplitude travel in opposite directions and interfere.
The string moves at the same speed as the vibration driver.
The wave reflects but does not interfere with the incoming wave.

2.For a string fixed at both ends, the second harmonic has two loops. State the number of nodes and antinodes. Include the two fixed ends when counting nodes.

3.At which point on a standing wave is the displacement always zero: a node or an antinode? Explain what happens at that point.

4.The vibrating length of Nathan’s string is 0.80 m. Calculate the wavelength of its fundamental mode. Show the relationship you use.

5.The string in Question 4 vibrates in its fundamental mode at 75 Hz. Use your wavelength from Question 4 to calculate the wave speed.

6.For a string fixed at both ends, how many times the fundamental frequency is the third-harmonic frequency? Explain using the harmonic number.

7.Keira increases the driver frequency while keeping the string length and tension unchanged. Which statement best describes the resonant patterns she may observe?

Only the amplitude changes; the number of loops cannot change.
Resonance occurs at particular frequencies, and higher harmonics have more loops and shorter wavelengths.
Every frequency produces a standing wave with the same pattern.
The wavelength increases at every higher harmonic.

8.Nathan doubles the tension in the string but keeps its length and mass per unit length unchanged. Use v = √(T/μ) to predict the factor by which wave speed changes. What happens to the fundamental frequency?

9.Draw the second harmonic for a string fixed at both ends. Mark both ends and the middle node with N, and mark each antinode with A.

10.A taonga pūoro produces a stronger sound when its air column resonates. Explain what resonance means here and why the sound becomes louder.