Standing Waves - PHYS143 (2024)

In this lab you will use the physics of standing waves and how waves propagate to make measurements of the speed of sound in air and the linear mass density $\rho$ of a vibrating string. You will also investigate the behavior of standing waves on the strings of a Ukulele.

Click here to get a copy of the lab notebook. Don't forget you need to be logged into your UChicago google account to access the document.

Experimental procedure


You will do experiments involving both transverse waves on a vibrating string and longitudinal sound waves traveling in a tube. It does not matter in which order you do these two experiments.

The part involving the Ukulele should be done after the string and tube parts.

Vibrating String

A vibrating string can be caused to vibrate with a relatively large amplitude if it is forced to oscillate at or near one of its “natural” frequencies. This phenomenon is called resonance.

For resonance to occur on a string fixed at both ends, there may be any number of nodes along the string, but the endpoints must be nodes. (See Fig. 1.) A string of length L vibrating at resonance must contain an integer multiple of half wavelengths,

$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$.

Figure 1: Standing waves on a string with both ends fixed.

Using the relation $f = v/\lambda$, we may write

$f_n = \dfrac{v}{\lambda_n}=l\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. (7)

The speed of propagation of a wave $v$ in general is given by,

$v = \sqrt{\dfrac{\textrm{“restoring force” factor}}{\textrm{“inertial” factor}}}$. (10)

For a transverse wave along a string, this becomes,

$v_{string} = \sqrt{\dfrac{T\vphantom{\large{T}}}{\rho}}$, (11)

where $T$ is the tension in the string and $\rho$ (rho) is the mass per unit length of the stretched string.

The apparatus for this experiment is illustrated in figure (2).

Figure (2). Diagram of the apparatus used for the vibrating string experiment.

The string is attached at one end to a fixed post, the other is tied to a mass which hangs over a pulley. A small speaker connected to a function generator is used to vibrate the string where it attaches to the post. The system should behave as a vibrating string with two fixed ends. The fixed ends being the points where the string attaches to the post and the point where is wraps over the pulley. If the string is vibrated at the resonant frequency, or an integer multiple of it, you should be able to see the appropriate standing wave patterns as shown in figure (1).

For this part of the lab you are expected to use your understanding of standing waves to predict the resonant frequencies of the first three vibrational modes for a string which is fixed at both ends, and then experimentally confirm your predictions.

Carefully consider what are your measured values and how will you estimate their uncertainties.

You must propagate these uncertainties through all of your calculations so that you can quantify the degree of agreement of your calculated and measured frequencies.

Interesting note.

You are already familiar with one example of where linear mass density $\rho$ is important. Stringed musical instruments. Guitars, violins, etc. use strings of different lengths and different $\rho$'s to produce vibrations at specific frequencies. For example the “A” string of a guitar in standard tuning has a frequency of 440Hz. The length, $\rho$ and the tension $T$ in the string have to be balanced according to the desired note to be produced.

Sound Tube

What we perceive as sound are pressure waves propagating through the air. This is an example of a longitudinal wave as opposed to the transverse wave of the vibrating string. By measuring the wavelength ($\lambda$) of the standing wave created by sound, of a known frequncy, propagating through a tube as a function of frequency one can determine the speed of sound in air.

The resonant frequencies for an air column with one end closed and the other end open is

$f_n = \dfrac{n v}{4L} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. (16)

The resonant frequencies for an air column with both ends open is

$f_n = \dfrac{n v}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. (17)

Velocity from wavelength measurement

Use the apparatus shown below to measure the wavelength of sound waves in the tube for a number of frequencies from 500hz to 1000hz. Plot the data in a way that allows you to extract the velocity of sound by fitting to a straight line.

A diagram of the apparatus is shown in figure (3).

Figure 3. Diagram of the apparatus used for the sound tube experiment.

It consists of a loudspeaker driven by a function generator as a source of sound waves. A hollow plastic tube with a plunger that can positioned anywhere inside the tube. And a sound pressure level (SPL) meter to measure the pressure of the sound waves which can also be displayed on an oscilloscope. The components are mounted on a long rigid rail.

With this apparatus you are able to create standing waves in the sound tube while varying the length of the sound cavity and the frequency of the waves.

Use the apparatus to measure the speed of sound at several frequencies between 500Hz and 1kHz.

Make sure to estimate uncertainties in your measured quantities and propagate them through your calculations.

Do you observe a significant dependence of the speed of sound on wavelength?

How do your measured values compare with the literature value for the speed of sound in Chicago? Note that the speed of sound in air varies with barometric pressure among other factors. As such the literature value is approximate, so do not expect precise agreement.

Time of flight measurement

The apparatus can also be used to produce short pulses of pressure by switching the function generator to output a square wave instead of a sine wave. The SPL meter will pick up both the initial pulse and the reflected pulse from the plunger. Thus the apparatus can be used to directly measure the speed of sound by using the scope to time how long it takes for the pulse to travel the length of the tube and back.

Use this technique to measure the speed of sound over a range of distances.

Finally, carefully study the polarities of the reflected pulses for two cases:

  • Far end of the tube closed.
  • Far end of the tube open.

Do the polarities of the reflected pulses behave as you would expect?

The speed of propagation of a sound wave in air is

$V_{gas} = \sqrt{\dfrac{\gamma k_BT}{m}}$, (13)

where $\gamma$  is the ratio of specific heats of the gas in which the wave is moving ( for a diatomic gas), $k_B$ is the Boltzmann constant, $T$ is the absolute temperature of the gas (in Kelvin), and $m$ is the mass of an individual molecule of gas. (The mass of an air molecule is about $4.8 \times 10^{-26}$ kg.).

Compare your measured values with the calculated value assuming the temperature in the lab is 300K.

Ukulele

Please handle the Ukulele with care. It is an inexpensive instrument and easily damaged. Do not play with them before reaching this part of the lab, after you have done both of the previous parts. Return them to the TA before you leave the lab at the end of the period.

If you play a musical instrument or even just enjoy listening to music, you are already familiar with one manifestation of waves in everyday life. What your brain interprets as sound is the result of vibrations of the eardrum in response to transverse pressure waves, such as the ones you worked with in the sound tube part of this lab. These pressure waves were ultimately produced by either the human voice or musical instruments.

Stringed instruments such as cellos, violins, guitars and Ukulele's are directly analogous to the vibrating string which you worked with in this lab. They are strung with strings whose length, linear mass density and tension have been chosen to allow you to tune each string to the appropriate note. Different notes can be played by using your finger to press a string against the neck of the instrument, thus shortening the length over which it is able to vibrate which in turn raises the resonant frequency.

As a mostly fun exercise there are Ukulele's which you can use to measure the resonant frequencies of the different strings. You have likely noticed that the sound produced by the speaker used for the sound tube part of the lab, is rather unpleasant and at frequencies above 1kHz can be incredibly annoying. This is the result of driving a speaker at a single frequency. However a note from a stringed instrument, tuned to that same frequency, is much more pleasing to the ear. It somehow sounds “richer” and less harsh than the tone from the function generator. This is because a plucked string vibrates at not just one frequency. Instead the plucked string will vibrate at both the fundamental resonant frequency, but also at frequencies associated with the higher order harmonics. Usually the fundamental frequency dominates, and the higher order harmonics are of small enough amplitude that they are not noticeable to the eye when you observe a plucked string. However these additional frequencies are there and they can be observed in the waveform associated with the sound pressure wave which they produce.

For this last exercise we want you to record the waveform produced by plucking one of the strings of a Ukulele, and try to observe it's higher order harmonic components. You could do this by using the sound pressure level meter from the sound tube apparatus to detect the wave and send it to the scope. However trying to get the triggering of the scope set up appropriately to obtain a clean waveform from the meter is not a trivial task. A much easier way to perform this measurement is to use the microphone detector built into the IoLab device that you have used in previous labs.

When you are ready for this part of the lab, after having completed the first two parts, get a Ukulele and an IoLab from your TA.

Setup the IoLab to measure the microphone signal. The location of the microphone is indicated on the front of the IoLab device.

With the IoLab software running, hold the Ukulele in front of the microphone in the IoLab and pluck one of its strings with your finger.

Stop the IoLab data collection and use the built in tools to zoom in on different parts of the recorded waveform. You should note that the nature of the tone varies. You should be able to observe that more than one frequency of oscillation is present in the waveform. Use the software to measure the different frequencies which you observe. You should be able to find frequencies corresponding to the fundamental and one or more harmonics. How many harmonics you observe may depend on whether you are looking at the beginning or at the end of the tone.

The software can perform a FFT of selected portions of the waveform. You can use this tool to also detect and measure the frequencies which are present in the waveform. If you have not covered Fourier Transforms in lecture, you may want to check with your TA regarding how to interpret what you are seeing in the FFT.

Report: Summary and conclusions


After the lab, you will need to write up your conclusions. This should be a separate document, and it should be done individually (though you may talk your group members or ask questions). Include any data tables, plots, etc. from the your lab notebook as necessary in order to show how your data support your conclusions.

The conclusion is your interpretation and discussion of your data. What do your data tell you? How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.) Your conclusions should always be based on the results of your work in the lab. It is not acceptable to evaluate the results of an experiment by comparison to known values or any other form of preconceived expectation. Your conclusions need to be supported by your data. If your data are inconclusive or in disagreement with regard to your expectations then your conclusion should reflect that.

REMINDER: Your report is due 48 hours after the lab. Submit a single PDF on Canvas.