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Chapter 13

Vibrations and Waves

Units of Chapter 13

Simple Harmonic Motion Equations of Motion Wave Motion Wave Properties Standing Waves and Resonance

13.1 Simple Harmonic Motion

Displacement (

x ) is the

directed distance of the objectfrom equilibrium. Amplitude (

A ) is the maximum

displacement. Period (

T ) is the time for one full cycle. Frequency (

f ) is the number of

full cycles per second.

13.1 Simple Harmonic Motion

SI unit of frequency: hertz, Hz 1 Hz = 1 cycle/second^ ANIMATION:

Simple Harmonic Motion

13.1 Simple Harmonic Motion

The total energy of an object in simple harmonicmotion is directly proportional to the square of theamplitude of the object’s displacement.

13.1 Simple Harmonic Motion

This allows us to calculate the velocity as afunction of position:^ and the maximum velocity (at

x^ = 0):

13.2 Equations of Motion

An equation of motion gives the position of anobject as a function of time. Simple harmonic motion can be representedas a component of uniform circular motion:

ANIMATION:

Uniform Circular Motion

13.2 Equations of Motion

The equation of motion for the oscillatingobject is given by: Here, the only possibility is that

y^ = 0 at

t^ = 0.

More likely, we would want

y^ =

A^ at

t^ = 0; that

is, at

t^ = 0 the object would have its maximum displacement. In that case,

13.2 Equations of Motion

This figure shows the displacement as afunction of time.

13.2 Equations of Motion

These graphs showthe form of theequation of motionfor different initialconditions.

13.2 Equations of Motion

In the real world, friction will cause anoscillating object to slow down and eventuallystop. This is called damped harmonic motion.

ANIMATION:

Damped Harmonic Motion

13.3 Wave Motion

A wave pulse is adisturbance thatpropagates througha medium. Ittransfers energywithout transferringmatter; the energyis a combination ofkinetic and potentialenergy.

13.3 Wave Motion

Such a wave will have a sinusoidal form inboth time and space. Amplitude: maximum displacement Wavelength: distance between points havingthe same phase Frequency: number of waves passing persecond Period: time for one complete wave to pass

13.3 Wave Motion

Relationship between wave speed,wavelength, period, and frequency: