W-4 WAVES IN A COIL SPRING
To investigate waves in a long coil spring, and to discover some of the laws obeyed by these waves.
Long coil spring, stopwatch or electric timer, meter stick.
Some aspects of this experiment cannot be measured or analyzed in a reasonable amount of time, such as the effect of spring sag, effect of amplitude of wave, etc. These will be noted, but ignored in the analysis, so the results cannot be expected to show strict conformity to idealized laws. It would therefore be futile to spend a lot of time getting highly accurate data, and a formal error analysis would be inappropriate. Still, lawful relations will be obvious even in this imperfect experiment, and you should have enough "feel for physics" by now to make reasonable estimates of the reliability and significance of your results, even to estimating how much the ignored factors could affect the results.
Definitions will be inserted where appropriate, and numbered D-1, 2, etc.
(1) Fasten one end of the coil spring to a rigid support. Stretch the spring to a length of 5 meters or more, as space permits. Throughout the first parts of the experiment you will keep the spring at some constant length, so mark the position of the free end and do not change it. Measure the straight-line distance between the spring's endpoints, and measure approximately the amount on sag at the center.
(2) Hold the free end in your hand. With the other hand, grasp the spring about a few decimeters from your hand, and pull this section back, compressing the section of spring between your hands. Continue to hold the free end fixed, but release your other hand suddenly. This causes a disturbance to travel from your hand down along the spring. It appears as a region of compressed coils moving along the spring.
D-1 A disturbance of brief duration is called a PULSE.
D-2 A disturbance of a medium in which all particles of the medium move parallel to the apparent motion of the disturbance, is called LONGITUDINAL.
The region of compression represents a longitudinal pulse. You will notice that it reflects from the other (fixed) end of the spring and returns to your hand. If your hand is held rigid, it may reflect from your hand back to the wall, and repeat several times before dying out.
(3) The pulse moves slowly enough that you can time it with a stop watch. Do so. The pulse velocity can now be calculated from v = L/t where L is the spring length, and t the time the pulse takes to travel that length. But you really can't easily calculate the length of the sagging spring, and fortunately it isn't needed for our analysis. Just carry along the letter L in lieu of a numerical value, and you will see that in the final analysis, L cancels out. So if it takes 2 seconds for the pulse to traverse L, you simply record the velocity as L/2.
(4) Now repeat procedures 2 and 3 with lateral jerks of the free end. Try up-jerks, and sidewise jerks separately, to see if the velocity is the same in all three cases.
D-3. A disturbance of a medium in which the particles of the medium move at right angles to the apparent motion of the disturbance, is called TRANSVERSE.
Observe whether an upward pulse remains upward after reflection from the fixed end. Note also what happens to a sidewise pulse after reflection.
(5) Now shake the free end of the spring in a slow, smooth, regular transverse motion.
D-4 A motion which repeats itself EXACTLY in equal time intervals is called PERIODIC.
D-5 The time between successive exact repetitions of a periodic motion is called the PERIOD of the motion. Units: seconds.
D-6 The reciprocal of the period is called the FREQUENCY. f = 1/T. Units: reciprocal seconds; named the "hertz" and abbreviated "Hz."
You will find that at most driving frequencies, the spring has a complicated motion, with no simple pattern. At certain particular frequencies the spring moves in a smooth fashion, in which every part of the spring falls into synchronism with your hand, clearly moving with the same frequency as your hand. There may be one or more points of the string which seem practically ar rest, but other parts move laterally with periodic motion.
D-7 These smooth, simple patterns are called STANDING WAVES.
D-8 In a vibrating body, the displacement from rest position at any point is called the AMPLITUDE at the point.
D-9 In a standing wave, the points where the amplitude is zero are called NODES.
D-10 In a standing wave, the points where the amplitude is maximum are called ANTINODES.
D-11 The amplitude at an antinode is called the AMPLITUDE OF THE WAVE.
D-12 The frequencies at which standing waves occur are called RESONANT FREQUENCIES.
You should notice that at the resonant frequencies, it takes very little work on your part to maintain the motion. This is very much like pushing a child on a swing: small pulses, properly timed, maintain a large amplitude motion.
D-13 The resonant frequencies of a body or system are called its NATURAL FREQUENCIES.
If you try to move your hand at a frequency which is not a natural frequency of the system, your hand experiences resistance, as if the spring were working against you (it is).
D-14 The lowest natural frequency of a system is called the FUNDAMENTAL FREQUENCY, or FIRST HARMONIC. The other natural frequencies are labeled in order from low to high: SECOND HARMONIC, THIRD HARMONIC, etc. (When describing the air vibrations of sound sources, musicians call the second harmonic the FIRST OVERTONE, the third harmonic the SECOND OVERTONE, etc.)
(6) Find the standing wave with the lowest frequency, the first harmonic of the spring. Draw a picture of what it looks like. At its fundamental frequency the string has no nodes except at its ends. Measure its frequency by timing a sufficient number of periods, say 10. The swing amplitude of your hand is the largest amplitude observed on the spring in this case, that is, an antinode is at the position of your hand.
(7) Find the frequencies of the first four or more harmonics in the same manner. In all cases be sure the swing amplitude of your hand is equal to the amplitude of the other antinodes, this ensures that your hand is always at an antinode. Avoid extremely large amplitudes.
(8) Arrange your data in a table, and perform the necessary calculations.
TIME FOR TIME/VIB VIB/TIME PICTURE OF WAVELENGTH FREQUENCY X ____VIBS. (PERIOD) (FREQ) PATTERN WAVELENGTH() sec sec Hz ( )L ( )L
D-15 In a standing wave, the WAVELENGTH is twice the distance between adjacent nodes, or four times the distance between a node and the nearest antinode.
In your table, express wavelength as a multiple of L, for example 0.51L, 0.24L, etc. (You won't necessarily get these particular values.) The product of frequency and wavelength is also a multiple of L.
(9) Now if things have gone well and you haven't blundered, you should notice that the entries in the last column are all approximately the same. If so, average these values. Compare the average with the velocity you calculated for transverse pulses in part 4. More accurate experiments should confirm the general relation:
v = fλ
You have discovered this mathematical relation in your data. This equation represents a neat "pattern" of nature which rivals the beauty of the visual patterns of the waves themselves. Other experiments verify that this relation is quite general, describing all waves: water, sound, light, etc.
Limitations of the experiment. In calculating the wavelength you probably assumed that an antinode was at the free end (at your hand). This is true if you took care to move your hand with an amplitude equal to the maximum amplitude of any part of the spring. But if your hand was actually somewhere between a node and an antinode, the actual location of the antinode may not be near the free end. Of course, one could locate and measure the node spacing directly, with a meter stick, for better results.
More accurate studies would show that the velocity in the spring is not exactly constant, but changes slightly with frequency, and also is affected by large amplitudes (because then the average spring length is longer, and the average spring tension higher.) The wave velocity will be most nearly constant when the spring amplitude is very small compared to the wavelength, so these effects will not show.
5. ANALYSIS: (may be done as homework)
(1) The observations of part 5 may be understood by noting that standing waves will be produced when the upswing of the wave is reflected back to the hand so that the reflected swing of the wave reaches the hand at just the right time, and is in just the same direction (up or down) that the hand is moving at that time. (Think of the child on the swing again). If the reflected wave tries to pull down, you feel "resistance" to upward motion of your hand. Only at the resonant frequencies does the return wave aid the motion of the hand rather than oppose it.
Discuss this, with reference to your data, and to diagrams, showing how your data supports (or contradicts) this explanation.
(2) The standing wave pattern you observed may be thought of as the sum of two waves of equal amplitude and the same wavelength traveling in opposite directions down the spring. (One is the reflected wave.) Show this with the aid of diagrams.
(3) Look at the ratio of frequencies you tabulated, compared to the fundamental frequency. Look at the ratio of wavelengths to the fundamental wavelength also. Are these ratios nearly integers? More careful experiments would show the ratios in this experiment to involve only the odd integers. Strings with both ends fixed (as in a piano, guitar, etc.) produce waves related to the fundamental by both even and odd integers. Explain, in the manner of Analysis (1), why even integers can't work in the situation you studied in lab.
(4) Compare your measured velocities of longitudinal, up-down, and sidewise pulses. Are differences real, or due to experimental error? Would you expect these three values to be the same in a more accurate experiment? Why, or why not? If not, which should be larger?
(5) If we kept the amplitudes constant, would the average spring lengths be different for different harmonics of a transverse wave? Why? How would the tensions compare? How would wave velocity depend on tension? Does your data show this?
© 1995, 2004, by Donald E. Simanek.