Introduction to Vibrations and Waves

Experiments with Slinky Springs

What is a wave?

A wave is a disturbance or oscillation that propagates (or travels) through space and time.

A wave can transport energy without carrying any mass with it. The energy passes through 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.

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?

Sound, heat energy and absorption from the holding objects, sap mechanical energy from the spring.

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 stays constant.

g. Does the speed of the pulse appear to depend on the size or shape of the pulse?

No.

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 = _________________________

time

lengthspeed =

b. Compute the speed of the traveling pulse.

c. Change the tension of the spring and repeat steps (a) and (b).

length = _________________________

time = _________________________

time

lengthspeed =

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.

For a stretch spring, no it does not. As a model for other media, changing tension could represent different density materials.

3. Interference. a. What is interference?

Interference occurs when one wave passes through another 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?

At any instant, the medium is the vector resultant for all points. The medium is shaped by the combination of both passing waves.

Figure 11-38Interference

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 lighter spring (physics demo 9-19 wave coupling.)

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.

Omit. Same answer as a.

1

Day #2: Standing Waves

Notes

{continued}

Vibrations

• Professor Julius Sumner Miller

Characteristics of a single-frequency continuous wave.

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

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 antinodes & nodes.

node = zero amplitudeantinode = maximum amplitude

f. Loops are caused by _________________ interference while nodes

are caused by ________________ __________________ interference.

constructive

completely destructive

g. Loop, are produced by waves that are _____ phase, nodes when

they are ______________ phase.

in

180o out of

h. What always occurs at the ends of the spring?

Each end is fixed, so the end has a node. This is where destructive interference occurs. i. There are always more _________ than ____________. How many more of one are there than the other?

nodes antinodes

one more node than antinode

j. Label the antinodes {A} and the nodes {N} on each of the drawings of standing waves. Give the length of each wave form in wavelengths.

N N

A

N N

NA A

N N N N

A A A

k. Generate a formula relating wavelength to the length of the string L, for any number of nodes, n.

2

nL

n

Ln

2

l. The n’s are called the

_________________

and n = 1 is referred to as the

__________________.

harmonics

fundamental

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 11-46The characteristics of a single-frequency wave at t = 0

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.

fT

1

Tf

1

secondhertzHz

1

fT

v

e. frequency of the nth harmonic standing wave:

n

Ln

2

nn fv

fn = frequency of the nth harmonic standing wave, corresponding to wavelength n.

L

nvvf

nn 2

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

propertyinertial

propertyelasticv

F = force of tension in the string

μ = mass per unit length of the string = “linear mass density” length

mass

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?

time

distv

s

m3100.50

75.2

sm0.55

fv v

f

mf s

m

270.0

0.55 Hz204

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

n

Ln

2

3

23

L

cm

cm0.60

3

0.9023 Hz

mf s

m

708600.0

425

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?

mkg

m

kg

L

m0300.0

00.2

0600.0

sm

mkg

NFv 129

0300.0

500

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.

F

v

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.

F

v 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:

Hz

L

v

L

v

L

nvf

short

short

short

short

short

shortn 60

22

1

21

Frequency of second harmonic of long wire:

Hz

L

v

L

v

L

nvf

short

short

short

short

long

longlongn 120

22

22

22

2,2

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?

m00.2

m50.1 m50.1

m

m

50.1

00.1cos

2.48

T TmgT sin2

sin2

mgT

19.48sin2

80.90.12 2smkg

T

NT 9.78

(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

L

nvvf

nn 2

L

vf

2

33

F

v mkg

N

0010.0

89.78 s

m281

L

vf

2

33

m

sm

00.22

2813

Hzf 2113