Chapter 15 Waves
I.
Wave Basics
A.
Water Waves
1) We can watch a single
wave move to shore
2) Motion is very regular
3) Water moves toward shore
but none accumulates on the beach: Complex Motion
4) Standing on beach, you can feel the KE of the waves
5) Waves in water are a familiar example
a) Light, radio, microwaves, infrared, sound, etc… are other examples
b) Wave motion explains may physics phenomena
B.
Wave Pulses
1) Simple example = slinky (spring)
2) Single, fast, compression at one end causes motion along the spring
a) Follow movement to opposite end; may even reflect back
b) Pulse = single wave
3)
What is really happening to cause and carry the Pulse?
a) Spring is in same position before and after the pulse
b) Parts of the spring move, taking turns as the pulse propagates
c) A local compression of the spring is what is moving
i. Compressed area switches places with relaxed areas
ii. Compressed area ends up at the opposite end
iii. Each individual loop moves forward and then back
C.
Features of Wave Motion
1) Medium = object or material through which a wave moves
a) A wave must use some kind of matter to move
b) Water waves use the water itself as the medium
2)
Disturbance within the material
a) Spring—compression of the loops
b) Water—up and down “displacement” of the water
Longitudinal Wave = displacement is parallel to
the direction of wave travel (spring compression,
sound)
Transverse Wave = displacement is
perpendicular to direction of wave travel (wave
on a string, light)
3)
Velocity = how fast the disturbance is traveling through the medium
a) Determined by the properties of the medium
b) Tension—tighter the spring, the faster the pulse travels
c) Density—denser the spring, the slower the pulse travels
4)
Transmission of Energy
a) Moving compression of a spring = KE + PE
b) Energy moves with the pulse
c) Water waves—noise, movement of the sand on the beach, etc…
E.
Periodic Waves
1) Wave = continuous set of pulses moving in the same way on the medium
2) Periodic Wave = equal times and distances between pulses
3) Period = time between to pulses of a periodic wave = T
4) Frequency = number of pulses per unit time = f = 1/T
a) Units: 27 per second = 27/s = 27 s-1 = 27 Hz
b) 1 Hertz = 1 Hz = 1 per second = 1/s-1
5) Wavelength = distance between pulses of a periodic wave = l (lambda)
6) Velocity of a periodic wave = frequency times the wavelength
d l
l
v  
 fl
t T 1/ f
II.
Small f = large l
Large f = small l
Waves on a Rope
A.
Rope similar to a very stiff spring
1) Hard to compress, so you don’t get longitudinal waves
2) Easy to move up and down, so you can make transverse waves
B.
Graphing Waves
1) At any given time, a single transverse pulse is easy to graph
2) Periodic waves can be more complicated
a) Complex periodic wave can be generated on a string by complex (but
still regular) motion of the hand up and down
b)
Simple Harmonic Motion gives a Harmonic Wave
i. Each point on the rope is moving in harmonic motion
ii. Simple sin graph, like in harmonic motion
iii. All complex waves can be broken down into the sum of several
harmonic waves = Fourier or Harmonic Analysis
C.
Velocity of a wave on a rope
1) Velocity does not depend on frequency (f) or the shape of the wave
2) Why do pulses move?
a) Initial Force, called Tension, starts the motion (acceleration)
b) Tension then acts on a neighboring area of the string, which gets
accelerated, and so on down the string
3)
Velocity depends on rate of acceleration at succeeding points on rope
a) F = ma or a = F/m
b) Large force causes a large acceleration to a fast velocity
mass
m
c) Large mass makes acceleration and velocity slower
m

Length
L
d) m = mass per unit length of the rope (similar to density)
e) Equation for velocity only includes F and m, not f or l
v
F
m
4)
v
Sample problem: A rope with a periodic wave on it has the following
properties L = 10 m, m = 2 kg, F = 50 N, f = 4 Hz
a) v = ?
F
50 N
50kgm / s 2



 250m2 / s 2  15.8m / s
m
m/ L
2kg / 10m
0.2kg / m
F
b)
l=?
v  fl  l 
v 15.8m / s

 3.95m
f
4/ s
III. Interference and Standing Waves
A.
B.
Interference
1) Waves on a rope reflect back and “interfere” with the input wave
2) Water waves reflect off of the beach and interfere with incoming waves
3) Interference = combination of two or more waves
Waves on a spliced rope
1) Principle of Superposition = when combining waves, the total
displacement is the sum of the individual wave displacements
2) Constructive Interference = combine waves of same f, l to give larger
total displacement (add together)
3) Destructive Interference = combine waves of same f, l to give smaller
total displacement (cancel out)
Constructive Interference
4)
5)
6)
B.
Destructive Interference
In Phase = two waves move the same way at the same time (constructive)
Out of Phase = two waves don’t move the same way at the same time
(destructive)
Amplitude = height of the displacement
a) Completely constructive—amplitude = 2 x individual amplitude
b) Completely destructive—amplitude = 0
c) Not quite in phase—amplitude somewhere between 0-2 x amplitude
Standing Waves
1) Above, we have multiple waves traveling in the same direction
2) Interaction of waves traveling in opposite directions
3) A periodic wave and its reflection are the simplest example (same f, l)
4) Apply the Principle of Superposition to see what happens to Amplitude
a)
At certain times, the two amplitudes
cancel out (+ and – added) for each
point
b)
Let the two waves move l/4 each,
and look again.
i. Points A still add to 0 amplitude
ii. Points B now have 2 x amplitude
Standing Wave = oscillating pattern in a rope when a periodic wave and
its reflection interfere to give a stable pattern of zero and 2 x amplitudes
c)
c)
d)
e)
f)
Node = point of no motion in a standing wave (points A)
Antinode = point of greatest motion in a standing wave (points B)
Node—Antinode distance = l/4
Node—Node distance or Antinode—Antinode distance = l/2