name____________________ period _______ lab partners___________________________________
Standing Waves
Introduction
When a tightly stretched string (or wire) is disturbed, standing waves are produced by interference of
waves reflecting back upon themselves in the confined region between the ends of the string. Figure#1 shows
the simplest modes of vibration. The higher frequency vibrations are multiples of a fundamental frequency-that is fn = n f1 , where n =1, 2, 3, etc. Together these compose the string's natural or resonant frequencies.
In Part A of this lab we shall attach a mechanical vibrator to a wire under tension and, by carefully
adjusting vibration frequency, set up standing wave patterns. Tiny segments of wire will oscillate with
amplitude that depends on their position: zero at nodes, biggest at anti-nodes. For the fundamental (n=1)
vibrational mode, the wavelength is twice the length of the wire, λ1 = 2L. Plugging this into the relationship
between wave velocity v, wavelength λ, and frequency f, written as f = v / λ, we see fundamental frequency
f1 = v / 2L. (equation 1). Before using this, we'll first need find velocity v--which depends on the tension FT
doing the stretching and on the mass per unit length of the wire m / L v = √ [FT / (m / L)] (equation 2)
Similar standing wave patterns involving sound waves can exist in columns of air. In Part B we shall use
a vibrating tuning fork to establish these as Figure #2 pictures. Here the wavelength for the fundamental
(n=1) vibrational mode for these (longitudinal) waves is four times the length of the air column: λ1 = 4L.
Plugging this into the relationship between wave velocity v, wavelength λ, and frequency f, written as f = v /
λ, we see fundamental frequency f1 = v / 4L. (equation 3). Here v refers to the velocity of the sound waves in
air--something we will experimentally determine based on the data we collect.
Figure #1: Standing Waves on a String or Wire With Fixed Ends
Figure #2: Standing Waves in an Air Column, With One Closed, One Open End
Objectives

To learn about standing waves set up on a wire and check the theoretical understanding of them by
comparing fundamental frequencies obtained from equations 1 and 2 with measured values, etc.
 To learn about standing waves set up in an air column. As a check on theory, we shall determine the
speed of sound from our data and compare it with the accepted (room temperature) value.
Materials: Part A: #21 gauge Cu wire, function generator, mechanical vibrator, pulley, support rods and
clamps, masses; Part B: tuning forks and striker, plastic tube, plastic container for water, metric ruler
Figure #3: Part A setup
Procedure
Part A: Note: this may be done as an instructor demo. The set up is shown in Figure #3.
1) Before starting, record length L of wire, and mass / length = M / L in Data Table A.
2) a) With hanging mass of 1.00 kg providing the tension, the vibrator frequency will be varied until
resonance at the fundamental frequency (n=1) is achieved. Record the frequency. b) Sketch the waveform.
c) Continue varying the frequency and find resonance associated with n=2, n=3, and n=4 harmonics.
Record the frequencies and sketch the waveforms. (Use Data Table A)
3) Repeat 2a) for mass of 1.50 kg. 4) Repeat 2a) for mass of 2.00 kg. 5) Repeat 2a) for mass of 2.50 kg.
Part B:
(Caution: Don't spill water!)
1) As suggested by Figure #2, position a vibrating tuning fork--one you have just struck--over the open end
of an air filled vertically positioned plastic tube. The other end of this tube will rest in water--which will
serve to close it. Move the plastic up and down in the plastic water container until the sound coming from
the tuning fork is noticeably amplified: this is resonance. Holding the tube in that spot, use the metric ruler
and measure the length of the column of air to the nearest tenth of a centimeter. Record this and the
frequency of the tuning fork (stamped on it) in Data Table B
2) Repeat the above step for another tuning fork of different frequency.
3) Repeat step 1) for a third tuning fork of different frequency.
Data
Data Table A length of wire between fixed ends, L = _______; M / L =_________________
sketches, part A2 -- include distances between nodes (for hanging mass 1.00 kg)
frequency
n=1
n=2
n=3
n=4
wavelength
Data Related to Fundamental Frequencies
hanging mass
m, kg
frequency, f1
experimental, Hz
tension, FT =
mg, N
√ FT
velocity v,
m / sec
frequency, f1
theoretical, Hz
difference in
frequency, Hz
% difference
Data Table B
tuning fork frequency, Hz
length L of air column @
resonance, meters
λ1 = 4L, meters
1 / 4L, meters-1
notes, as needed
Data Handling / Analysis / Questions
1) From Part A data, using equations 1, and 2, calculate theoretical frequencies, and % differences between
experimental and theoretical frequencies. Put results (and intermediate results) in Data Table A.
Briefly discuss the results of this comparison, identifying possible sources of error.
2) Using data in Data Table A, on the sheet of graph paper provided, plot experimental frequency f1 (in Hz
plotted vertically) vs. √ FT (plotted horizontally). Draw a best fit line through the data points. Comment on
how well a line fits this data. (Bonus, do on separate sheet: determine the slope. What does it represent?)
3) Using data in Data Table B, with your graphing calculator (TI 83, TI 84, etc.) enter (try STAT, then
EDIT) 1 / 4L in List 1 and tuning fork frequency f in List 2. Do a linear regression (look on STAT, CALC
menu) and record the equation for the associated best fit line when the calculator plots 1 / 4L horizontally
vs. frequency (vertically). Record the correlation coefficient r and / or r2. Most importantly, record the slope
(the value of a in y= ax + b). This value is your experimental determination for the speed of sound.
4) How does your value for the speed of sound compare with the expected value at room temperature?
Compute % error. Briefly discuss the results of this comparison, identifying possible sources of error.