In this lab, we will study standing waves on both stretchy and rigid string.^{Hoveroverthese!}

• 1 Koolertron function generator (https://www.koolertron.com/)
• 2 Banana cables, red & black
• 1 Pasco Mechanical Wave Driver, WA-9855
• 2 Strings: 1 Stretchy (White), 1 Non-Stretchy (garish green)
• 1 Pulley + 1 table clamped string support
• Various Masses (can combine to 50-250 grams in steps of 50)
• 1 Meter Stick
• Access to a Scale
• Record data in this Google Sheets data table

Basics of Standing Waves

Suppose you have a long horizontal string.¹ Suppose the string is under tension \(F_T\) and has a linear mass density (mass per unit length) of \(\mu\) (both of which are independent of your position along the string).

If the string is "free" (or you look far away from the endpoints), then you will find waves that "propagate" along the string with some speed \(v\).¹ This speed can be calculated in terms of the parameters of the string:²

$$v=\sqrt{\frac{F_T}{\mu}}$$

We will not be looking at this case, but the velocity \(v\) is a key parameter of the string regardless. We will instead be looking at standing waves, which is what happens when we hold the string still at both ends.²

A special class of solutions in this case are the harmonics. These are sine functions that oscillate with a frequency \(f\) and a wavelength \(\lambda\). This means that your vertical³ oscillations make the string have a vertical position \(y(x,t)\) of:

$$y(x,t)=\sin\left(\frac{2\pi x}{\lambda}\right)\left[A\sin(2\pi ft)+B\cos(2\pi ft)\right]$$

The velocity is then related to the frequency and the wavelength according to the simple formula:

$$v=f\lambda\label{veqfl}$$

The zeroes of the above function, \(x=\frac{m}{2}\lambda\) (for \(m\) any integer), are known as the nodes of the wave. The "flat" points (maxima/minima), \(x=\frac{m+\frac{1}{2}}{2}\lambda\), are known as the antinodes.

Note that the endpoints of the string, \(x=0\) and \(x=L\), are both always nodes (since we hold those points still), meaning that there is some integer \(n\) for which \(L=\frac{n}{2}\lambda\). (We use \(n\) to refer to the specific integer \(m\) that gives the zero corresponding to \(x=L\).)

In general, we know that \(\frac{L}{\lambda}\) is the "number of wavelengths" that fit into our string length. This means that exactly \(n/2\) waves fit into our string, which means that there are \(n\) "half-waves," or equivalently, \(n\) antinodes.

Hence, if we fix \(L\), we can state what wavelengths are possible, because they always have to correspond to some integer number \(n\) of antinodes, and these wavelengths take the form:

$$\lambda_n=\frac{2L}{n}$$

Combining this with formula \eqref{veqfl}, we can find a formula for the \(n\)th fundamental frequency:

$$f_n=\frac{v}{2L}n$$

You may (quite fairly) wonder about the general behavior: what about solutions that are not these special harmonics? It turns out that these can always be written as a sum (or, more accurately, linear combination, or superposition) of these harmonics: a general vibration is a combination of these oscillations.

Thus, studying these special cases allows us to understand general vibrations, too.⁴

Experimental Setup

Schematically, the experimental apparatus that we will use to do this looks as follows:

Click to expand images of the apparatus

The hanging mass \(M\) will determine the tension \(F_T\) in the string. Note that in the last part (with the fixed-length string), we will vary this mass, where we should observe the relation:

$$f_n^2=\left(\frac{n}{2L}\right)^2\frac{g}{\mu}M\label{fnsq}$$

Note that the length of the string, \(L'\), is just for the measurement of mass density. The oscillations themselves don't care about the string \(\delta L\) beyond the end of the pulley; we treat the pulley as a fixed point!

Click here for some discussion about the role of the wave driver in this experiment

Part I: Predicting Stretchy-String Wave Velocity

Begin by taking your (white) stretchy string. Weigh it (the full length) on a scale to determine the mass, \(m\).

Identify if one end has a loop. If one end does, grab the other end; if neither end does, grab either end. Push the string into the black plastic piece on top of your Mechanical Driver. ¹

If your string does not have a loop, tie one on the other end.²³

Put the loop of stretchy string over your pulley. Hang 150g of mass from the end of the loop (and record this mass as \(M\), with an uncertainty of 0.1g). This mass should neither be at the pulley nor sitting on the ground; adjust your driver position if necessary to fix this.

Now, measure and record the lengths \(L\) and \(\delta L\), with appropriate uncertainties.³

A schematic showing \(L\) and \(\delta L\)

Part II: Measuring Stretchy-String Wave Velocity

Now, attach the function generator to the wave driver. To do this, use a red banana cable to connect the red terminal on Channel 1 of the function generator to one of the banana jacks on the wave drive. Use a black banana cable to connect the black terminal of Channel 1 to the other banana jack on the wave drive. Turn the function generator On (sliding switch on the side of the unit). It will put itself in default settings. To change these to what we want:

1. Press the blue button 2 to select Channel 2. Press the blue button 2 again to turn output 2 Off. We are not using it.
2. Press the yellow button 1 to select Channel 1. Press the top gray button to choose Frequency. The knob now controls Frequency; the two arrow keys choose which digit the knob adjusts. Set the Frequency to ~30 Hertz (Hz).
3. Press the second gray button to select Amplitude. Use the knob to set the amplitude to 20 Volts.
4. Press the top gray button (again!) to select Frequency.

Next, vary the frequency: you will see different nodal patterns appear and disappear as the drive frequency changes. Observe what values of n you can reach with \(f\) between 5 and 80 Hertz. You definitely want at least four different possible n, and ideally at least seven. The data table has room for eight, so you don't need more entries than that regardless, although you're welcome to take them for better results. If you can't even get four different n , consult your TA.

Now, take the frequency back down to the value at which you found the least \(n\) you can get out (ideally \(n=1\). Slowly adjust, while watching the string, until you maximize the amplitude of that mode. You may wish to adjust by \(0.1\) Hz to be as precise as (reasonably) possible.

Record \(n\) and the freqency which gives maximum amplitude for \(n\) anti-nodes.⁴

Repeat this measurement (varying the drive frequency) for a variety of \(n\) - as many as you can get.

For each \(f\) assign an uncertainty \(\sigma f\) based on your judgement. This is not the uncertainty of \(f\) from the function generator (this is very small) but, rather, is the range of \(f's\) you could vary over and still be at about the maximum amplitude for that \(n\).⁵

Part III: Measuring \(\mu\) of the Fixed-Length String

Now, take the fixed-length (bright green) string and attach it to your wave driver.

Hook the string over the pulley, attach a \(100g\) mass (to make it taut), adjust your wave drive position if needed, and measure \(L\) and \(\delta L\) again, as before.

Choose a value of \(n\) to work with over this experiment - something you think you can measure easily, that was showing up consistently in the previous part. (\(n=3\) usually works.) Record this \(n\).

Now, we take off the 100g mass, and put a 50g mass there instead. Record this as \(M\), and adjust the frequency of the function generator to make a pattern with n antinodes (n loops) and maximum amplitude. Record that set frequency. Repeat, adding 50g each time up until a total \(M\) of 250g. (Neglect uncertainties in \(M\) in this part.)

Part I: Predicting Stretchy-String Wave Velocity

Calculate tension \(F_T\) in the string and total used length of string \(L'\). To find mass density \(\mu\) of the string in use (\(L'\)), use \( \mu= (m/L' \). Propagate the uncertainty in \( (m/L_{total}) \).

From tension and mass density, calculate wave velocity \(v\). Propagate uncertainties.

Part II: Measuring Stretchy-String Wave Velocity

Make a plot of \(f\) vs. \(n\). Since we neglected all uncertainties, you don't need error bars.

From the slope and \(L\), calculate wave velocity. Propagate uncertainty when you do so.

Answer the question about agreement with expectation.

Part III: Measuring \(\mu\) of the Fixed-Length String

Calculate used length of string \(L'\). Get mass density \(\mu\) using \( \mu= (m/L' \). Take the uncertainty in \(\mu\) to be the uncertainty in \( (m/L' \).

For each frequency period \(f\), calculate the corresponding frequency \(f^2\). Then, make a plot of \(f^2\) vs. \(M\). No uncertainties are needed here, since we neglected all of them.

From the slope and appropriate other quantities, calculate mass density (without using the mass \(m\) or the used length \(L'\). Propagate uncertainties.

Answer the question about agreement with expectation.

Your TA will ask you to discuss some of the following points (they will tell you which ones):

• Comparison to "Known" Quantities: The unstretched mass density of the white string is ~6g/m. The unstretched mass density of the green string is ~1.8g/m.⁵ Discuss whether your measurements make sense with these numbers.
• Derivation of Equation \eqref{fnsq}: Do the algebra to show that equation \eqref{fnsq} holds.
• Systemic Error: Distributed Weight: Suppose you had a large loop, and decided to account for some of the lost length by measuring the height of the loop and adding that to \(\delta L\). This would do a reasonable job reducing the error in \(L'\) for the fixed-length string, but there's still more issues for the stretchy string.
  • What would be the tension in the part of the string in the loop? (Each string, not the total.)
  • Suppose we were to model the string as a spring - i.e., the stretch is proportional to the tension. Why does your above conclusion indicate that, for the stretchy string, there is an error in \(L'\) from the loop that cannot be accounted for by just adding the appropriate extra string length?
• Wear and Tear in the Stretchy String: You may have noticed that some pieces of white string are a bit frayed and have lost their coating in some places.
  • In This Experiment: What assumption of our experiment does this violate? What are the implications of this?
  • Difference Between Coated and Uncoated: Suppose half our string were coated, and half our string were uncoated. If we had a travelling wave in both regions, what would be the main difference?
  • Boundaries Between Sections of String (I): Suppose the coating was really thick (most of the mass density of the string), and you sent the wave from the coated part to the uncoated part. This means that the end of the thick part is "almost" free (since it's really easy to push the super-light string). What should happen at this boundary? (Hint: what happens at a free end? Almost that.)
  • Boundaries Between Sections of String (II): Same question as above, but now consider a wave from the uncoated part towards the coated part. Note that this means you're entering a region that's really heavy, so your string will be almost fixed, instead of almost free.
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