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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 17.5 Waves on a String

Activity 17.5.1. Warm-up: Modeling a String.

A primitive model for a string is a straight line of molecules connected by identical springs, as shown in the figure below. Imagine you create a wave through this string by oscillating one end of this line of molecules back and forth.
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A horizontal row of small circles representing molecules, each separated by a representation of a spring.
Figure 17.5.1. A series of connected molecules approximating a string.
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(a)

Do you think the speed of the wave depends on the mass of the molecules? If so, do you think the wave speed is greater or less if the mass is greater? Explain your reasoning.
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(b)

Do you think the speed of the wave depends on how big the spring force between the molecules is? If so, do you think the wave speed is greater or less if the spring force is greater? Explain your reasoning.
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(c)

Do you think the speed of the wave depends on the frequency that you oscillate the end of the line? If so, do you think the wave speed is greater or less if the frequency is greater? Explain your reasoning.
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(d)

Consider a real stretched string, such as you might find in a musical instrument such as a guitar. Given your answers above, what properties of the guitar string do you think might affect the speed of the wave?
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Consider a stretched string of length \(L \) and mass \(m \) with tension \(T_s \text{,}\) and linear density \(\mu_s\) where
\begin{equation*} \mu_s = \frac{m}{L} \end{equation*}
is the mass-to-length ratio of the string. The wave speed \(v \)
\begin{equation*} v_{\text{string}} = \sqrt{\frac{T_s}{\mu_s}} \end{equation*}
depends on the tension that serves as a restoring force for the medium and the linear density of the string. Interestingly, the wave speed does not depend on the frequency of the wave!
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A bell-shaped pulse moves from left to right on a horizontal string.
Figure 17.5.2. A wave pulse passing through a string. The wave propagates with wave speed \(v_{\text{string}}\text{.}\)
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Subsubsection Activities

Activity 17.5.2. Sensemaking: Covariational Reasoning.

How, if at all, does the speed of the wave depend on each of the following?
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  1. The amplitude of the wave
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  2. The frequency of the wave
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  3. The properties of the medium through which the wave is traveling
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Activity 17.5.3. Sensemaking: Units of wave speed.

Explicitly check the units of \(v_{\text{string}} \) to confirm its relationship holds.
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Answer.
\begin{equation*} m/s = \sqrt{N/(kg/m)} = \sqrt{\frac{(kg*m/s^2)}{(kg/m)}}=\sqrt{\frac{m^2}{s^2}} = m/s \end{equation*}
Indeed, the units are consistent.
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The diagram below shows a wave pulse at \(t = 0\) moving to the left on a string with wave speed 2 mm/s. Each grid box represents 1 mm.
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A horizontal string on a gridded background, with a wave pulse on the right moving to the left. The pulse is serrated, with one 2-mm peak, a 1-mm valley, and a 3-mm peak.
Figure 17.5.3. A wave pulse moves to the left.
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Activity 17.5.4. Practice: Moving Wave Pulse.

Sketch the wave pulse at \(t = 3 \text{ s}\text{.}\)
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Answer.
The peak from the previous image has moved 6 mm to the left.
Figure 17.5.4. The wave pulse has moved to the left.
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Activity 17.5.5. Comparing Waves.

Assume all the waves below have the same amplitude. Make all graphs below by hand (without a graphing calculator).
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(a)
Sketch two waves that have different frequencies. What kind of graph (history or snapshot) would you want to draw?
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(b)
Sketch two waves that have different wavelengths. What kind of graph (history or snapshot) would you want to draw?
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(c)
Sketch two waves with the same frequency and the same wavelength but different phases. What kind of graph (history or snapshot) would you want to draw?
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(d)
Which waves do you think would be easiest to add together?
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(e)
Explain why you agree or disagree with each of the following statements:
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1: “Two waves that have the same frequency, wavelength, and amplitude will always look the same when you graph them.”
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2: “If two waves have different phases, they will start at different points in their cycle even if they have the same shape.”
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References References

[1]
Animations courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State. licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. For more information see https://www.acs.psu.edu/drussell/demos.html
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