# PHY 123 Lab 8 - Standing Waves

(updated 10/29/13)

The purpose of this lab is to study (transverse) standing waves on a vibrating string.

If you need the .pdf version of these instructions you can get them here.

## Video (Prof. Matt Dawber)

CHANGES FROM WHAT'S IN THE VIDEO: You are given the (unstretched) mass densities of both strings (see Equipment below) so you do not need to measure the masses of the strings. However, you still must determine the linear mass density of the stretched yellow string. The first part of the video was shot with a hanging 50 gram mass. As specified in this Lab 8 manual, for Parts I and II you must use, instead, an M = 150 gram mass; it gives better results. The last part of the video mentions using the n = 3 standing wave, but, for Part III you must use the n = 1 standing wave as this Lab 8 manual specifies.

## Equipment

• electric motor with flag that blocks the photogate beam once per turn
• photogate connected to the interface box that is connected to the computer with suitable software
• variable power supply with current-limit knob and both “coarse” and “fine” voltage knobs to control the motor speed
• (thick, yellow) rubber band (your “stretchy” string): unstretched linear mass density $\:\:\large \mu_y = 2.70 \times 10^{-3}$ kg/m (You must determine its stretched linear mass density $\:\:\large _s\mu_y$.)
• (thin, white) “non-stretchy” string: linear mass density $\:\:\large \mu_w = 0.18 \times 10^{-3}$ kg/m
• pulley
• meter stick
• various weights (marked as masses in grams; take their uncertainties to be zero)

## Introduction

In this lab you will study transverse standing waves on two different kinds of vibrating “string”, one that is “stretchy” (a yellow cord with a rubber-band core) and another that is non-stretchy (a piece of thin white string). “Transverse” means that the actual vibrations of the string are perpendicular to the axis defined by the non-vibrating string. “Standing” means that the locations of the antinodes (“maxima” in the amplitude of the transverse oscillations) and nodes (“minima” – ideally zeroes – in the amplitude of the transverse oscillations) do not move. These transverse standing waves occur only under certain conditions, which you can control experimentally.

## Review the physics for this Lab

You should review relevant material from your course notes and from a textbook, e.g., Knight, Jones and Field (2nd ed.),(abbreviated KJF2), to get ready for this lab. In the course notes here, open ch10.pdf and review the treatment of standing waves on a string in slides 18 through 23'. If you don't have a copy of the textbook KJF2, you will find one bolted to the table in the Help Room for this course in room A-131 of the physics building. In KJF2 you should review the material in Chap. 15.2, especially Eq. (15.2) on p. 481. Then read the material in Chap. 16.1 through 16.3. Pay particular attention to Figs. 16.5, 16.6, 16.7, 16.8, 16.12, 16.13, and 16.14 and the worked Example 16.2 on p. 514, which follows from Eq. (15.2).

## Playing harmonics on standing-wave-stringed musical instruments

Every time you play (or watch someone else playing) a stringed musical instrument, you are seeing someone control the frequency of the standing waves being excited by the fingers, a pick, or a bow. Particularly skilled players can even control the number of nodes in the vibrations of the strings and create striking musical effects. (Rock and heavy metal guitarists invented the “pinch harmonic” technique, which became popular. Learn how to do it at this active link.) Maybe you have seen a guitar player press a finger very lightly at a particular position along a plucked string. Without the lightly pressed finger, the string is free to vibrate over its full length in its “lowest order mode”, which has the lowest frequency (musical “note”) for the length, tension, and mass density of that particular string. When the lightly-pressed finger forces a “node” to occur away from the ends of that string, it can no longer vibrate in the lowest order mode over its full length. Instead, it vibrates predominantly at a higher harmonic with a delicate sound. If you have (access to) a guitar, try it after you've done this lab. Then explain how it works to a family member or friend!

### Two videos worth watching (optional)

Here is an active link to a YouTube video, with sound, showing lightly-pressed-and-released string plucks being used as an aid for “harmonic-tuning” a guitar. At 0:55 Mike Lace does this on the low E string at the 5th fret, which is 1/4 of the distance along the string from the “nut” (its tuning-peg end at the right) to the bridge (its “body” end at the left). (Stop the video at 0:26 and put a ruler up to the computer screen. Measure the distance yourself (in mm on the screen) and do a simple calculation to see that the nut-to-5th fret distance is 1/4 of $\:\:L$, the “nut” to “bridge” distance.) Mike's technique emphasizes the $\:\:n=4$ harmonic vibration of the string, which now plays a note two octaves above ($2^2 = 4$ times) the $\:\:n=1$ frequency $\:\:f_1$ of the “open” string. For more details, see this active link.

Here is an active link another YouTube video, without sound, showing a high-speed movie of a lightly-pressed-and-released 2nd-harmonic guitar-string pluck. Notice that the player's left index finger touching the string at a particular position along it (half the distance $\:\:L$ from the “nut” to the “bridge”) forces a node (zero amplitude) of oscillation to occur there, and after (carefully!) lifting the finger off the vibrating string, the node “stays there”. You can actually see the string continuing to vibrate freely in its 2nd-harmonic mode.

## What you will do in the lab

In this Lab you will first predict what the traveling wave velocity should be on the stretchy string from the properties of the rubber band. Then you will excite standing waves with experimentally-controllable numbers of antinodes and nodes on this rubber band, determine the traveling wave velocity from them, and compare what you determine with what you predict. Finally, you will investigate the dependence of the wave velocity on the tension in the string and use your results to determine the gravitational acceleration $\:\:g$.

## Part I: Determination of the Traveling Wave Velocity from the Tension and Linear Mass Density

In this part you will calculate the traveling wave velocity $\:\:v$ from the tension $\:\:T$ in the stretched rubber band and its stretched linear mass density $\:\:_s\mu_y$. When a string is “clamped” (i.e., is forced to have nodes) at both ends, at certain vibration frequencies traveling waves going “left” and “right” (which add amplitudes by “superposition” and, therefore, can interfere “constructively” and “destructively” (see KJF2, Chap. 16.1) to create standing wave patterns with distinctive nodes and antinodes. An example with 11 nodes (not counting the nodes at the ends), which corresponds to $\:\:n=12$ “half-waves” in the standing wave pattern, is shown in Fig. 3.

The equation for the velocity $\:\: \large v$ of a traveling wave on a string is [KJF2, Eq. (15.2)]

$\Large{ v=\sqrt{\frac{T}{\mu}}}\:\:.$
(8.1)

The linear mass density $\:\:\mu$ for string is the ratio of its mass $\:\:m$ and length $\:\:L\:^{\prime}$,

$\Large{ \mu=\frac{m}{L'}}\:\:.$
(8.2)

The tension $\:\:T$ in the string is given by the gravitational force

$\Large{ T=Mg}$
(8.3)

of the weight that is “pulling” on it in Fig. 3. If the string is “stretchy”, then its stretched length $\:\:L\:^{\prime}$ depends on $\:\:T$. Since the stretching of the string does not change its total mass $\:\:m$, it is obvious that the linear mass density (mass divided by length) $\:\:_s\mu_y$ (pre-subscript $\:\:s$ for stretched and post-subscript $\:\:y$ for yellow) of the stretched yellow string is not equal to the linear mass density $\:\:\mu_y$ of the unstretched yellow string: $\:\:_s\mu_y \neq \mu_y$.

Determine the mass $\:\:m$ and the length $\:\:L\:^{\prime}$ of the rubber band while it is under tension with a suspended weight of $\:\:M = 150$ grams. Assign an absolute uncertainty of 0.1 gram to $\:\:m$ and an absolute uncertainty of 1 cm to $\:\:L\:^{\prime}$, which accounts for loops or knots in the string. Enter your values into the table on your worksheet.

Calculate the values of $\:\:T$ and $\:\:\mu$ and enter these on your worksheet. The largest relative uncertainty is in the length of the string; therefore, from Eq. (E.7) of the Lab 1 manual, Uncertainty, Error and Graphs, you can approximate the relative uncertainty in $\:\:_s\mu _y$ to be the same as the relative uncertainty in $\:\:L\:^{\prime}$.

$\Large{ \frac{(\Delta _s\mu _y)}{(_s\mu _y)}=\frac{\Delta L'}{L'}}\:\:.$
(8.4)

Use this to calculate the absolute uncertainty in $\:\: _s\mu _y$ and write that on your worksheet.

Calculate the value of the traveling wave velocity $\:\:v$ on the rubber band and enter it on your worksheet. In calculating the uncertainty in the velocity you can neglect the uncertainty in the tension because its relative uncertainty is much less than the relative uncertainty in $\:\:_s\mu _y$. Using Eq. (E.8) of the Lab 1 manual, you can say that

$\Large{ \frac{\Delta v}{v}=\frac{1}{2}\;\frac{(\Delta _s\mu _y)}{(_s\mu _y)}}\:\:.$
(8.5)

Use this to find the relative uncertainty, and then the absolute uncertainty, in $\:\:v$, which you should record on your worksheet.

## Part II: Measurement of the Traveling Wave Velocity Using the Frequencies of the Observed Standing Waves on the Stretched Rubber Band

In this part you stretch the yellow rubber by hanging the $\:\:M=150$ gram mass on it and excite standing waves at resonance frequencies $\:\:f_{n}$. The integer $\:\:n$ is the number of half-waves fitting into the string; see Fig. 3. Then from a graph of $\:\:f_{n}$ versus $\:\:n$ you'll determine the traveling wave velocity and compare it to the value $\:\:v_{\small PI}$ you obtained in Part I.

The equation for the frequencies $\:\:f_{n}$ of the standing waves on a string is (see, e.g., the worked Example 16.2 on p. 514 of KJF2)

$\Large f_{n}=n\;\frac{v}{2L}\:\:.$
(8.6)

You can see that this equation has the form $\:\:f_{n}$ = […]$n$. The term […] is the slope $\:\:k$ of the corresponding linear graph you should obtain if you plot $\:\:f_{n}$ versus $\:\:n$.

The yellow stretchy string should be strung over the pulley with the 150 gram mass $\:\:M$ hanging on it. A small electric motor on the other end of the yellow string is driven by a variable power supply, and the “connecting rod” attached to it provides the oscillatory force that drives a traveling wave initially moving “left” on the string in Fig. 3. However, when that traveling wave “hits” the pulley, it is reflected back along string, now traveling to the “right”. There it is reflected again, etc. When the second reflected wave (traveling to the left) is in phase with the original wave traveling to the left, conditions are right for a standing wave pattern to be produced on the string as sketched in Fig. 3. The distance between the nodes of the standing wave is one-half of the wavelength of the wave. “Nodes” are points of destructive interference where the string (ideally) remains still and anti-nodes are points of constructive interference where the string oscillates with maximal amplitude.

Sketch on the grid on your worksheet a standing wave pattern for a wave with several half-waves fitting into the length $\:\:L$. Label your sketch with the length $\:\:L$ of the rubber band, its nodes, and the number, $\:\:n$, of half-waves.

Measure the length $\:\:L$ (see Fig. 3) with your meter stick, estimate its absolute uncertainty $\:\:\Delta L$, and enter both on your worksheet. Note that this is not the same as L', which you measured earlier!

• Place the motor in the photogate so that the flag attached to the cylinder in the center of the motor blocks the light beam of the photogate when the flag is raised.
• Connect the photogate output to the interface box by plugging its cable into the top socket (labeled “DIG/SONIC 1”) of the black interface box (“LabPro”).
• Turn on the computer and check the system by following these instructions: Double click the icon “Exp8_Period”. A window with a spreadsheet on the left (having “State” and “Pulse Time” columns and a graph labeled “Pulse Time”) comes up. On top is a window “Sensor Confirmation”.
• Click “Connect”, again “Connect”.
• Test the photogate: Block the photogate beam with your finger and see the red light on the cross bar of the photogate turn on.
• Click OK.

You are now ready to take data.

Vary the frequency of the driving motor by varying the voltage from the power supply; note that there are both “coarse” and “fine” adjustment knobs. You must be sure that the current knob is turned up (clockwise) enough that the power supply can supply enough current to run the motor. If the current knob is set too low (not clockwise enough), turning up (clockwise) the voltage knobs will not increase the speed of the motor. Using first the coarse voltage knob to set the voltage approximately and the fine voltage knob (which should be started out in the halfway-up position so that you can fine-adjust the voltage either up or down) for careful adjustment, observe standing waves as sketched in Fig. 3 whenever the driving frequency is equal to a frequency that should obey Eq.(8.6), which gives the allowed resonance frequencies of the stretched yellow string. Once you have a stable standing-wave pattern, click the green icon “Collect” on the PC screen to record a few measurements. The times you see on the left of the monitor screen are measured periods for the motor revolution. They should vary very little, thereby showing that the motor speed is remaining constant at the value to which you adjusted it. The frequency $\:\:f_n$ at which the stretched yellow string is shaken to produce the $\:\:n$th standing wave is the reciprocal of the period $\:\:\Large \tau_n$, viz., $\:\:\Large{f_n=\frac{1}{\tau_n}}$. (Don't confuse the symbol $\:\:\Large \tau$ for the period of oscillation with the symbol $\:\:T$ used for the tension in the string in Part I of this manual and Part I of the worksheet.) You see on the right a horizontal line because all periods are equal as time goes on.

Find the conditions for, and observe, at least six resonance frequencies $\:\:f_{n}$ for consecutive values of $\:\:n$ and enter the values of all seven, together with the number of half waves $\:\:n$, on the table in your worksheet. Use the plotting tool below to plot $\:\:f_{n}$ versus $\:\:n$; no uncertainties are required. You should check the box that asks if you the fit goes through (0,0). (Think about why you should do this!)

<form method=“post” action=“http://mini.physics.sunysb.edu/~mdawber/plot/lab6plot.php” target=“_blank”>

x axis label (include units):
y axis label (include units):
Check this box if the fit should go through (0,0).
(Don't include (0,0) in your list of points below, it will mess up the fit.)
What kind of uncertainties are you entering below?
x1: +/-    y1: +/-
x2: +/-    y2: +/-
x3: +/-    y3: +/-
x4: +/-    y4: +/-
x5: +/-    y5: +/-
x6: +/-    y6: +/-
x7: +/-    y7: +/-
x8: +/-    y8: +/-
x9: +/-    y9: +/-

## Results

Record the value of the slope $\:\:k$ and its uncertainty on your worksheet. Use it to calculate the velocity $\:\:v$ using $\:\:\Large{ k=\frac{v}{2L}}$. Calculate the uncertainty in $\:\:v$ from the uncertainty in $\:\:L$ and the uncertainty in the slope using Eq. (E.7) from the Lab 1 manual, Uncertainty, Error and Graphs. Are the two values you obtained for $\:\:v$ consistent, i.e., do they agree to within their combined uncertainties?

## Part III: Verification of the Dependence of the Traveling Wave Velocity on the String Tension T and a Measurement of the Acceleration of Gravity g

In this part you will measure the dependence of one of the frequencies $\:\:f_{n}$ on the tension $\:\:T$ in a non-stretching string. You will vary the tension by suspending various weights of mass $\:\:M$ from the string. You should keep the number of half waves fixed at $\:\:n = 1$.

By combining equations (8.1), (8.3) and (8.6) you can obtain an equation for the frequency $\:\:f_{1}$ as a function of the mass $\:\:M$ that is hung from the string:

$\Large (f_{1})^2=\frac{(1)^2}{(2L)^2}\;\frac{g}{\mu}\;M\:\:.$
(8.7)

Replace the “stretchy” rubber band with the thicker, not-very-stretchy white string. Because this string does not stretch noticeably when put under tension, it retains an approximately fixed linear mass density for various tensions $\:\:T$. Its linear mass density, $\:\:\mu_w$, is given to you in the Equipment section of this Lab 8 manual.

Vary the tension $\:\:T$ by suspending masses $\:\:M = 50$, 100, 150, 200, and 250 grams in turn. Measure the frequencies $\:\:f_{1}$ for the standing wave with $\:\:n=1$ half-waves for each mass $\:\:M$ and enter them with their squares (i.e., as $\:\:(f_{1})^2$) in the table on your worksheet.

Plot $\:\:(f_{1})^2$ vs $\:\:M$ using the plotting tool below (no uncertainties). Do you know why you should check the box that asks if you the fit goes through (0,0)?

x axis label (include units):
y axis label (include units):
Check this box if the fit should go through (0,0).
(Don't include (0,0) in your list of points below, it will mess up the fit.)
What kind of errors are you entering below?
x1: +/-    y1: +/-
x2: +/-    y2: +/-
x3: +/-    y3: +/-
x4: +/-    y4: +/-
x5: +/-    y5: +/-
x6: +/-    y6: +/-
x7: +/-    y7: +/-
x8: +/-    y8: +/-
x9: +/-    y9: +/-

## Results

Write down the slope $\:\:k$ of the graph and its uncertainty $\:\:\Delta k$. Use this to calculate $\:\:g$, the acceleration due to gravity. (From Eq. (8.7), $\:\:\Large{k=\frac{(1)^2}{(2L)^2}\;(\frac{g}{\mu_w})}$. To calculate the uncertainty in your measurement of $\:\:g$, neglect the uncertainty in $\:\:L$ and and use the fact that the relative uncertainty in $\:\:g$ will be due to the relative uncertainty in the slope, $\:\:\Large{\frac{\Delta k}{k}}$, and the relative uncertainty in $\:\:\Large \mu_w$, $\Large{\frac{\Delta \mu_w}{\mu_w}}$, and from Eq. (E.7) of Uncertainty, Error and Graphs, it will be

$\:\:\Large{ \frac{\Delta g}{g}=\sqrt{(\frac{\Delta k}{k})^2+(\frac{\Delta \mu_w}{\mu_w})^2}}$.

Once you have your value for $\:\:g$ and its uncertainty $\:\:\Delta g$, compare it to the accepted value for $\:\:g$. Are they consistent? 