Standing Waves on a String PURPOSE

San Diego Miramar College
Physics 197
NAME: _______________________________________ DATE: _________________________
LABORATORY PARTNERS : ______________________________________________________
Standing Waves on a String
PURPOSE: To study the relationship among stretching force (FT), wavelength (𝝺), vibrational
frequency (f), linear mass density (µ), and wave velocity (v) in a vibrating string; to observe
standing waves and the harmonic frequencies of a stretched string.
THEORY: Standing waves can be produced when two waves of identical wavelength, velocity,
and amplitude are traveling in opposite directions through the same medium. Standing waves
can be established using a stretched string to create a train of waves, set up by a vibrating body,
and reflected at the end of the string. Newly generated waves will interfere with the old reflected
waves. If the conditions are right, then a standing wave pattern will be created. A Stretched
string has many modes of vibration, i.e. standing waves. It may vibrate as a single segment; its
length is then equal to one half of the wavelength of the vibrations produced. It may also vibrate
in two segments, with a node at each end in the middle; the wavelength produced is then equal to
the length of the string. It may also vibrate in a larger number of segments. In every case, the
length of the string is some integer multiple of half wavelengths. So, if a string is stretched
between two fixed points, the ends are constrained not to move; hence, these are the nodes. In a
standing wave the nodes, or the points that do not vibrate, occur every half the wavelength; thus,
the ends of the string must correspond to nodes and the whole length of the string must
accommodate an integer nymber of half wavelengths. Roughly midway between the nodes are
antinodes, which is where the standing wave has maximum amplitude.
Remember that a wave traveling along a string will have a velocity given by
Where FT is the tension in the string and µ is the linesar density of the string. If the wave
velocity is “right”, then the standing wave will have a wave velocity of
Thus the fundamental frequency for a particular string (where FT and µ are constant) would be
Notice that the length of the string is L and that a standing wave of the fundamental frequency is
½ 𝝺, so 𝝺 = 2L. The other allowable frequencies (that generate standing waves) would be
an integer multiples of this fundamental frequency. This can be written as
It is fairly easy to measure the vibrating length of the string L and, hence, determine 𝝺. It is also
possible to measure the “taut: length of the string and its mass to determine the linear mass
density, µ. Thus we can solve for the experimental value of tension as
APPARATUS: PASCO SF -9324 Mechanical vibrator, 2 PASCO universal Table Clamp,
support rod, pulley, two types of string, electronic balance, meter stick, 500 gram hooked mass
or 200 gram hooked mass, 2 jumper cables, PASCO PI -9587B digital function generator.
PROCEDURE: Set up the equipment as illustrated in the figure below ( but do not attach the
string to the resonator yet).
1. First take a piece of string about 2.5 meter long, mass it and measure its length when
“taut”. Using the measured value, calculate the linear density of the string. Also record
the mass and length in the following table.
2. Tie one end of the string to the bar, pass it over the vibrator, then ovr the pulley and
attach it to the 500g mass. Be sure the string is paralled to the table top. Measure the
length of the string from the resonator to the pulley and record.
3. Turn on the power supply and start the frequency at 10 Hz. Increase the frequency until
you establish a 1 segment (2 nodes) standing wave pattern. To adjust the frequency, just
turn the knob. If you believe you have one segment but might not have maximum
amplitude, you might have to move the pulley/resonator backward or forward from it’s
clamp for “tuning” or adjust the frequency until you get maximum amplitude. Record
your frequency value in the table.
4. Measure and record the segment’s length between the resonator and the pulley. This can
be done by holding a 2-m stick to measure the distance from one node to the next
consecutive node. Remember that one segment of a standing wave pattern is 1/2 𝝺.
Calculate and record the wavelength (by doubling the length of one segment). Also
calculate the experimental tension, and the percent difference of that tension with
the theoretical tesional value.
a. To get a standing wave that has 2 segments, should you increase or decrease the
frequency?
b. By how much should you change the frequency?
String1:
Mass of the string______________
Total length of the string ______________
Linear density of the string _______________________
Tension (theoretical) ________________________
Segment
1
2
3
4
5
6
7
8
Frequency(Hz)
Calculation of experimental tension:
Wavelength(m)
Tension(N)
%difference
Calculation of % difference of tension:
5. After finishing this measurement, repeat the process for 2, 3, 4, 5, 6, 7 and 8 segment
standing waves by adjusting the frequency. Besides recording the frequency and
measuring the length of one segment for each standing wave, calculate and record the
experimental tension and % difference of tension.
a. Calculate the wave velocity in the string when you have a standing wave pattern of 3
segment.
b. For 7 segment.
c. What was the frequency value for the fundamental frequency?
6. Now repeat this process for the second string. Is the Linear density larger or smaller?
Do you expect the string to have a higher fundamental frequency or less?
String2:
Mass of the string______________
Total length of the string ______________
Linear density of the string _______________________
Tension (theoretical) ________________________
Segment
1
2
3
4
5
6
7
8
Frequency(Hz)
Wavelength(m)
Tension(N)
%difference
NAME: _______________________________________ DATE: _________________________
LABORATORY PARTNERS : ______________________________________________________
Standing Waves
Resonance
OBJECTIVE: We are going to examine the relationship between wave speed, frequency, and
wavelength using standing waves.
THEORY: When two waves traveling in opposite directions produce a standing wave. We are
studying standing sound waves in an air column. The length of the pipe determines the
wavelength and the frequency of the various harmonics. For the first harmonic, speed of sound
can be found by v = f𝝺 and L, the length of the tube is odd multiple of 𝝺/4.
EQUIPMENT: spatial standing tubes, meter stick, tuning forks
PROCEDURE: Examine the tube and draw the standing wave for the tube for harmonics up to
at least the third harmonic.
Describe how you determine the speed of the sound in air using the tube.
Follow the procedure you have described above using at least three tuning forks and for each
tuning fork repeat the process three times, for at least nine entries.
Observer
Frequency
Harmonic
Number
Resonance
length
Wavelength
Wave speed
Calculate the average wave speed from your data. Calculate percent error from the theoretical
speed of sound at 25° C of 346 m/s.