The two questions to be pursued in these experiments are:
What are the characteristic or allowed resonant frequencies (let's call them the eigenfrequencies of the normal modes while we are at it!) of a mechanical system comprising a stretched string of a certain length $L$, stretched with a given tension $T$, with a given linear mass density $\mu$? The boundary conditions are ``closed-closed". [1]
Is it possible to measure the phase velocity of traveling waves in or with such a system using standing waves?
We wish to predict the frequencies of the normal modes of allowed standing waves (task #1), and to figure out how the speed of traveling waves on a stretched string depends on the tension in the string (task #2). To predict is to make appropriate use of a model, in this case, the theory of standing waves [1], which are coherent destructive and constructive interference effects, arising from the superposition of transverse traveling waves, subject to boundary conditions. The researcher is called on to render an opinion regarding just how well the one dimensional simplified theory corresponds to, or 'models', the reality of the actual experiments. We will use a `diving board' style vibrator to excite
transverse oscillations near one boundary of the stretched
string. A simple function generator is used to drive the vibrator which allows us to vary the frequency of the vibration. A partial set up is shown in figure 1.
Figure 1. A driver of some sort, at one end of a string is shown exciting a resonance, or standing wave, on a string stretched by a mass dangling at its other end.
3.2.1 Task #1: Determine the frequencies of the normal modes (i.e. the standing waves) of the vibrating string
To begin taking data, you will have to play with the apparatus a little bit, making an initial choice of mass, with the power on, with a low amplitude [2]. What is the effective length of the vibrating system? Record in an appropriate place estimate the effective length of the string, $L$, and its uncertainty, being careful to explain your rationale. Look for modes. We find that with our particular apparatus that weird rattling sets in at too high of a frequency and too large of an amplitude. This seems to alter the mode structure near the diving board. You'll know by the funny sound. Try backing off of the amplitude with this happens and try to sneak up on the mode. We'll have to do the best we can. But be keenly observant of the mode structure everywhere once a resonance is found. Just get used to the apparatus. Make notes about what to watch out for.
Make a sketch of a diagram for this experiment and a separate block diagram indicating interconnections of the instruments used, including all settings of those instruments. Annotated cartoons of the things you measure convey and encode, even visually define, LOTS of critical information for experimental work. Indicate those quantities that must be measured, which varied, and so forth, how things are electrically connected, and so forth. For a particular example of such things, have a look at what they do at Caltech in one of their lab courses (see esp. page 41 of the lab notebook). Our principal instrument is a function generator. Be sure to record all the settings. Imagine trying to repeat a data set and one of the buttons on the control panel of function generator has changed, the feeling of disequilibrium one has. At some point you will want to measure the linear mass density of the string. I am on the fence about whether to have you do this or to supply it. We'll see. Get a number for this one way or the other and write this down too.
Now, begin taking data as follows: attempt to excite the string at a frequency so as to produce not the fundamental resonance, but the second one, $f_2$, the frequency of the second normal mode. Sketch in your lab notebook what you expect that to look like. How is $\lambda$ related to $L$, the effective length of the string? Remember the boundary conditions! Where do you expect the nodes to be? Sketch and write down your thinking.
Set up this resonance by carefully adjusting the coarse and fine frequency adjustments (probably best to have the fine adjustment knob in the middle of its travel to begin with). Going from too low to too high in $f$, watching carefully to see the antinodes grow and wane, back and forth (i.e. high to low, low to high, back and forth), is called 'dithering', or 'dithering the frequency'. Measure $f_2$, estimating its uncertainty. You'll need to set up a table for the data set you are about to take, 3 columns (the same drill as last time), one column for mode number, a column for frequency, another for the uncertainty in that frequency (remembering to JUSTIFY IT, I mean, how well can you determine the frequency by dithering it to make sure you've found the amplitude?). Make sure that each column heading has a descriptor and units. Record your judgment about the uncertainty, how it was estimated and so forth. After this step, you'll have one row of data. It will be useful to create a spreadsheet file (.csv or .xslx, etc.) from which to cut an paste into the input buffer for the on-line curve fitting interface environment (fitteia.org). Name the file and record where it is saved.
Before taking any more data, make a model calculation of the frequency you expect to be the case for $n=3$. Predict $f_3$. What distance do you expect there to be between the nodes? Sketch the mode structure you expect according to your model, and calculate the next frequency, given the one you just measured. Then carefully hunt for this mode. What is the discrepancy between what you found and what you predicted? Is the discrepancy equal to, greater than, or less than the uncertainty in the quantity $f_3-f_2$? That would be really bad if it were that big yes? To what should we compare it? Interpret. This is an intermediate check on progress.
Now find as many normal modes as you can. Indicate the mode number, the frequency, and so forth. Take the data down, build the table, perhaps add columns to aid in keeping track of the effective length of the string, remember to record uncertainty in the measured thing. Find the biggest `n' you can!
How do you know your data is any good? We have a sort of local check on progress, but not an overall, or global sense. Make a plot of frequency vs. mode number, perhaps you have already set up the graph and you built this graph up as you were filling up the data table (best practice!). Make sure you've indicated the units on the axes labels. Comment: is the pattern that emerges the one you expected? Pursue this further using fitteia to make a modeling plot of $f_n$ vs. mode number. Check the goodness of the fit.
Write down the model you will use for the fitting. Compare the best fit parameter and its uncertainty (using the Goldilocks plot analysis) with your experimental estimate of that same fitting parameter and its uncertainty (which will involve propagation of the experimental error in measured quantities that go into it, discussed below). A portion of what goes in the abstract is now complete. One now probably has an idea of the speed of the waves traveling on the stretch string for a given mass. But how does the speed of the waves vary with mass? This is the subject of the next and final task.
3.2.2 Task #2: Determine the speed of traveling waves on the string, and how this can be done with the apparatus at our disposal
Now we'll try to keep the mode structure the same, and vary the tension in the stretched string, measuring the normal mode frequency as we go. Prepare a table (again, include a column for measured and expected distance between nodes, including your estimate for the uncertainty in that quantity. I'm not sure that it's critical which mode you pick, only that the fundamental is hard to see, and if the frequency gets too high one runs into problems, so maybe choose $n = 2$ or $n=3$. The main task here is to vary $m$, and measure $f_n$ making sure to keep the mode structure the same. You probably already have one data point yes? Make sure you reverify it. Record it.
Before adding extra mass of your choosing, before altering the frequency so as to excite the same mode, predict the new $f_n$ frequency. Will the new frequency be higher, lower, or will it stay the same? Write down your reasoning. Then make a model calculation. Then carefully tune the function generator to the new resonance frequency, assessing uncertainty and discrepancy. This is a check on your progress.
Following this, try as many different masses that you can, plotting your results and filling in your table as you go. Again, you may wish to create a spreadsheet file, blah, blah, to make throwing the data into the fitteia's input buffer easier. Anyway, the model may be emerging before your eyes. Is it linear or nonlinear? What do you expect it to be? Once the plot is reasonably complete it's time to make a modeling plot. Write down the mathematical model that you are using, the meaning of the fitting parameter in terms of the theory of standing waves and normal modes, and so on, and toss your data into the fitteia input buffer, edit the fitting function, etc. and make a a modeling plot of the expected dependence of the normal mode frequency on mass. Check the goodness of the fit.
Compare the fitting parameter gotten from the best fit to your experimental results both quantitatively and quantitatively. Again, you'll have to propagate error. This is explained below. Interpret your results. You should now have an opinion on the speed of traveling waves on stretched strings arrived at using the theory of normal modes and standing waves subject to given boundary conditions. What qualitative idea about the phase velocity of traveling waves is being tested here? What do the results indicate?
Still, this result is somewhat abstract. What about a direct measure of phase velocity using the product of wavelength and frequency, the resonance frequency in this case, when the tension in the string is increasing? What would a plot of the phase velocity, obtained directly from the product of the wavelength multiplied by the frequency, vs. $\sqrt{T}$, where $T$ is the tension in the string, look like? Briefly write down what you think that plot would look like. Then,
Add a column to the table you've created with heading phase velocity (m/s), and enter the product of the measured resonance frequency and the wavelength, another column for $\Delta v$, the uncertainty in the velocity (more on this presently), and another column which is $\sqrt{T}$, calculated directly from the masses (column 1 above). Then make a second matrix with $x$, $y$, and $\Delta y$, where $x$ is $\sqrt{T}$, etc. etc., and so make a modeling plot of phase velocity vs. the square root of the tension in the string. What is your mathematical model for the dependent and independent variables? Record this. If your fitting model is of the form $ y = b*x,$ what is $b$, physically? What relationship do you expect to emerge between the variables, how well does the model fit, and how does the uncertainty in the best fit value of $b$ compare with the experimental value of $b$? What is its uncertainty, and what is the discrepancy? Do they agree within error? Interpret your results. Now you have an defensible opinion about how to measure the speed of traveling waves on a stretched string using the theory of standing waves (interference effects associated with the superposition of traveling waves subject to given boundary conditions). Print and tape your best modeling plots into your lab notebook.
In order to complete the modeling assignments, there will be a comparison of a fitting parameter gotten from modeling (and a Goldilocks plot), with the experimentally determined value of that same parameter. And the experimental value of that parameter may need to be calculated from measured things, each of which has its own uncertainty. What is the uncertainty of a quantity that is a product of quantities, each with an uncertainty, say,
\begin{equation}
c = (a \pm \Delta a)(b \pm \Delta b),
\end{equation}
what is $\Delta c$? Rather than enter into an extended discussion here, please read the updated error propagation page here.
Come to lab with questions!
Compose an abstract describing your finding regarding the two questions pursued in these experiments. Of course, the answer to the second question is ''yes'', however, one must support that answer as physicists do. What would you give as evidence to support such an answer? Does a satisfactory defense involve quoting results, comparing uncertainties and discrepancies, using significant figures appropriately, capturing the essence of the methods, and interpreting what conclusions are supported by your work? In any case, this abstract should be the last thing prepared.
References:
T. Moore, 6 ideas that shaped Physics, Unit Q: Particles
Behave Like Waves, (3rd. Ed. McGraw Hill, NY, 2017) Chapter Q2, sections Q2.3. and Q2.4. It will be assumed that students are conversant with the concepts and theories worked out there. Remember the work we did with Eq. (Q2.4),
\begin{equation} y(x,t) = 2A \sin{(kx)} \cos{(\omega t)}, \end{equation} or,
\begin{equation} y(x,t) = 2A \cos{(\omega t)} \sin{(kx)}, \end{equation} or even,
\begin{equation} y(x,t) = A(t)\sin{(kx)}, \end{equation} \end{equation}
where $y(x,t)$ is the wiggle function for standing waves, $y$ is the transverse displacement, say, of a stretched string, $x$ is the coordinate parallel to the line passing through the boundaries, without forgetting that Eq's. (Q1.11,12) pertain to this work as well.
Low amplitude means the physical height of the string measured from a horizontal line through the nodes is vanishingly small compared with the effective ``L''. That's the meaning. But how do you control it experimentally? With the outpout knob of the functions generator (another setting to record!). All the way CW is probably too much. If you try it (try it!) you might hear a funny noise...there is a mechanical reason for that effecting the boundary condition at the diving board end... back off of that....don't use a setting that is ``too CW''.
