1. Vibrations and Waves

2. Periodic Motion
Periodic motion is that motion in which a body moves back and forth (oscillates) over a fixed path
An oscillating system exhibits SHM when
the restoring force is proportional to the
displacement (from equilibrium)

3. Simple Harmonic Motion, SHM
Simple harmonic motion is periodic (repeating) motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed.
A restoring force, F, acts in the direction opposite the displacement of the oscillating body.
F = -kx x F

4. Hooke’s Law (for a mass/spring system)
– the amount of stretching or compressing of a spring (displacement) is proportional to the force applied.
F = -kx
F = force (Newton)
k = spring constant (N/m)
x = displacement (m) from
equilibrium
The negative sign
means that the direction
of the force is opposite
the direction of motion.

5. SHM Simple Harmonic Motion

6. Energy in a system Etotal = ½ kx2 + ½ mv2 Etotal = mgh + ½ mv2
Work is done in stretching or compressing a spring (or displacing a pendulum). So potential energy is stored in the system.
The total energy is the sum of the potential and kinetic energy.
Etotal = ½ kx2 + ½ mv2
Mass/spring
Etotal = mgh + ½ mv2
Pendulum

7. Etotal = ½ kx2 + ½ mv2 Energy changes: Etotal = PEel + KE Note:
The first, third and last
diagrams indicate max
PE. (v = 0)
The second and fourth
indicate max KE.
(a = 0)
Etotal = ½ kx2 + ½ mv2

8. Properties of Waves

9. What are waves?
A wave is a transfer of energy from one place to another.
Wave-types take many forms.
Wave Characteristics include:
Period
Frequency
Wave speed
Amplitude
Wavelength

10. Measured in units of time – seconds for SI
Period (T) – the time needed for one full
cycle (back and forth)
Measured in units of time – seconds for SI
Frequency (f ) - the number of complete cycles (waves) that pass a given point in the medium in 1 second. Measured in Hertz (Hz) 1 Hz (s-1) = 1 cycle/second NOTE: Frequency is the inverse of the period It is set by the vibration source so it stays the same as the wave moves from one medium to another

11. Pendulum T = period (sec) l = length of pendulum (m) g = 9.81 m/s2
Note: the restoring force is proportional to the displacement. It is equal to mgsinθ FT FT
Restoring force mg mg
T = period (sec)
l = length of pendulum (m)
g = m/s2

12. Example. What must be the length of a simple pendulum for a clock which has a period of two seconds (tick-tock)? L
T2 = 4π2 L/g
L = m
L = (T2g)/(4π2)

13. Mass and spring system F and Accel at max in the + direction
T = period (sec)
m = mass (kg)
k = spring constant (N/m)
F and Accel = 0
velocity at max
F and Accel at max
in the + direction
velocity = 0
F and Accel at max
in the neg. direction
velocity = 0

14. Example: The suspended mass makes 30 complete oscillations in 15 s
Example: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? x F
Or 2.0 s-1

15. Amplitude
Amplitude - the maximum displacement of the medium measured from the rest (equilibrium) position.
Proportional to the amount of energy carried by the wave
Emax = 1/2kx2 so the energy ~ amplitude2 or x2
Crest - highest point on waveform
maximum positive displacement of medium
Trough - lowest point on waveform
maximum negative displacement of medium

16. Energy is proportional to x2 -- where x is the displacement from equilibrium (amplitude)

17. Mechanical Waves
Waves that require a medium to transfer energy (a physical disturbance in a medium)
May be either transverse or compressional (longitudinal).
Examples:
Sound waves, water waves, earthquake waves, waves in a string

18. Wave Types Two types of periodic waves –
Transverse – the vibrations are perpendicular to the direction of motion
Longitudinal (compression) – the vibrations are parallel (in the same direction) as the direction of motion

19. Transverse Waves
particles in the medium vibrate perpendicular to the direction of wave motion or speed
Motion of particles
Motion of wave

20. Longitudinal Waves (Compression Waves)
particles in the medium vibrate parallel to the wave velocity
one wave is made up of alternating areas of high pressure (compressions) and low pressure (rarefactions) v
Motion of wave
Motion of particles

21. Comparison of transverse and longitudinal waves:
The “crest” on a transverse wave (max amplitude)
corresponds to the compression point on a
longitudinal wave.
The “trough” on transverse corresponds to the
rarefaction point on a longitudinal wave.

22. Wavelength The symbol “λ” for wavelength is “lambda”
The distance between corresponding points on consecutive waves.
The symbol “λ” for wavelength is “lambda”
This is a distance measurement (meters).

23. v = lf Wave Speed velocity of a wave as it travels through a medium
Mechanical wave: medium determines the speed
EM wave: velocity constant (“c”) in a vacuum
v = lf

24. Example 2: A vibration sends waves down a string
Example 2: A vibration sends waves down a string. The vibrator makes 600 complete cycles in 5 s. For one complete vibration, the wave moves a distance of 20 cm. What are the frequency, wavelength, and velocity of the wave?
f = 120 Hz
The distance moved during a time of one cycle is the wavelength; therefore:
v = fl
v = (120 Hz)(0.20 m)
l = 0.20 m
v = 24.0 m/s

25. Wave Speed in a String
The wave speed v in a vibrating string is determined by the tension F and the linear density μ, or mass per unit length
v = speed of the transverse wave (m/s)
F = tension on the string (N)
m or m/L = mass per unit length (kg/m)

26. F = (0.20 kg)(9.8 m/s2) = 1.96 N (weight of mass)
Example 1: A 5 g section of string has a length of 2 m from the wall to the top of a pulley. A 200 g mass hangs at the end. What is the speed of a wave in this string?
200 g
F = (0.20 kg)(9.8 m/s2) = 1.96 N (weight of mass)

27. Wave Interactions

28. Superposition and Interference patterns
When two or more waves occupy the same space in a medium at the same time they interfere with each other.
The medium’s displacement will be the vector sum of the displacements caused by the individual waves.
After the waves pass each other
they return to original form
There are two types of interference patterns.

29. Constructive Interference
When two waves interfere in a manner such that the amplitude of the resultant wave is greater than the amplitude of the individual waves.

30. Destructive Interference
When two waves interfere in a manner such that the amplitude of the resultant wave is smaller than the amplitude of the individual waves.
Note: Complete cancellation does not always occur in destructive interference.

31. Reflection/Refraction and Boundaries
When a wave encounters the boundary between two media some will be reflected and some will be transmitted.
How much goes each way is determined by the relative densities of the media.
The transmitted pulses will always be upright (right side up)
NOTE: A BOUNDARY is a change between media. A BARRIER is a fixed
point that causes a reflection – no transmission.
(See Standing Waves)

32. Crossing Boundaries More Dense to Less Dense
When a wave is moving from a more dense medium to a less dense medium, the reflected pulse is upright. v
Before
More dense medium
Less dense medium v v
After

33. Crossing Boundaries Less Dense to More Dense
When a wave is moving from a less dense medium to a more dense medium, the reflected pulse is inverted. v
Before
Less dense medium
More dense medium v
After v

34. Cool link for boundaries Scroll down to the last two animations

35. Diffraction The bending of a wave around a barrier, such as an obstacle or opening.
Occurs when a wave passes an edge, passes through a narrow gap or goes past an object.
None of the properties of a wave are changed by diffraction. The wavelength, frequency, period and speed are same before and after diffraction.
The only change is the direction in which the wave is traveling.

36. Resonance
A condition that exists when the frequency of a force applied to a system matches the natural frequency of vibration of a second system that causes a dramatic increase in amplitude. Example: a guitar body. The string vibrates and produces a sound wave. This energy causes a “sympathetic” reaction in the sounding board of the guitar and amplifies the sound.

37. Doppler Effect
See “The Physics Classroom” –
Wave Interactions

38. Standing Waves

39. Standing Waves
Caused by a wave and its reflected wave interfering in the medium
Reflection caused when waved encounters a fixed barrier.
Wave pattern of alternating nodes and antinodes.
Nodes - areas of no displacement of the medium caused by destructive interference
Antinodes - areas of maximum displacement of the medium caused by constructive interference.
Note: The wavelength of a standing wave consists of 2 antinodes. 1 wavelength is shown.
node
Antinode

40. Only particular frequencies will produce a standing wave on a string of a certain length. The ends produce nodes.
Lowest frequency:
1 antinode = ½ λ
Note: 1 λ = 2L
Next possible frequency:
2 antinodes = 1 λ
Note: 1 λ = L
Next frequency:
3 antinodes = 1.5 λ
Note: 1 λ = 2/3 L

41. Possible Wavelengths for Standing Waves
General equation to find the
wavelength on a given length
of string.
n = 1, 2, 3, 4 etc
(or the number of antinodes)

42. Possible Frequencies f = v/l :
f = 1/2L
f = 2/2L
f = 3/2L
f = 4/2L
f = n/2L
General equation to find
the specific frequency that
will produce a standing
wave on a given length.
n = 1, 2, 3, 4 etc
(number of antinodes)
v = speed of a wave on the
string

43. Electromagnetic Waves
Waves that consist of an electric field and a magnetic field working to propagate it through free space (a vacuum)
No medium required
Best modeled using transverse waves
All EM waves travel at the speed of light, “c” c = 3 x 108 m/s
Examples:
light waves, microwaves, ultraviolet, infrared, radar, radio waves, gamma rays, x-rays, etc.