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.

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.

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

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

For a transverse wave along a string, this becomes,

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).

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

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

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.

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.

The speed of propagation of a sound wave in air is

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.

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.

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).

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? 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.