WAVES AND THE VIBRATING STRING
OBJECTIVE: To study waves.
TERMINOLOGY:
wave
interference
node
frequency (f)
constructive
anti-node
wavelength ()
destructive
linear mass density ()
THEORY:
Waves are a common phenomena. We are all aware that when a rock is thrown into water, it
creates a wave that travels out from the source (the splash). For water waves, the wave moves
rather slowly so that it is easily seen. Most other waves move rather fast and so are hard to
see.
In Parts 1 and 2 we will use just any old "pulse" as a wave on a slinky in this experiment (since
the wave on a slinky moves relatively slowly) to see how the wave moves and then how it
reflects (bounces).
In part 3 we will use a vibrator to create a "nice" wave, a sine wave, on a string. We will not be
able to see the wave itself move since it moves so fast, but we will be able to have it reflect
back and forth and INTERFERE with itself so that it sets up what are called standing waves (for
reasons that should become evident) that we can see.
The speed of a wave depends quite naturally on what the wave is moving in. For a wave along
a stretched string the speed of the wave depends on the tension in the string and the string's
linear mass density, i.e. its mass per length, and is given by the equation:
v  F /
(1)
where v is the speed of the wave, F is the tension in the string, and  is the mass per unit length
or linear mass density ( = m / L) of the string.
For "nice" waves (sine waves), the wave repeats itself in both space and time. The distance
over which it repeats itself is called the wavelength (); the time over which it repeats itself is
called the period (T). Instead of period, however, we often talk about the frequency (f) of a
wave which is defined simply as 1/T. [Note: period has units of seconds/cycle, while frequency
has units of cycles/second.] By definition, the speed of a wave (v) is the distance traveled
divided by the time it takes to cover that distance:
v = /T =  f .
(2)
Note that the speed depends on the tension and the mass density of the string (Eq. (1)) and the
speed relates the wavelength to the frequency (Eq. (2)). The tension and the mass density
determine the speed, and the speed and frequency then determine the wavelength (or the
speed and wavelength determine the frequency).
If one end of a string is held fixed and the other end is attached to a vibrator so its direction of
vibration is at right angles to the direction of the string, wave disturbances will travel along the
string with the speed v of Eq (1). At the fixed end the waves will be reflected back along the
Waves and the Vibrating String 2
string. In general, the reflected waves will interfere with the incoming waves to produce a rather
jumbled pattern on the string. But if the tension and the length are adjusted so that the number
of half-wavelengths on the string is an integer, then the initial and reflected waves will add
together to set up a stationary wave pattern. NOTE: for standing waves, the theory predicts that
the wavelength should be twice the distance between the nodes.
Part 1: Waves on a slinky
PROCEDURE:
1. Stretch the slinky to some distance and try to measure the speed of a pulse that you put on
the slinky. Remember that speed is simply the distance traveled divided by the time it takes
to go that distance. Try to estimate the speed of the pulse as best you can.
2. Stretch the slinky to some other distance and again try to measure the speed of the pulse
that you put on the slinky. Did the speed change when you changed the conditions on the
slinky?
REPORT: Record your measurements and observations as described in the Procedure.
Part 2: Reflection of waves
PROCEDURE:
1. Stretch the slinky to some distance and put a pulse on the slinky so that the pulse is up.
Then watch to see if the pulse reflects (bounces) off of a fixed end (your partner?). If it does
reflect, notice whether the pulse stays up or becomes inverted (pulse is down).
2. Now stand on something high (the desk?) and hold the slinky up so that the other end does
NOT touch the ground (the end is free). Now put a pulse on the slinky (pulse is away from
you). Notice whether there is a reflection in this case, and if so whether the pulse stays
away from you or becomes inverted (points toward you) on reflection.
REPORT: Record your observations as described in the Procedure.
Part 3: Interference and standing waves on a string
PROCEDURE:
1. Note that the frequency of the vibrator is 120 cycles/sec (also called Hertz). This is
determined by the construction of the vibrator.
2. Measure the total length of the string with a meter stick, and weigh it on a balance. (The
string should be at least 2 meters long.) Calculate the linear density ().
3. Connect one end of the string to the vibrator and the other end to a weight hanger hung over
a pulley.
4. Start the vibrator, and increase the tension (F) in the string by adding weights until there are
six or seven loops in the string. When in proper adjustment the amplitude of vibration will be
a maximum. The place where the maxima occur are called anti-nodes; the places where the
Waves and the Vibrating String 3
string does not vibrate are called nodes. Record the tension, number of loops, and the
length of the string from vibrator to pulley.
5. Repeat the above procedure, each time increasing the tension so as to reduce the number
of loops, until at least three sets of measurements have been obtained.
6. Calculate the speed of the wave from Eq. (1) and then wavelength of the wave from Eq. (2)
for each set.
REPORT:
1. Record all of your data and show your calculations as described in the Procedure.
2. Compare your calculated wavelengths from step 6 above to the length of the string (from
vibrator to pulley). The wavelengths should be such that an integer number of wavelengths
equals twice the length. Is this true in all your cases? What could account for any slight
discrepancies?
Application:
For a guitar, the speed of the wave on a guitar string is determined by the mass density of the
string (notice on a guitar that the six strings have different diameters) and by the tension in the
string (notice that to tune a guitar you must adjust the tension in the strings). When you play a
certain note on a guitar string, you must press the guitar string at a certain position. This
position then determines the length of the string, and hence the possible wavelengths that will
sound (resonate). Then from Eq. (2), there will be only certain frequencies coming from the
string, and these certain frequencies combine to form that certain note on the guitar.