1. Chapter 14 Notes
Vibrations and Waves

2. Section 14.1 Objectives
Use Hooke’s law to calculate the force exerted by a spring.
Calculate potential energy of an elastic spring.
Identify objects in simple harmonic motion.
Determine variables that affect the period of a pendulum.
Describe the affect of resonance on an object.

3. Periodic Motion Examples – clock pendulum, vibrating guitar string
Motions which repeat in a regular cycle
In these examples there is one point in the motion where the net force on the object is zero (equilibrium)
When the object is pulled away from its equilibrium position, the net force is no longer zero and the object is pulled back to the equilibrium position.

4. Simple Harmonic Motion
Described by two quantities:
Period
Amplitude
A motion that occurs when the restoring force on an object is directly proportional to the object’s displacement from equilibrium.
Restoring force – force bringing the object back to equilibrium
Period (T)—the time needed for an object to repeat one complete cycle of the motion
Amplitude—the maximum distance that the object moves from equilibrium

5. Pendulums
Consists of a massive object (bob) suspended by a string or light rod of length l.
The string exerts a tension force and the force of gravity pulls downward on the bob.

6. Example Problem
How long must a pendulum be on the Moon, where g=1.6 m/s2, to have a period of 2.0 s?
L=0.16 m

7. Resonance
Occurs when small forces are applied at regular intervals to a vibrating or oscillating object and the amplitude of the vibration increases.
The additions of small amounts of force at specific times in the motion of an object cause a larger and larger displacement
Examples: swinging, jumping on trampoline
Watch Mechanical Universe “Harmonic Motion” – Tacoma Narrows Bridge collapse

8. Section 14.2 Objectives
Differentiate between transverse and longitudinal waves.
Determine wave speed, wavelength, and frequency using corresponding equations.

9. Mechanical Waves Types Transverse Longitudinal Surface
Wave—a disturbance that carries energy through matter or space
Mechanical waves require a medium to travel through (air, water, string, etc.)
Transverse waves
Longitudinal waves
Surface waves

10. Transverse Waves
Wave that vibrates perpendicular to the direction of the wave’s motion
Wave pulse—a single bump or disturbance that travels through a medium
Periodic wave—wave that moves up and down at the same rate

11. Longitudinal Waves
The disturbance is in the same direction as the wave’s motion
Compressions and rarefactions

12. Surface Waves
Has characteristics of both transverse and longitudinal waves
Particles move parallel and perpendicular to the direction of the wave

13. Measuring a Wave Speed Amplitude Wavelength Phase Period Frequency
Speed—depends upon the material the wave moves through
Measure displacement of the wave’s peak and divide by the time interval
Amplitude—the maximum displacement of the wave from equilibrium
Depends upon how it is generated
Determines the amount of energy
Wavelength—the shortest distance between points where the wave pattern repeats itself
Wavelength = v/f
Phase—any two points on a wave that are one or more whole wavelengths apart
Period and Frequency
Only apply to periodic waves
Frequency—the number of complete oscillations per second
f = 1/T

15. Example Problem
A sound wave has a frequency of 192 Hz and travels the length of a football field, 91.4 m, in s.
What is the speed of the wave?
What is the wavelength of the wave?
What is the period of the wave?
If the frequency was changed to 442 Hz, what would be the new wavelength and period?
337 m/s
1.76 m

16. Section 14.3 Objectives
Relate a wave’s speed to the medium in which the wave travels.
Describe the motion of a wave as it encounters boundaries.
Apply the principle of superposition to wave interference.

17. Waves at Boundaries
Wave speed depends upon the properties of the medium it passes through
Water waves—depth of water affects speed
Sound waves—temperature of air affects speed
Waves on spring—spring’s tension and mass per unit length affects speed

18. Waves at Boundaries Free boundary Fixed boundary
Fixed boundary
Free-boundary—waves are reflected
when a pulse travels down a rope whose end is free to slide up the post, the pulse is reflected from the free end.
Fixed-boundary—waves are reflected and inverted
when a pulse travels down a rope that is fixed at one end, the reflected pulse is inverted.
Energy transmitted by the wave is reflected back after it hits the wall
Wave will have almost exactly the same amplitude as before

19. Superposition of Waves
Interference
Constructive
Destructive
Principle of Superposition—the displacement of a medium caused by two or more waves is the algebraic sum of the displacements caused by the individual waves
Interference—the result of the superposition of two or more waves
Constructive interference – completely in phase
Destructive – completely out of phase
Longitudinal waves—in a compression particles are moved closer together. In a rarefaction particles are moved farther apart.

20. Standing Waves
A wave that appears to be standing still, produced by the interference of two traveling waves moving in opposite directions.
Wave will be larger than vibrations due to constructive interference.
If you double the frequency of vibrations you will produce an additional node.
Node—The points at which complete destructive interference occurs
Antinode—midway between two adjacent nodes, vibrations found with the largest amplitudes
Standing wave demo