chapter 15 - waves traveling waves –types –classification –harmonic waves –definitions...
TRANSCRIPT
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