Chapter 15 - Waves• Traveling Waves

– Types

– Classification

– Harmonic Waves

– Definitions

– Direction of Travel

• Speed of Waves• Energy of a Wave• Standing Waves

– Reflection and Transmission

– Superposition and Interference

• (Refraction and Refraction)

Types of Waves

• Mechanical Waves - Those waves resulting from the physical displacement of part of the medium from equilibrium.

• Electromagnetic Waves - Those wave resulting from the exchange of energy between an electric and magnetic field.

• Matter Waves - Those associated with the wave-like

properties of elementary particles.

Requirements for Mechanical Waves

• Some sort of disturbance

• A Medium that can be disturbed

• Physical connection or mechanism through which adjacent portions of the medium can influence each other.

Classification of Waves• Transverse Waves - The

particles of the medium undergo displacements in a direction perpendicular to the wave velocity– Polarization - The orientation

of the displacement of a transverse wave.

• Longitudinal (Compression) Waves - The particles of the medium undergo displacements in a direction parallel to the direction of wave motion.– Condensation/Rarefraction

Waves on the surface of a liquid

3D Waves

Sound Waves

http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Harmonic Waves• Transverse displacement looks like:

At t = 0

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6

x (m)

y (m

) A

2y x Asin x

Let the wave move

Traveling Wave

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8

x (m)

y (m

)

vt

2y x, t Asin x vt

Standing at the origin• Transverse displacement looks like:

At x = 0

-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6

t (sec)

y (m

)

Dm

2 2 2 v 2 2y x, t Asin 0 vt Asin 0 t Asin 0 t

T

Phase Velocity

distance moved in one cyclev f

time required for one cycle T

• Wave velocity is a function of the properties of the medium transporting the wave

That negative sign

• Wave moving right

• Wave moving left

2 2y x, t Asin x t

T

2 2y x, t Asin x t

T

Alternate notation

2 2y x, t Asin x t

T

y x, t Asin kx t

2k

2

T

Wave number

Angular frequency

2v

T 2 T k

Definitions• Amplitude - (A, ym) Maximum value of the displacement of a particle in a

medium (radius of circular motion).

• Wavelength - () The spatial distance between any two points that behave identically, i.e. have the same amplitude, move in the same direction (spatial period)

• Wave Number - (k) Amount the phase changes per unit length of wave

travel. (spatial frequency, angular wavenumber) • Period - (T) Time for a particle/system to complete one cycle.

• Frequency - (f) The number of cycles or oscillations completed in a period of time

• Angular Frequency - Time rate of change of the phase.

• Phase - kx -t Time varying argument of the trigonometric function.

• Phase Velocity - (v) The velocity at which the disturbance is moving through the medium

Velocity of transverse wave in a cord

yF t p

T

y

F v

F v

T

vF t vt v

v

TFv

General rule for wave speeds

Elastic Propertyv

Inertial Property

Young's modulus Ev

density

Bulk modulus Bv

density

Longitudinal wave in a long bar

Longitudinal wave in a fluid

Superposition

• Waves in the same medium will add displacement when at the same position in the medium at the same time.

• Overlapping waves do not in any way alter the travel of each other (only the medium is effected)

Superposition in reverse

• Fourier’s Theorem – any complex wave can be constructed from a sum of pure sinusoidal waves of different amplitudes and frequencies

Interference (Superposition of equal amplitude waves)

Constructive Destructivehttp://www.kettering.edu/~drussell/Demos/superposition/superposition.html

Interference of harmonic waves• Constructive - Waves are

in phase. Amplitude doubling occurs

• Destructive - Waves are 180 degrees out of phase. Amplitude cancellation occurs

Reflection

Fixed Boundary“Flips”

Free BoundaryDoesn’t flip”

http://www.kettering.edu/~drussell/Demos/reflect/reflect.html

Standing Waves - Resonance

ry x, t Asin kx t

ly x Asin kx t

y x, t Asin kx t Asin kx t

1 2 1 21 2sin sin 2sin cos

2 2

y 2Asin kx cos t

Nodes and Antinodes• Node – position of no

displacement• Antinode – position of

maximum displacement

y 2Asin kx cos t 2

kx x 0, , 2 ,3 ...

3x 0, , , ,.....

2 2

3 5 7x , , , .....

4 4 4 4

2 3 5 7kx x , , , ...

2 2 2 2

Natural frequencies

y 2Asin kx cos t

2kL L 0, , 2 ,3 ,4 ,....

n

2L n=1,2,3,....

n

nn

v nvf n=1,2,3,4,....

2L

TFv

Energy in a Wave

2 2 2maxP 2 vAf s

Intensity

2 2 2max

PI 2 vf s

Area

Two dimensional wave reflection

i r