Make-Up Lab: Waves

This is the make-up lab assignment for students who missed an in-person lab session this quarter. This lab can be done remotely (no in-person activity required) and it is to be completed individually. It should take you approximately 2-3 hours to finish, but it does not need to be completed in one sitting.

For this lab, we will use an interactive simulation from PhET in order to explore wave motion. You will experimentally determine what properties affect the speed of a traveling wave and then observe (and quantify features of) standing waves.


A wave is a disturbance (usually in some medium) which travels away from its source. Some familiar examples are sound, water, and string waves. There are transverse waves and longitudinal waves, and there are traveling waves and standing waves (among others). We'll define these quantities more in a minute, but first let's take a look at some neat standing wave phenomena courtesy of YouTuber Physics Girl to get us motivated.


This lab is intended to serve as a completely remote make-up assignment for students who missed a normal in-lab day of lab this quarter. You will work on this assignment alone and submit your report directly via email to your TA for grading.

Lab notebook template

Click on the link below to start your individual lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.)

Getting oriented

For this lab we will use the Wave on a String PhET simulation: When the simulation loads, you should see a screen that looks like Fig. 1.

Figure 1: The home screen of the “Wave on a String” PhET Simulation

Take a minute to get oriented. (No need to record these answers in your report.)

  • What parameters or options can you change? What are the physical meanings of these things?
  • What measurement tools do you have access to?
  • Is there anything you can do in this simulation that would be impossible to achieve in real life?

Notice how the simulation can be controlled.

  • In the lower right corner is a round orange circle with an arrow. This resets the entire simulation and brings you back to the starting state.
  • In the upper left corner a button marked “Restart”. This returns the string to its flat state without clearing any other settings.
  • You can move your measurement tools (and even start or stop your stopwatch) while the simulation is paused.
  • When the simulation is paused, you can advance one step at a time by clicking the forward button ().


In a one-dimensional body of finite size – such as the stretched string in our simulation – waves can travel in either direction along the body. A disturbance which travels along a string and hits the end (either clamped in place or free to move) is reflected and will travel back along the string in the opposite direction it came.

When different disturbances encounter each other – for example, when a pulse going forwards on a string collides with a pulse which has reflected and is traveling backwards – the result is just the sum of the individual amplitudes at that moment. This is called the superposition principle; any two arbitrary wiggles will add together to give a single wiggle equal to the sum.


While in manual mode, move the wrench to create wave pulses.

  • Can you make a positive pulse (where the bump goes up)? A negative pulse (where the pulse goes down)?
  • What happens to the wave when it reflects off a fixed end? Off a loose end?
  • Can you create two pulses that cross paths? What happens?

Wave speed

What determines the speed of propagation of a wave, though? Generically speaking, wave velocity is given by the form

$v = \sqrt{\dfrac{\textrm{"restoring force" factor}}{\textrm{"inertial" factor}}}$. (1)

For a transverse wave along a string, what are these two factors? The “restoring force” is the tension in the string that wants to keep the string taut and flat. If there's higher tension, there is more “snap” to the spring. The “inertial” term has to be related to the mass of the string. A heavier string will resist motion more than a light string.

Putting these ideas together (without derivation), Eq. (1) is more explicitly

$v_{\textrm{string}}=\sqrt{\dfrac{T}{\rho}}$, (2)

where $T$ is the tension in the string and $\rho$ (rho) is the mass per unit length of the stretched string.

What about the speed of light?

For a long time, physicists thought that because light is a wave, it must travel through a medium (just like sound or water waves). They termed this medium the “aether” and thought that it permeated the whole universe. Everything – you, me, the planets, and light – moved through the aether.

A clever experiment in 1887 by Albert A. Michelson and Edward Morley ( done at Case Western Reserve University in Cleveland in 1887) tried to look for the aether and measure its properties, but in the end found no evidence that it existed at all. This work – coupled with James Clerk Maxwell's earlier work in 1850 on electrodynamics – spurred Einstein on to develop his Theory of Special Relativity and to postulate that speed of light is a constant and that this speed is same when measured by all observers – no matter whether they are moving or not.

There is no equivalent formula to Eq. (1) for light, but since light is a traveling wave made up of an electric field and a magnetic field, it does make sense that the speed depends on properties of both: $c = 1/\sqrt{\epsilon_0\mu_0}$, where $\epsilon_0$ is the permittivity of free space (a property of electric fields in a vacuum) and $\mu_0$ is the permeability of free space (a property of magnetic fields in a vacuum).


Staying in manual mode…

  • What parameters in the simulation affect wave speed? How do you make the pulse move faster or slower?
  • Is it possible to make the wave move faster by wiggling the wrench faster? (Why or why not?)

Now switch to pulse mode where you can create more uniform pulses.

  • Do any of the new options (amplitude and pulse width) affect wave velocity?

Using either mode above, measure the wave velocity (with uncertainty) for two cases where you believe the velocity should be different. Describe your measurement technique(s) (with pictures!) and how you estimate the uncertainty.

A reminder about uncertainties

In some experiments this year, we have emphasized rigorous uncertainty analysis (e.g. statistical methods and propagation of uncertainties), while in others we instead just pushed more for a general understanding (e.g. identification of possible sources, or classification into statistical uncertainties versus systematic biases). Let us do a quick recap of different techniques so that we will draw your attention to what level of estimate we are looking for.

Since you're working remotely, you may have to get creative or make a judgement call. Estimating uncertainties from a photo or a video isn't always as straightforward as when you're in the lab with the apparatus in your hand, and the “tools” available in the simulations might have quirks of their own that are hard to understand. Remember that determining uncertainty is an estimation; don't get too worried if you think your method isn't rigorous enough.

Check out the expandable categories below if you need reminders.

Direct measurement uncertainty

Whenever you measure a quantity yourself, there is some inherent uncertainty. Think of all the times that you have measured a distance, time or mass. You may find that each time you measure that quantity, the value fluctuates a bit, maybe because…

  • …your measurement tool shows some variation from measurement to measurement (e.g. the flickering of the last digit when using a digital scale).
  • …your measurement technique changes subtly (e.g. by choosing slightly different end points when measuring a distance with a ruler).
  • …there is some inherent scatter (e.g. variations in reaction time when using a stopwatch).

To estimate a direct measurement uncertainty, you may need to repeat a measurement a few times to see how much the values vary (1 cm, 10 cm, 0.1 cm?), or repeat the measurement many times (say 20 or more) and find the standard deviation of the mean. In rare cases where you find that there is no fluctuation when repeating a measurement, you can use the measurement device's resolution as your uncertainty. (E.g. if a ruler has only 1 mm tick marks, you can estimate the uncertainty as either ±1 mm or ±0.5 mm (depending on how confident you are about interpolating between tick marks).

Propagation of uncertainties

Whenever you calculate a quantity from measured values, the uncertainties in that measured values will lead to a final overall uncertainty in the calculated value. For example, if you calculate a velocity from $v = x/t$, then your uncertainty in $v$ must depend on the uncertainties in both $x$ and $t$. The calculation of the uncertainty in the final value is called propagation of uncertainties.

There is a “rigorous” way to propagate uncertainties and there is a “rough” way. Use whichever method you are comfortable with unless otherwise specified in the directions.

“Rigorous” propagation of uncertainties The rigorous way to determine the uncertainty in a calculated quantity $f(a,b,\dots)$ where each variable has its own estimated measurement uncertainty ($a\pm\delta a$, $b\pm \delta b$, …) is to calculate contributions using partial derivatives and to sum up those contributions in quadrature. Mathematically, the uncertainty in the quantity $f$ is given by

$\delta f = \sqrt{\bigg(\dfrac{\partial f}{\partial a}\delta a\bigg)^2 + \bigg(\dfrac{\partial f}{\partial b}\delta b\bigg)^2 +\cdots}$.

This general formula simplifies in some of the most common cases:

  • Multiplication and division: If $f = AB$ or $f = 1/AB$, then $\delta f/f = \sqrt{(\delta A/A)^2+(\delta B/B)^2}$.
  • Addition and subtraction: If $f = A + B$ or $f = A- B$, then $\delta f = \sqrt{(\delta A)^2 + (\delta B)^2}$.
  • Exponent: If $f = A^n$, then $\delta f/f = n(\delta A/A)$.

“Rough” propagation of uncertainties If computing the rigorous propagation will take too long or is otherwise too burdensome, you can estimate uncertainty by looking how much spread there is in the computed value at the extreme values. This is sometimes called the “(max - min)/2” method.

For a function $f(a,b,\dots)$, let's assume the dominant uncertainty is in the variable $a\pm \delta a$ and the uncertainties in all other variables can be ignored. The “(max-min)/2” method estimates the uncertainty in $f$ as half the difference between the largest possible value (taken when $a\rightarrow a+\delta a$) and the smallest value (taken when $a\rightarrow a+ \delta a$). Mathematically, this gives

$\delta f = |f(a+\delta a, b, \dots) - f(a−\delta a,b,\dots)|/2$

For example, if you are measuring many periods $T$ of a fixed-length pendulum in order to estimate the acceleration due to gravity, $g = 4\pi^2L/T$, then the uncertainty becomes $\delta g = 2\pi^2L(1/T^2_{min} − 1/T^2_{max})$ where $T_{max} = T + \delta T$ and $T_{min} = T - \delta T$.

Comparison of values

When we ask you to compare two numbers – for example, “How does your measurement compare to the literature value?” – you should be as quantitative as you can be. How large is the difference between the two values? How large is you uncertainty (on one or both of the items you are comparing)? Do the two values overlap within error bars?

You may remember the $t^{\prime}$ test we learned in PHYS 131: $t^{\prime} = \drac{A-B}{\sqrt{\delta A^2 +\delta B^2}$.

  • If $|t^{\prime}| \leq 1$, $A$ and $B$ agree.
  • If $|t^{\prime}| \geq 3$, $A$ and $B$ disagree.
  • If $1 < |t^{\prime}| < 3$, it is inconclusive.

Traveling waves and standing waves

Theory and definitions

It's possible to go over a lot of mathematics about standing waves before we try to observe then, but we suggest you try the simulation below first. After you've played a bit, feel free to come back and look at this material if you need help grounding what you see in mathematics or need to check different definitions.

Traveling and standing waves

A wave is a disturbance (usually in some medium) which travels away from its source. Some familiar examples are sound, water, and string waves. A transverse wave is a wave in which the disturbance causes a momentary displacement in the medium in a direction normal to the direction in which the wave propagates (for example, a water wave). A longitudinal wave is a compression wave in which the disturbance causes a momentary displacement of the medium along the same direction in which the wave propagates (for example, a sound wave).

A traveling wave takes the mathematical form

$y_1 = A\sin (kx - \omega t)$, (1)

where $A$ is the amplitude, $\omega = 2\pi f$ is the angular frequency (with units of radians/sec), $f$ is the normal frequency (with units of hertz), $k = 2\pi/\lambda$ is the wavenumber (in units of radians/meter) and $\lambda$ is the wavelength (in units of meters).

Such a wave is called a traveling wave because any point on the wave will move along at fixed amplitude with speed given by

$v=\lambda f = \omega /k$. (2)

(Think, for example, of a surfer riding the crest of a wave as it comes into shore.)

Imagine now that we have a wave which is identical but is traveling in the opposite direction. Such a wave has mathematical form 

$y_2 = A\sin (kx + \omega t)$. (3)

By the superposition principle, if these two waves travel along the same body, the resulting wave is the sum of the two waves,

$y = y_1 + y_2 = y = 2A\sin(kx)\cos(\omega t)$, (4)

where we made use of the trigonometric identity

$\sin A + \sin B = 2 \sin \left(\dfrac{1}{2}(A+B)\right)\cos \left(\dfrac{1}{2}(A-B)\right)$.

This sort of wave – Eq. (4) – is called a standing wave. Notice that at any fixed position $x$, the string undergoes simple harmonic motion as time goes on. Notice also that all points along the wave oscillate at the same frequency. The amplitude, however, depends on the position $x$. This characteristic is quite different from a traveling wave in which the amplitudes of all points along the wave are equal.

Nodes and anti-nodes

By Eq. (4), the amplitude of a standing wave is a maximum when

$kx = \dfrac{\pi}{2},\dfrac{3\pi}{2},\dfrac{5\pi}{2},\dots$

Since $k = 2\pi/\lambda$, this corresponds to

$x = \dfrac{\lambda}{4},\dfrac{3\lambda}{4},\dfrac{5\lambda}{4},\dots$ (5)

These positions of maximum amplitude are called anti-nodes.

Similarly, there are positions along the wave where the amplitude equals zero, namely where

$kx = 0,\pi ,3\pi ,\dots$


$x = 0,\dfrac{\lambda}{2},\lambda ,\dfrac{3\lambda}{2},\dots$ (6)
Resonant frequencies

A system 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. Let's look at several specific systems.

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.) Equation (6) states that the distance between adjacent nodes is $\lambda/2$. Therefore, 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)}.$

Using the relation $f = v/\lambda$, we may re-write this as

$f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$ (7)
Figure 1: Standing waves on a string with both ends fixed. (Click here to see an animated version.)

For a string with one end fixed and the other end free, the fixed end must be a node and resonances will occur when the free end is an anti-node. (See Fig. 2.) Here, the string (or air column) length must contain an odd integer multiplier of quarter-wavelengths,

$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$

such that the resonant frequencies will be

$f_n = \dfrac{v}{\lambda_n} = \dfrac{nv}{4L}  \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. (8)
Figure 2: Standing waves on a string with one free and one fixed end. (Click here to see an animated version.)

If both ends of the string are free, resonance will occur when there are anti-nodes at both ends. (See Fig. 3.) At resonance, the string length will contain integer multiples of half wavelengths,

$\dfrac{n\lambda_n}{2} = L \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$

with resonant frequencies

$f_n = \dfrac{v}{\lambda_n}=\dfrac{nv}{2L} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. (9)

This is the same result as Eq. (7) for a string with both ends fixed.

Figure 3: Standing waves on a string with both ends free. (Click here to see an animated version.)


Change the simulation to oscillate mode. Think back to the standing waves created on the slinky in the motivational video… can we find standing waves in our simulation like that?

  • What should you look for to identify when you are at a standing wave resonance?
  • What happens when you are close to a standing wave resonance, but slightly off?
  • When you think you've identified a standing wave frequency, how can you estimate the uncertainty?
  • Which parameters (e.g. amplitude, damping, tension, end conditions) set the resonance frequency?

Some tips

The theory above talks about ideal waves, but our simulation can produce some quite realistic waves.

  • Our simulation allows us to add damping. If the damping is large, then the wave will die out before it reaches the end of the string and reflects. If there's no reflection, there's no standing wave, so we want to work with zero or low damping.
  • Our simulation allows us to drive the wave with arbitrarily large or small amplitudes. If the driving amplitude is much smaller than the amplitude of the waves on the string, then we can say that the end is approximately “fixed” and we can use the equations derived above. On the other hand, if the driving amplitude is comparable to the wave amplitude, then the driving end is neither fixed nor open… it's something else. So, we want to keep the driving amplitude small (but not zero).
  • When we get close to resonance, the reflected waves add to the incoming waves and amplitude of the anti-nodes grows bigger. If we are at resonance (and there is no damping), then this amplitude will keep growing as long as we keep driving it. But if we are just a little bit off from the correct frequency, we can see the amplitude start to grow, then start to shrink, then start to grow, etc. This is a phenomenon known as beats (which we covered in the Interference lab), but it is one sign that you aren't quite at resonance.

Can you find a standing wave in the simulation? Describe your criteria for determining whether you have found a standing wave resonance and think about how you will estimate uncertainty.

If you are able to find a resonance, please record the following:

  • What are the general parameter settings (e.g. driving amplitude, damping, tension, and end conditions).
  • What frequency do you find resonance at? Is it the fundamental frequency ($n = 1$) or a higher frequency mode?
  • How could you estimate the uncertainty on this frequency?
I still can't find a resonance!

If you are unable to get resonance on your own, here's a method that pretty reliably works. Before we get to the details, let's highlight the two key features:

  • The driving amplitude needs to be small so that the left boundary condition is nearly “fixed”.
  • The damping should be zero (or at least very small) so that waves have enough energy to reflect off the far end and produce standing waves.

Here is the procedure:

  • When you start the simulation, switch into oscillate mode and turn damping off and amplitude low (say 0.10 cm).
    • For the example that follows, we use fixed end and high tension, but the same procedure works for other parameter settings.
  • Hit reset to clear any bumps from before adjustments were made (since there's no damping to get rid of them).
  • Now, if you watch the string (with the default frequency of 1.50 Hz), you see the amplitude never grows above 0.10 cm. (You can put a reference line on screen at the top of the driving amplitude if it helps illustrate this.) And if you watch for a while, you even sometimes see the wave completely flatten out.
  • So, this means you are not on resonance. You can try going up or down in frequency a little bit, and repeating. (Hit reset to clear between changes.)
  • Now, let's say you're at 1.20 Hz. This looks promising! The second and fourth green dots look almost stationary. The amplitude is growing way above 0.10 cm. But as you watch, it seems to grow, then fade, grow, then fade. You are close, but not there.
  • Changing to 1.25 Hz and resetting, we now see growth that seems to increase without bound. Maybe this is resonance! (You'd want to check to see if this is as close as you can get… try 1.24 Hz, try 1.26 Hz, etc. Use this as a way to estimate uncertainty.)
  • Once you have found one resonance, you can check for resonances at other $n$ values. (If you found $n$ = 2, then doubling frequency will give $n$ = 4, or halving frequency will give $n$ = 1.)

Making careful measurements

Let's now make some careful, quantitative measurements.

Pick two “conditions” to study. A condition is your choice of high, medium, or low tension plus choice of fixed end or loose end. (So, for example, “high tension + fixed end” or “medium tension + loose end”.)

For each condition…

  • Measure the fundamental ($n$ = 1) resonance frequency and at least two higher frequencies (and the corresponding $n$ value).
  • Use your measured resonance frequencies to estimate the wave speed. How does this compare to the directly measured value (either from the exercise above or from a new measurement)?

Remember to describe your technique for identifying when you've reached resonance and to include uncertainty values and how you estimated those values.

Submitting your notebook

For this make-up assignment, you do not need to write a summary or conclusion report. Instead, just make sure that you fill out all the sections of the notebook in enough detail and answer all the questions asked.

When you are finished, save the notebook as a PDF and email it to your TA. (Do not submit the notebook on Canvas.)

Your TA will grade the notebook out of 8 points and use this score in place of the scores you normally would have received for your missed lab.

2023/04/05 10:56 · mccowan