THE VIBRATING STRING
OBJECTIVE: To observe waves on a slinky, to experimentally determine if and how waves reflect, and to see the phenomenon of resonance as it applies to waves on a string.
THEORY: When a force sets up a disturbance ( y ) on a string, Newton's Second Law of ;;；motion () can be used to get a wave equation which predicts that the disturbance will Fma;
propagate down the string with a velocity ( v ) that depends on the tension in the string ( T) s
and the linear mass density of the string ( ？ ) according to:
where the linear mass density is defined as
？ = m/ L . (2) total total
This speed of the disturbance on a string is too fast to allow one to see the actual motion of the disturbance along the string. However, we can see the propagation of a disturbance (or pulse) on a slinky since the speed is much lower. We will observe this pulse propagation in Part One. We refer to a disturbance or pulse that travels as a wave.
If the pulse hits an end of the string (or slinky) that is tied down, then the pulse does reflect from
that tied down end in such a way that the incident wave and the reflected wave add together at the tied down point so as to give a zero displacement at the end. What this means, in effect, is that the wave undergoes a reversal of amplitude, i.e. it “flips” or becomes inverted. We will see this in Part Two for a wave on a slinky.
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 continuously travel along the string with the velocity v of Eq. (1) and start interfering (adding) with previous
parts of the wave that have reflected back from the tied down end. Since the other end is also constrained by the vibrator, then waves will reflect from that end as well causing many reflected waves to interfere with the incident wave. Normally, this interference between incident and the many reflected waves will cause the resulting wave to have about as many up (or positive) contributions as down (or negative) contributions causing approximately zero resultant amplitude. However, when the cause of the disturbance is periodic (sinusoidal) and when the length of the string (from vibrator to pulley) is at (or very close to) an integer number ( n ) of
half-wavelengths ( ？=wavelength),
L= n(？ / 2) , (3)
then we get constructive interference at certain points along the string resulting in what is called a standing wave pattern. Also, when the waves are sinusoidal (or harmonic), the wavelength is
related to the frequency in a simple way:
？ f = v . (4)
This relation comes from the basic definition of speed as distance (here ？) divided by time
(here the period which is 1/f ).
Vibrating String 2
In the Part Three of the experiment we will employ a sinusoidal oscillator with a set frequency of 120Hz. We can measure both the mass and length of a string and use Eq. (2) to get ？. We
will control (and measure) the tension in the string (T) by using weights hung over a pulley. s
Then by observing when we get standing waves, we can use Eq. (1) to determine the velocity of
the wave and hence determine the wavelength of the waves on the string from Eq. (4). We can then test to see if Eq. (3) holds true for standing waves.
Part 1: Wave Propagation
1. Stretch the slinky to some distance and try to measure the speed of a pulse that you put on the slinky by flicking your hand up and down. 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?
3. In Steps 1-2, you were examining a transverse wave. The medium supporting the pulse (the
slinky) moved up and down, perpendicular or transverse to the direction that the pulse traveled. A wave that travels in a medium where the medium moves back and forth in the same direction that the wave travels is called a longitudinal wave. You can send a longitudinal wave along a
slinky by stretching the slinky while it is lying on a desk or the floor and rapidly pushing and pulling on one end. Produce such a longitudinal wave on the slinky and observe its motion.
1. Record your estimates of the speed of the pulse in Steps 1-2. What did stretching the slinky do to the speed of the wave?
2. Is light a transverse or longitudinal wave? What about a sound wave? What about a wave on a string?
Part 2: Reflection of Waves
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.
Vibrating String 3
Part 3: Interference and Standing Waves On a String
1. Note that the frequency of the vibrator is 120 cycles/sec or 120 Hz. This is determined by the construction of the vibrator and that the voltage from an outlet in the U.S. changes polarity 60 times a second.
2. Measure a 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. Measure the length of the string from vibrator to pulley. This length is L. It should be
at least 1.5 meters.
4. Start the vibrator, and increase the tension (T) in the string by adding weights until there s
are several loops in the string. When in proper adjustment the amplitude of vibration will be a maximum. Even though points on the string are moving up and down, it appears as if the string is „standing‟ in mid-air with several loops along its length. The place where the maxima occur are called anti-nodes; the places where the string does not vibrate are called nodes. Each loop
corresponds to a half-wavelength. Because the number of loops must fit between the end points of the string, the condition for a standing wave is given by Eq. (3), L= n(？ / 2), where n
.) is the number of loops. (If you think about it, n must also be the number of nodes plus one
Record the hanging mass (m) and the number of nodes. Note that the tension in the string is equal to the weight (mg) of the hanging mass. Why?
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. If possible, repeat this whole standing wave process for waves on a different type of string or on a wire.
1. For each standing wave, calculate the speed of the wave using Eq. (1). Then calculate the wavelength using Eq. (4). Finally, calculate the length of the string using Eq. (3). (Recall that n
equals the number of loops or the number of nodes plus one.) Compare this calculated length to that measured with the meter stick by finding the percent difference between the two values. Consider possible sources of uncertainty and see if these are sufficient to explain the percent difference. To do this, consider your possible sources of uncertainty as percentages. [Note: when multiplying two numbers, the percent uncertainties of the two numbers add; also recall that a number to a power is simply that number multiplied by itself that many times so that the percent uncertainty of the number should also be multiplied by the power.]
2. Define the following physical quantities: (a) frequency, f; (b) period, T; (c) linear density, ？;
(d) tension, T; (e) velocity of propagation, v. s
3. What does an increase in tension do to the following quantities: speed of wave, wavelength, number of nodes ?
Vibrating String 4
4. State the necessary conditions for the production of standing waves.
5. What tension would be required to produce vibration in the fundamental mode (n = 1) in
each of the strings or wires of this experiment.
6. Two strings, A and E, on a musical instrument are the same length. The frequency of A is ? the frequency of E. The diameter of A is twice the diameter of E, and the volume density of
2the material (！=m/V where V=，rL) of A is ? the density of the material in E. Calculate the wire
ratio of the tension in the A string to the tension in the E string.
7. Why do some strings of musical instruments have wire wrapped around the full length of the string?
For a guitar, the velocity of the wave on a guitar string is determined by the linear mass density of the string (notice on a guitar that strings have different thicknesses) 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 Eqs. (3)&(4), there will be only certain frequencies supported on the string, and these certain frequencies combine to form that certain note on the guitar. If a string is plucked, the lowest such frequency (called the fundamental frequency) is usually heard.
Higher frequencies can be selected by lightly touching the string with a finger string (not pressing it down all the way to the fret bar) while plucking it. These higher frequencies are called the harmonics.