Experiment with the Slinky Introduction to Waves Experiment with the Slinky

What is a wave? A wave is a disturbance or oscillation that propagates (or travels) through space and time. A wave can transport energy from one place to another without carrying any mass with it. The energy passes through the mass of the material carrying the wave.

A. TRANSVERSE PULSE: 1. A transverse pulse or wave is: A transverse wave is a pulse or oscillation that vibrates perpendicular to the direction the wave travels.

a. Diagram a transverse pulse.

The amplitude or size of the wave gets smaller. b. Make a statement about a transverse pulse relating the motion of the separate coils of the spring to the path traversed by the pulse. As the energy of the wave passes through the material, the material is made to oscillate in a direction perpendicular to the wave’s direction of motion. c. How does the shape of a short pulse change as it moves along the spring? The amplitude or size of the wave gets smaller. d. Can you suggest a reason for the loss? There is friction or drag between the coils of the slinky and the floor, which saps energy from the wave as it tries to move the coils across the floor.

There does not appear to be any dependence on the shape of the wave. e. Upon what does the initial amplitude depend? The amplitude depends on the size of the initial disturbance that creates the wave. f. Does the speed of the pulse appear to change with its shape? The speed of the wave appears to stay constant, not affected by the amplitude. g. Does the speed of the pulse appear to depend on the size or shape of the pulse? There does not appear to be any dependence on the shape of the wave.

length = _________________________ 2. Speed of the pulse. a. Measure the length of the stretched slinky and the travel time of a pulse generated at one end. length = _________________________ time = _________________________ b. Compute the speed of the traveling pulse. speed =

length = _________________________ c. Change the tension of the spring and repeat steps (a) and (b). length = _________________________ time = _________________________ speed =

d. Is the speed of propagation dependent on or affected by the tension in the spring? If yes, is the speed greater or less with increased tension? When coils from the end of the spring were pulled out from the oscillations, the tension was increased and the speed of the wave increased. e. Does the slinky, under different tensions, represent the same transmitting media? Explain. In general, no. Since coils were removed from the slinky for this case, there was less mass distributed along the length of the slinky. This also affects wave speed.

Interference occurs when one wave passes through another wave. a. What is interference? Interference occurs when one wave passes through another wave.

The pulses add constructively to produce a larger amplitude wave. b. How does the pulse amplitude during interference compare with the individual amplitudes before and after superposition when... (1) the pulses are on the same side of the spring? The pulses add constructively to produce a larger amplitude wave. (2) the pulses are on opposite sides of the spring? The pulses add destructively to produce a smaller amplitude wave. c. What conclusions can you draw about the displacement of the medium at a point where two pulses interfere? The two waves pass through one another without affecting one another. The medium follows the combination of the amplitudes.

Figure 11-38 Interference Graphs showing two waves, and their sum, as a function of time at three locations. In (a) the two waves interfere constructively, in (b) destructively, and in (c) partially destructively.

4. Reflection From Fixed and Free End Terminations. a. How does the amplitude of a reflected single pulse compare to its original pulse? The pulse keeps the same amplitude. No energy is lost to the “collision”.

b. What is the phase of the reflected pulse relative to the transmitted pulse when the spring has a fixed end termination? From drawing (a), the wave switches sides, corresponding to a 180 degree phase change. c. What is the phase of the reflected pulse relative to the transmitted pulse when the spring has a free end termination? From drawing (b), the wave does not switch sides, corresponding to a 0 degree phase change.

5. Wave Behavior Between Two Media. a. What happens to the pulse when it reaches the junction between the two springs? Some of the pulse will pass through to the other spring, but some will reflect back from the junction. b. How does the speed of propagation in the slinky compare with that in the heavier spring? The pulses travels faster in the light weight slinky and more slowly in the heavy spring.

c. Describe what happens when a pulse is transmitted from the slinky to the heavier spring. The transmitted pulses remains on the same side of the spring, but the reflected pulse travels back on the opposite side of the spring. Behaves like reflection from a fixed end. d. Describe what happens when a pulse is transmitted from the heavier spring to the slinky. The transmitted pulses remains on the same side of the spring as before, but the reflected pulse travels back on the same side of the spring. Behaves like reflection from a free end.

B. LONGITUDINAL WAVES. a. Why is this called a longitudinal wave? A longitudinal wave is a pulse or oscillation that vibrates parallel to the direction the wave travels.

b. Make a statement about the longitudinal pulse relating the motion of the separate coils of the spring to the path traversed by the pulse. As the energy of the wave passes through the material, the material is made to oscillate in a direction parallel to the wave’s direction of motion.

Day #2: Standing Waves Notes {continued}

C. STANDING WAVES. A standing wave is produced by the interference of two periodic waves of the same amplitude and wavelength traveling in opposite directions. a. How does the motion of a standing wave compare to that of a transverse wave? The standing wave seems to “stand still”, oscillating in the same place on the spring.

b. What is the effect of frequency on a standing wave? The higher the frequency of the oscillations, the more “loop” patterns appear in the spring. c. Draw a diagram of a standing wave with low and high frequency. low frequency high frequency

WAVELENGTH: The wavelength of the wave is the literal length of the wave or the length of the repeated pattern.

Figure 11-23 Characteristics of a single-frequency continuous wave.

Figure 11-46 The characteristics of a single-frequency wave at t = 0 The characteristics of a single-frequency wave at t = 0 (just as in Fig. 11-23).

“loop” = antinode = maximum amplitude d. Do all of the parts of the spring move equally? Describe any variation, if any. Some parts of the wave have large amplitude, other points have zero amplitude. e. Compare the motion of loops and nodes. node = zero amplitude “loop” = antinode = maximum amplitude

Each end is fixed, so the end has a node. constructive f. Loops are caused by _________________ interference while nodes are caused by ________________ __________________ interference. completely destructive in g. Loops are produced by waves that are _____ phase, nodes when they are ______________ phase. 180o out of h. What always occurs at the ends of the spring? Each end is fixed, so the end has a node. nodes antinodes i. There are always more _________ than ____________. How many more of one are there than the other? one more node than antinode

A N N N A A N N harmonics A A A fundamental N N N N j. Label the loops {L} and the nodes {N} on each of the drawings of standing waves. Give the length of each wave form in wavelengths. k. Generate a formula relating wavelength l to the length of the string L. A N N N A A The n’s are called the _________________ and n = 1 is referred to as the __________________. N N harmonics A A A fundamental N N N N

D. STANDING WAVES ON A STRING D. STANDING WAVES ON A STRING. The main purpose of this section is to find a relationship between the wavelength of the standing wave on a string and the frequency of standing wave on the string. 1. Diagram a representative standing wave and define each of the following terms: a. standing wave b. wavelength

Figure 11-23 Characteristics of a single-frequency continuous wave.

Figure 11-46 The characteristics of a single-frequency wave at t = 0 The characteristics of a single-frequency wave at t = 0 (just as in Fig. 11-23).

c. period and frequency: The period (T) is the time for the wave to complete one cycle. The frequency (f) is the number of cycles a wave completes per unit of time, specifically one second. d. wave speed: A traveling wave will move a distance equal to one wavelength in a time of one period.

f. frequency of the nth harmonic standing wave: fn = frequency of the nth harmonic standing wave, corresponding to wavelength ln.

F = force of tension in the string 2. Factors to influence the wave speed: In general, the speed of a wave through a string will be independent of the wavelength or frequency of the waveform. What does contribute to the velocity of a wave through a string is the tension force in the string and the inertia of the string. As shown earlier, raising the tension in the string increases the speed of the wave. Increasing the density of the string has the effect of slowing the wave. In general, the wave speed in any medium is given as: F = force of tension in the string μ = mass per unit length of the string = “linear mass density”

E. Examples. Ex. #1: When a single pulse (wave) travels through a string, the pulse covers 2.75 m in a time of 50.0 ms. What would be the frequency of a standing wave with a wavelength of 27.0 cm on this string?

Ex. #2: A string is tied between two fixed ends 90 Ex. #2: A string is tied between two fixed ends 90.0 cm apart and vibrates as show in the diagram. The speed of the wave along the string is 425 m/s. Determine the wavelength and frequency of this wave. 3rd harmonic

Ex. #3: a. What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?

b. What is the speed of a transverse wave in a rope of equal length and same material, but twice the diameter as the previous rope? A rope with twice the diameter and same material would have four times the volume and four times the mass per unit length as the original rope. If the mass per length was four times bigger, the speed of the wave in the new rope would be half of the speed in the original rope.

Same cross section means same mass per unit length. Ex. #4: Two pieces of steel wire with identical cross section have lengths L and 2L. The wires are each fixed at both ends and stretched so that the tension in the long wire is four times greater than in the shorter wire. If the fundamental frequency in the shorter wire is 60 Hz, what is the frequency of the second harmonic in the long wire? Same cross section means same mass per unit length. Longer wire has four times the tension, so the longer wire has twice the wave speed as the shorter wire. Fundamental frequency of short wire:

Frequency of second harmonic of long wire:

Ex. #5: A 12-kg object hangs in equilibrium from a string of total length L = 5.0 m and linear mass density μ = 0.001 0 kg/m. The string is wrapped around two light, frictionless pulleys that are separated by the distance d = 2.0 m (diagram, below left). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate in order to form the standing-wave pattern shown in figure below right?

(b) At what frequency must the string between the pulleys vibrate in order to form the standing-wave pattern shown in figure below right? 3rd harmonic