Waves appear all around us and show up in many different systems. In this experiment, we provide the following three examples for you to study:

- transverse waves along a string,
- longitudinal waves along an air column, and
- longitudinal waves along a spring.

For each system, you will *nominally* be tasked with measuring wave speed, but in the process you will explore wave properties more generally and will gain familiarity with both travelling and standing wave conditions. You will have freedom to design your own experiments and make your own choices about what quantities to measure (and how).

There are multiple setups of each system in the room, and you may complete them in any order you wish.

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 experiments 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.

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.

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

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

or

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

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.) |

The speed of propagation of a wave is given by a formula of the form

$v = \sqrt{\dfrac{\textrm{“restoring force” factor}}{\textrm{“inertial” factor}}}$. | (10) |

For a transverse wave along a string, Eq. (10) is more explicitly

$v_{string} = \sqrt{\dfrac{T\vphantom{\large{T}}}{\rho}}$, | (11) |

where $T$ is the tension in the string and $\rho$ (*rho*) is the mass per unit length of the stretched string.
For a stretched spring, Eq. (10) becomes

$v_{spring} = \sqrt{\dfrac{k_sL}{\rho}}$, | (12) |

where $k_s$ is the spring constant, *L* is the length of the stretched spring and $\rho$ is the mass per unit length of the stretched spring.
The speed of propagation of a sound wave is

$v_{gas} = \sqrt{\dfrac{\gamma k_BT}{m}}$, | (13) |

where $\gamma$ is the ratio of specific heats of the gas in which the wave is moving ($\gamma = 1.4$ for a diatomic gas), $k_B = 1.38 \times 10^{-23}$ J/K is the Boltzmann constant, *T* is the absolute temperature of the gas (in Kelvin), and *m* is the mass of an individual molecule of gas. (The average mass of an air molecule is about $m = 4.8 \times 10^{-26}$ kg.)

We may use Eqs. (11)-(13) to express the predicted frequencies in terms of variables we can control and measure in the laboratory. Substituting Eq. (11) into Eq. (7) (which is the same as Eq. (9)), gives the resonant frequencies for a **string, fixed at both ends or free at both ends** as

$f_n = \dfrac{n}{2L}\sqrt{\dfrac{T\vphantom{\large{T}}}{\rho}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$, | (14) |

and for a **spring, fixed at both ends or free at both ends** as

$f_n = \dfrac{n}{2L}\sqrt{\dfrac{k_sL}{\rho}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (15) |

Similarly, substituting Eq. (13) into Eq. (8) gives the resonant frequencies for an **air column with one end closed and the other end open** as

$f_n = \dfrac{n}{4L}\sqrt{\dfrac{\gamma k_BT}{m}} \mathrm{\;(for\;}n\mathrm{\;= 1,3,5,\dots)}$. | (16) |

Substituting Eq. (13) into Eq. (9) gives the resonant frequencies for an **air column with both ends open** as

$f_n = \dfrac{n}{2L}\sqrt{\dfrac{\gamma k_BT}{m}} \mathrm{\;(for\;}n\mathrm{\;= 1,2,3,\dots)}$. | (17) |

**NOTE**: There are three distinct experiment stations – string, spring, and air column. These stations may be completed in any order.

For each system, your task is to **experimentally determine the wave speed (with uncertainties)** and compare this value to a predicted value. You will be told whether to use a traveling wave or a standing wave technique, but the details of the experiment are up to you!

In each case, however, you must use a method that **incorporates multiple data points**. That is to say, don't just measure *one* quantity and use *one* equation to solve for *one* wave speed… you must collect data that can be plotted (and fit to a function) or which can be averaged or combined in some other way.

One member of the group should click on the link below to start your group lab notebook. (You may be asked to log into your UChicago Google account if you are not already logged in.) Make sure to share the document with everyone in the group (click the “Share” button in the top right corner of the screen) so each member has access to the notebook after you leave lab.

We provide a Google Colab notebook that you can use to plot and fit your data.

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.

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.

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).

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 $f(a,b...)$ where each variable has its own estimated measurement uncertainty ($a\pm \delta a$, $b\pm \delta b$, …) Mathematically, the uncertainty in the quantity $f$ is given by

$\delta f = \sqrt{\bigg(\frac{\partial f}{\partial a}\delta a\bigg)^2 +\bigg(\frac{\partial f}{\partial b}\delta b\bigg)^2 + \ldots}$. |

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

**Multiplication and division**: If $f=AB$ or $f=A/B$, then $\frac{\delta f}{f} = \sqrt{\bigg(\frac{\delta A}{A}\bigg)^2 +\bigg(\frac{\delta B}{B}\bigg)^2}$.**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 $\frac{\delta f}{f} = n\frac{\delta A}{A}$.

More details about this rigorous method (and examples of how to use it) are given here.

**“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 $f(a,b,...)$, 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

$\delta f = |f(a+\delta a,b,\ldots) - f(a - \delta a, b, \ldots)|/2$. |

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 $\delta g = 2\pi^2L(1/T_{min}^2-1/T_{max}^2)$ where $T_{max} = T + \delta T$ and $T_{min} = T - \delta T$.

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' test we learned in PHYS 131: $t^{\prime} = \dfrac{A-B}{\sqrt{\delta A^2 + \delta B^2}}$.

- 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.

In this part of the experiment, a stretched string is vibrated mechanically by a small speaker driven by a function generator. (See Fig 4). The speaker cone has a hook cemented to it and it transforms the sinusoidal electrical energy to sinusoidal mechanical motion at the same frequency. Since the amplitude of the speaker motion is very small, the end of the spring attached to the speaker is considered fixed in comparison with the much larger motions of the string observed at resonance.

The string used for this apparatus has linear mass density, $\rho = (1.37 \pm 0.05) \times 10^{-3} \mathrm{\;kg/m}$.

Use a standing wave technique to determine wave speed. Consider the following questions. (You may need to play around a little bit with the apparatus before you can answer some of these… that's OK!)

- Is your system one where you have a node at both ends? An anti-node at both ends? A node at one end and an anti-node at the other?
- What will you look for to determine when you are “on resonance”? (That is to say, when you have satisfied the standing wave condition.)
- What variables determine wave speed? Which variables are fixed? Which variables can you adjust?
- What variables determine the resonant frequency of the standing wave? Which variables are fixed? Which variables can you adjust?

Once you are oriented, decide on an experimental technique to measure wave speed. Remember that your technique must make multiple measurements that can be put together in a table or plot and be used together to get a best estimate (with uncertainty).

In addition, calculate your predicted value for wave speed. Compare your measured value to this prediction.

For a wave on a string, the speed depends on tension. Choose **one** value of tension, and measure the wave speed only for that one value of tension.

**CAREFUL**: The speaker input connector which plugs into the function generator (and the resistor attached to it) can become hot if the function generator has been running for a long time. Be careful not to burn yourself.

In this part of the experiment we observe standing waves along a stretched spring as in Fig. 5. One end of the spring is attached to the speaker. Since the amplitude of the speaker motion is very small, the end of the spring attached to the speaker is considered fixed in comparison with the much larger motions of the spring observed at resonance. Likewise, the end of the spring connected to the string can also be considered fixed. *Therefore, we have a node at each end.*

Use a standing wave technique to determine wave speed. Consider the following questions. (You may need to play around a little bit with the apparatus before you can answer some of these… that's OK!)

- Is your system one where you have a node at both ends? An anti-node at both ends? A node at one end and an anti-node at the other?
- What will you look for to determine when you are “on resonance”? (That is to say, when you have satisfied the standing wave condition.)
- What variables determine wave speed? Which variables are fixed? Which variables can you adjust?
- What variables determine the resonant frequency of the standing wave? Which variables are fixed? Which variables can you adjust?

Once you are oriented, decide on an experimental technique to measure wave speed. Remember that your technique must make multiple measurements that can be put together in a table or plot and be used together to get a best estimate (with uncertainty).

In addition, calculate your predicted value for wave speed. Compare your measured value to this prediction.

For a wave in a spring, the speed depends on spring length. Choose **one** value of length (set by how much mass is in the pan), and measure the wave speed only for that one value of length.

**HINTS:**

- You will need to know the linear mass density of the spring, $\rho = m/L$. You can measure the mass using the balances in the lab.
- You will need to know the spring constant of the spring, $k_s$. To determine this, recall that the spring constant is defined by the equation $F = k_sx$, where a spring's length changes by $x$ when a force $F$ is applied. Can you do a quick side experiment to determine $k_s$ using the tools at your disposal?

The equation for the resonant frequency depends explicitly on the length of the spring as $f_n = (n/2L)v_{spring}$. However, the wave speed itself also has factors of length hidden inside it: $v_{spring} = \sqrt{(k_sL/\rho)} = \sqrt{(k_sL^2/m)}$, where we have used $\rho = m/L$. Putting these two together, we find that the resonant frequency is actually *independent of length*: $f_n = (n/2L)\sqrt{(k_sL^2/m)} = (n/2)\sqrt{k_s/m}$. Maybe this is important to you, maybe not… it depends on what measurement method you choose.

Fig. 6 shows the apparatus for observing standing sound waves in an air column. A speaker is used to excite the resonances, and a microphone and oscilloscope are used to detect the amplitude near one end of the tube.

For this setup, you will determine the speed of sound in air using both a standing wave method and a traveling wave (e.g. pulse) method.

We will determine the sound wavelength by sliding the plunger along the plastic tube and looking for resonances as we produce standing waves. Set the frequency generator to produce a sine wave of approximately 2-3 kHz. Move the plunger slowly while observing the signal on the microphone connected to the oscilloscope.

Use a standing wave technique to determine wave speed. Consider the following questions. (You may need to play around a little bit with the apparatus before you can answer some of these… that's OK!)

- Is your system one where you have a node at both ends? An anti-node at both ends? A node at one end and an anti-node at the other?
- What will you look for to determine when you are “on resonance”? (That is to say, when you have satisfied the standing wave condition.)
- What variables determine wave speed? Which variables are fixed? Which variables can you adjust?
- What variables determine the resonant frequency of the standing wave? Which variables are fixed? Which variables can you adjust?

Once you are oriented, decide on an experimental technique to measure wave speed. Remember that your technique must make multiple measurements that can be put together in a table or plot and be used together to get a best estimate (with uncertainty).

Pull the amplitude knob outwards to cut the amplitude by a factor of 10 for this experiment, and please turn off the function generator when you aren't making measurements. Otherwise your TA and fellow students may be very, very annoyed with you.

In addition, calculate your predicted value for wave speed. Compare your measured value to this prediction.

You will use the same apparatus as shown in Fig. 6. This time, however, choose the square wave output of the function generator. In this mode, the speaker provides a fast “pulse” of sound pressure to one end of the air column. Part of the pulse is detected immediately by the nearby microphone, while part of the pulse travels down the air column and is reflected back to the microphone at a slightly later time. By measuring the distance the pulse travels and the time between the immediate and reflected pulses, one may determine the speed of sound in air.

To make this measurement, we need to set our apparatus up carefully as follows:

- Set the function generator for a
**square**wave frequency of between**1 and 5 Hz**. (Rather than read a value off the function generator, listen for the pulses from the speaker. A frequency of 1 Hz means one pulse per second. A frequency of 5 Hz means five pulses per second.) - Trigger the scope
**externally**from the TTL output of the function generator. - Position the plunger about
**half-way down the tube**. **Zoom in**the time scale near the trigger point. You should see a messy initial pulse followed by a lot of oscillations. (These oscillations are caused by the loudspeaker continuing to vibrate after the square pulse hits it).**Cup your hand**around the speaker, microphone, and tube area to block out some of the room noise. You should see the amplitude of your signal increase a bit.- (
*Optional*) To dampen the ringing of the speaker, you can also try*lightly*place your**finger**on the center of speaker. You should see the oscillations following the initial pulse reduce in amplitude. (Don't push too hard! Just enough to stop the extra vibrations.) - Do you now see a
**second pulse**several milliseconds after the initial pulse? As you slide the plunger up and down the length of the tube, the time position of this pulse should change. You are now seeing the reflected sound pulse off the end off the tube!

Once you understand what you are looking at, use the scope to measure the time between the immediate and reflected pulses. For several positions of the plunger, you should be able to plot time versus distance traveled, and use this information to determine the speed of sound in air (with uncertainties).

Calculate the expected speed, and compare it to your measured quantity.

Make sure to submit your lab notebook by the end of the period. Download a copy of your notebook in PDF format and upload it to the appropriate spot on Canvas. **Only one member of the group needs to submit to Canvas, but make sure everyone's name is on the document!**

When you're finished, don't forget to **log out** of both Google and Canvas, and to close all browser windows before leaving!

Write your conclusion in a **new** document (not your lab notebook) and submit that as a PDF to the appropriate assignment on Canvas when you are done. You should write your conclusion *by yourself* (not in a group), though you are welcome to talk to your group mates to ask questions or to discuss.

The **conclusion** is your *interpretation* and *discussion* of your data. In general, you should ask yourself the following questions as you write (though not every question will be appropriate to every experiment):

- What do your data tell you?
- How do your data match the model (or models) you were comparing against, or to your expectations in general? (Sometimes this means using the $t^{\prime}$ test, but other times it means making qualitative comparisons.)
- Were you able to estimate uncertainties well, or do you see room to make changes or improvements in the technique?
- Do your results lead to new questions?
- Can you think of other ways to extend or improve the experiment?

In about *one or two paragraphs*, draw conclusions from the data you collected today. Address both the qualitative and quantitative aspects of the experiment and feel free to use plots, tables or anything else from your notebook to support your words. Don't include throw-away statements like “Looks good” or “Agrees pretty well”; instead, try to be precise.

**REMINDER**: Your post-lab assignment is due 24 hours after your lab concludes. Submit a single PDF on Canvas.