1. Chapter 13
Vibrations and Waves

2. Periodic motion Periodic (harmonic) motion – self-repeating motion
Oscillation – periodic motion in certain direction
Period (T) – a time duration of one oscillation
Frequency (f) – the number of oscillations per unit time, SI unit of frequency 1/s = Hz (Hertz)
Heinrich Hertz
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3. Motion of the spring-mass system
Hooke’s law:
The force always acts toward the equilibrium position: restoring force
The mass is initially pulled to a distance A and released from rest
As the object moves toward the equilibrium position, F and a decrease, but v increases

4. Motion of the spring-mass system
At x = 0, F and a are zero, but v is a maximum
The object’s momentum causes it to overshoot the equilibrium position
The force and acceleration start to increase in the opposite direction and velocity decreases
The motion momentarily comes to a stop at x = - A

5. Motion of the spring-mass system
It then accelerates back toward the equilibrium position
The motion continues indefinitely
The motion of a spring mass system is an example of simple harmonic motion

6. Simple harmonic motion
Simple harmonic motion – motion that repeats itself and the displacement is a sinusoidal function of time

7. Amplitude
Amplitude – the magnitude of the maximum displacement (in either direction)

8. Phase

9. Phase constant

10. Angular frequency

11. Period

12. Velocity of simple harmonic motion

13. Acceleration of simple harmonic motion

14. The force law for simple harmonic motion
From the Newton’s Second Law:
For simple harmonic motion, the force is proportional to the displacement
Hooke’s law:

15. Energy in simple harmonic motion
Potential energy of a spring:
Kinetic energy of a mass:

16. Energy in simple harmonic motion

17. Energy in simple harmonic motion

18. Chapter 13
Problem 11
A simple harmonic oscillator has a total energy E. (a) Determine the kinetic and potential energies when the displacement is one-half the amplitude. (b) For what value of the displacement does the kinetic energy equal the potential energy?

19. Pendulums Simple pendulum: Restoring torque:
From the Newton’s Second Law:
For small angles

20. Pendulums
Simple pendulum:
On the other hand

21. Pendulums
Simple pendulum:

22. Pendulums
Physical pendulum:

23. Chapter 13
Problem 32
An aluminum clock pendulum having a period of 1.00 s keeps perfect time at 20.0°C. (a) When placed in a room at a temperature of –5.0°C, will it gain time or lose time? (b) How much time will it gain or lose every hour?

24. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

25. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

26. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

27. Simple harmonic motion and uniform circular motion
Simple harmonic motion is the projection of uniform circular motion on the diameter of the circle in which the circular motion occurs

28. Damped simple harmonic motion
Damping
force
Damping
constant

29. Forced oscillations and resonance
Swinging without outside help – free oscillations
Swinging with outside help – forced oscillations
If ωd is a frequency of a driving force, then forced oscillations can be described by:
Resonance:

30. Forced oscillations and resonance
Tacoma Narrows Bridge disaster (1940)

31. Wave motion A wave is the motion of a disturbance
All waves carry energy and momentum

32. Types of waves
Mechanical – governed by Newton’s laws and exist in a material medium (water, air, rock, ect.)
Electromagnetic – governed by electricity and magnetism equations, may exist without any medium
Matter – governed by quantum mechanical equations

33. Types of waves
Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:
Transverse – if the direction of displacement is perpendicular to the direction of propagation
Longitudinal – if the direction of displacement is parallel to the direction of propagation

34. Types of waves
Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as:
Transverse – if the direction of displacement is perpendicular to the direction of propagation
Longitudinal – if the direction of displacement is parallel to the direction of propagation

35. Superposition of waves
Superposition principle – overlapping waves algebraically add to produce a resultant (net) wave
Overlapping solutions of the linear wave equation do not in any way alter the travel of each other

36. Sinusoidal waves
One of the most characteristic solutions of the linear wave equation is a sinusoidal wave:
A – amplitude, φ – phase constant

37. Wavelength
“Freezing” the solution at t = 0 we obtain a sinusoidal function of x:
Wavelength λ – smallest distance (parallel to the direction of wave’s travel) between repetitions of the wave shape

38. Wave number
On the other hand:
Angular wave number: k = 2π / λ

39. Angular frequency Considering motion of the point at x = 0
we observe a simple harmonic motion (oscillation) :
For simple harmonic motion:
Angular frequency ω

40. Frequency, period
Definitions of frequency and period are the same as for the case of rotational motion or simple harmonic motion:
Therefore, for the wave velocity

41. Wave velocity
v is a constant and is determined by the properties of the medium
E.g., for a stretched string with linear density μ = m/l under tension T

42. Chapter 13
Problem 41
A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along the rope in 10.0 s. What is the wavelength?

43. Interference of waves
Interference – a phenomenon of combining waves, which follows from the superposition principle
Considering two sinusoidal waves of the same amplitude, wavelength, and direction of propagation
The resultant wave:

44. Interference of waves If φ = 0 (Fully constructive)
If φ = π (Fully destructive)
If φ = 2π/3 (Intermediate)

45. Reflection of waves at boundaries
Within media with boundaries, solutions to the wave equation should satisfy boundary conditions. As a results, waves may be reflected from boundaries
Hard reflection – a fixed zero value of deformation at the boundary – a reflected wave is inverted
Soft reflection – a free value of deformation at the boundary – a reflected wave is not inverted

46. Questions?

47. Answers to the even-numbered problems
Chapter 13
Problem 2
1.1 × 102 N
The graph is a straight line passing through the origin with slope equal to k = 1.0 × 103 N/m.

48. Answers to the even-numbered problems
Chapter 13
Problem 8
575 N/m
46.0 J

49. Answers to the even-numbered problems
Chapter 13
Problem 12
2.61 m/s
2.38 m/s

50. Answers to the even-numbered problems
Chapter 13
Problem 16
0.15 J
0.78 m/s
18 m/s2