Table of Contents
Standing Waves - PHYS143
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.
Teaching Goal
The quantities which you are measuring, the speed of sound and the linear mass density of a piece of string, are not in and of themselves particularly physically interesting quantities. As with the other labs in the PHYS140's sequence our goal is to teach you Experimental Physics in parallel with the physics concepts you are learning in lecture. The primary learning goals for this lab are twofold.
Continuing to give you experience with applying your growing knowledge of physics concepts to performing experimental investigations in a lab setting.
And.
When doing experiments an important question arises which is how do you know if your results are right? How to answer this question is a central theme of the labs associated with the PHYS140's because it is an important part of scientific research. As we have stated multiple times in previous quarters, no one does experiments for the purpose of measuring things which we already know, or to confirm what we already know from theory. Experimental physics is the art of determining what we do and do not know. As such learning how to test your results in order to be confident that they are correct is part of what we are trying to teach.
For this experiment there are two independent ways to build confidence in your understanding of your data and final conclusions.
- Testing for consistency by performing multiple experiments using experimentally independent methods and techniques. If you measure the same thing multiple different ways you expect to find the same result, within experimental uncertainties,
- Making measurements which also test functional relationships. For example force depends on both mass and acceleration according to $F=ma$. You could determine the gravitational force on a falling object by simply measuring it's acceleration for a given mass. However if you think about it, the functional form of $F=ma$ also tells you that the force should vary in a linear fashion with respect to the mass of the object. So if you determine the force for a range of different masses and plot the data you expect to see a linear relationship. If you do see a linear relationship then you have some confidence that the system is behaving as expected. However if your data are not linear this tells you that something is amiss, either in your experimental technique or possibly your understanding of the physics of the phenomena which you are investigating.
Pretty much all physical theories and models are built upon assumptions. In the classroom these assumptions frequently take the form of frictionless surfaces, or spherical massless elephants. When these assumptions break down you can no longer trust the theory or model. The real world is non-ideal meaning our surfaces have friction ( or air resistance in the case of a falling object) and our elephants are not really massless spheres. What we do as experimentalists is to either minimize the effects of non-deal factors in our experiments, or figure out how to correct for them. In any case one must always convince oneself that the experiment is behaving as expected. Data which deviate from the expected functional form could indicate that a simplifying assumption is not valid, or that the experiment is not being performed correctly.
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 Task
For this lab you will be making use of your knowledge of the basic principles of standing waves, nodes & anti-nodes, and the speed of propagation of waves through a medium in order to design and execute experiments to measure the linear mass density $\rho$ of a piece of string and the speed of sound in air.
The speed $v$ of a wave is defined as
$v = f \lambda$,
where $f$ is the frequency of the wave in Hz and $\lambda$ is its wavelength in m. Thus if one can produce a wave with a known frequency (which we will do with a function generator) and then measure its wavelength you can calculate its speed.
Thus one of the experimental problems to be solved is the measurement of the wavelength. To do this we will make use of the fact that the maximum amplitudes for a standing wave with both ends fixed for a vibrating string (or both ends closed for sound waves in a tube) occur when the length $L$ of the string (or tube) contains an integer number $n$ of half-wavelengths $\lambda$,
$L = \frac{n\lambda}{2}$.
Figure 1: Standing waves on a string with both ends fixed.
Above we have simply stated material which is typically covered in detail in lecture. Thus it is expected that you have seen traveling and standing waves in class as well as a discussion of nodes and anti-nodes for various combinations of boundary conditions such as both ends fixed, or one end fixed and one end free. If you are not clear where the above relationships come from you should consult your text. More details can also be found on this wiki page which provides a more complete review of the properties of Wave Motion and Propagation.
Experimental procedure
You will do two independent experiments for this lab, one involving transverse waves on a vibrating string and the other involving longitudinal sound waves traveling through a tube. It does not matter in which order you do the two experiments.
Vibrating String
The goal of this experiment is straight forward, to measure the Linear Mass Density of a piece of string using what you know from lecture about standing waves on a string with both ends fixed.
It may not have been covered in lecture, but in general the speed of propagation of a wave in a given medium is
$v = \sqrt{\dfrac{\textrm{“restoring force”}}{\textrm{“inertia”}}}$.
For a transverse wave along a string the restoring force turns out to be the tension $T$ in the string and the inertial factor is the linear mass density $\rho$
$v_{string} = \sqrt{\dfrac{T}{\rho}}$.
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).
The setup allows you to vary the mass (and therefor the tension in the string), the length of the string between the fixed points, and the driving frequency. By varying the frequency you should be able to achieve resonance for multiple nodes $n$.
Given what you know about the physics of how standing waves are formed, and the velocity of the wave along the string, you should be able to plan and execute at least two experiments for measuring the linear mass density of the string.
Hint : all of the parameters in bold above can be either held constant or varied. Find a function which contains these parameters and $\rho$. There will be multiple ways to arrange the terms so that you can measure the value of one parameter as a function of a second one with the others remaining fixed. Just choose two of these arrangements to measure.
Perform both of these experiments in a manner which allows you to test the expected functional relationship between independent variables in addition to providing a value for the linear mass density.
Note that there are three obvious ways that you can perform this measurement, you can perform all three measurements if you like but we are only asking for you to do two due to time constraints. Also note that the different parameters which you can control all have associated uncertainties which must be carefully assessed and propagated through subsequent calculations. Some of the uncertainties will be more significant than others.
Finally you can also make a rough measurement of the linear mass density from a direct measurement of it's length and mass. While this method is more direct, the small mass of the string is difficult to measure accurately and will result in a large uncertainty. The vibrational methods are capable of making a more precise measurement. Nonetheless the results from all three measurement techniques should be in agreement to within experimental uncertainties.
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 in the string have to be balanced according to the desired note to be produced.
Sound Tube
The phenomena of standing waves created by sound propagating through a tube with closed or open ends can be used to measure the speed of sound in air. 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 what you know about the physics of standing waves for sound in a tube with two closed ends to measure the speed of sound.
As in the vibrating string experiment you should conduct the experiment in a manner which allows you to test the functional form of the relationship between relevant parameters. Unlike the string experiment here we will recommend that you do the measurement at one frequency for the sound waves somewhere in the range of 3kHz to 5kHz. The reason we are specifying the value of this parameter is that the speaker we are using only reproduces sound waves accurately at frequencies over about 1kHz, so there is no use in trying to vary the frequency.
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.
Compare your results for the speed of sound using the two different techniques. You can also lookup the literature value for the speed of sound.
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?