In this experiment, you will investigate standing waves on a string of a certain length, with a measurable linear mass density, and held taught by a user-determined weight, with the aim to discover
the characteristic or allowed standing wave resonance frequencies of the system that satisfy certain boundary conditions (``closed-closed'') [1].
the phase velocity of the (counter-propagating) traveling waves of the system.
We wish to predict the frequencies of the normal modes of allowed standing waves (task #1), and to figure out how to estimate the speed of traveling waves that give rise to the standing wave resonances on a stretched string, principally by investigating the effects of the tension in the string (task #2). Prediction involves making 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 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
Make an initial choice of mass to determine the tension in the string. Turn up the frequency and see if you can observe a progression of standing wave normal modes. Try adjusting the output amplitude so that the normal modes are visible, but not so great that the system becomes unstable--keep the wave amplitude low. [2] Make notes for the record about your choices.
Measure the effective length, $L$, of the vibrating spring and its uncertainty.
Make a sketch of a diagram for this experiment and a separate block diagram indicating electrical interconnections of the instruments used, including all settings of those instruments (here is an example from a relevant lab at Caltech). Annotate the cartoon of the setup, visually defining things like $L$, $T \: (tension)$, etc. Indicate in your lab notebook those quantities that must be measured, which varied, and so forth, how things are electrically connected, and so forth. Be sure to record all the settings..
Measure the linear mass density of the string. You will need this value for your modeling work. Get a number for this one way or the other and write this down too.
Now, begin taking data by playing with -- frequency tuning, along with optimizing the output voltage for the vibrator -- the set up. For comments about this, see Ref [2]. Get some experience making normal modes. Attempt to excite the string at a frequency so as to produce not the fundamental resonance, but many. See and note what it takes, to change from one normal mode to the next, noticing carefully the uncertainty in the frequency associated with the $n_{th}$ normal mode, $\Delta f_n$. Decide, justify, record. This is an investigative exploration. Once you have an answer for the experimental uncertainty, then,
Find the first harmonic higher than the fundamental resonance, $f_2$. This should be easiest to `see' and understand since $L$ and $\lambda$ should be the same. Sketch what you see. Is it? Explain how this mode satisfies the boundary conditions.
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 space distance do you expect there to be between the nodes? Sketch the mode structure you expect for $f_3$ according to your model, and also estimate $f_3$ given what you found for $f_2$. How much higher in frequency should it be? Does the model suggest a ratio? Exhibit this, make an estimate of the shape and frequency of $f_3$, and then see -- find out experimentally -- how well the model predicts its shape and frequency. This is an intermediate check on progress.
Now get lots of data on as many normal modes as you can identify. Set up a data table for your results, including columns for mode number, frequency, uncertainty in frequency, wavelength (how to calculate that??), etc. I mustn't specify all of this, you must make choices and justify them with comments in your record about the choices you have made. 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, and what data is in it.
Plot $f_n$ vs $n$, both the data, and your modeling curve using FITTEIA. Find out
the goodness of the fit and (with the Goldilocks analysis) the uncertainty and best value of the fitting parameter. What claims about the correspondence model and the data is supported by this analysis?
the physical quantity (or quantities) in the physical model that corresponds to that fitting parameter. Make sure you have a separate measurement of those quantities so that you make compare the fitting parameter (and its uncertainty) with your calculation of the same from the physical quantities. Is the quantitative magnitude of the discrepancy equal to, less than, or greater than the sum of the uncertainties? Again, what claim about the correspondence between the data and model is supported by this analysis?
Make sure the ``how to '' is well documented as this makes the analysis above more straightforward.
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
Choose on particular normal mode ($n = 2$?) and, keeping $n$ fixed, vary the tension in the string. How will $f_n$ vary with $T$ and what does this tell us about the phase velocity of the traveling waves that create the normal modes (standing wave resonances)? Prepare another table, deciding the column headings, being sure to include uncertainties, and so forth, for fixed $n$, vary $T$. Surely there must be a column for $m$ We can suppose that $m$ is without experimental uncertainty. 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 for use with FITTEIA, etc. Record all filenames (with path information) and a comment about the contents, and so forth.
Plot $f_2$ (or $f_3$; the $f_n$ for the $n$ you chose) vs $m$ (or $T$), both the data, and your modeling curve using FITTEIA. Find out
the goodness of the fit and (with the Goldilocks analysis) the uncertainty and best value of the fitting parameter. What claim about the correspondence model and the data is supported by this analysis?
the physical quantity (or quantities) in the physical model that corresponds to that fitting parameter. Make sure you have a separate measurement of those quantities so that you make compare the fitting parameter (and its uncertainty) with your calculation of the same from the physical quantities. Is the quantitative magnitude of the discrepancy equal to, less than, or greater than the sum of the uncertainties? Again, what claim about the correspondence between the data and model is supported by this analysis?
Make sure the ``how to '' is documented as this makes the analysis above more straightforward.
Some general comments about both tasks:
Are the models as they apply to each task linear or nonlinear? What do you expect it to be? This internal conversation needs to be a part of the record.
You'll have to propagate the errors in the quantities of physical interest in order to compare your experimental estimate of the fitting parameter in each case to the fitting parameter that FITTEIA produces. 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! In any case, propagating error is vital to the quantitative comparison between discrepancy and uncertainty and in turn vital to any claim you might make about your results. Again, a good ``how to'' record makes this straightforward.
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?
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), (4th Ed.)
\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}
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 output 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''. We find that with our particular apparatus that weird (loud) rattling sets in at too high of a frequency or too large of an amplitude. This seems to alter the mode structure near the diving board, which is meant to be still (not obviously vibrating).
