PHYSICS 149: Lecture 23
• Chapter 11: Waves
–
–
–
–
Lecture 23
11.5 Mathematical Description of a Wave
11.6 Graphing Waves
11.7 Principle of Superposition
11.8 Reflection and Refraction
Purdue University, Physics 149
1
Midterm Exam 2
•
•
•
•
•
•
Wednesday, November 17, 6:30 PM – 7:30 PM
Place: PHY 114
Chapters 5 - 8
The exam is closed book.
The exam is a multiple-choice test.
There will be ~15 multiple-choice problems.
– Each problem is worth 10 points.
• Note that total possible score for the course is 1,000 points (see the
course syllabus)
• The difficulty level is about the same as the level of textbook
problems.
• You may make a single crib sheet
– you may write on both sides of an 8.5” × 11.0” sheet
Lecture 20
Purdue University, Physics 149
2
Simple Harmonic Motion
• Occurs when having linear restoring force F= -kx
– x(t) = [A] cos(ωt)
– v(t) = -[Aω] sin(ωt)
– a(t) = -[Aω2] cos(ωt)
• Springs
– F = -kx
– U = ½ k x2
– ω = sqrt(k/m)
• Pendulum (small oscillations)
– ω = sqrt(L/g)
Lecture 23
Purdue University, Physics 149
3
Types of Waves
• Transverse: The medium oscillates perpendicular to the
direction the wave is moving.
– Water (more or less)
– Slinky
energy transport
• Longitudinal: The medium oscillates in the same direction
as the wave is moving.
– Sound
– Slinky
energy transport
Lecture 23
Purdue University, Physics 149
4
Waves on a String
Lecture 23
Purdue University, Physics 149
5
Speed of Transverse Waves
• The speed of a transverse wave on a string is
where
No!
F: tension
m: mass of string
L: length of string
μ: mass density
• The speed of a transverse wave depends on
mechanical properties of the wave medium.
More restoring force makes faster waves; more
inertia makes slower waves.
Lecture 23
Purdue University, Physics 149
6
Periodic Waves
• A periodic wave is a wave that repeats the same
pattern over and over.
amplitude: A
• Period T: the time for a single point to repeat itself (that is, the time
for a single point to move from crest Æ equilibrium Æ trough Æ
equilibrium Æ crest). Or, the time for a pulse to move from a crest
to the next crest.
• Frequency f: the number of cycles (for a single point) per unit time
• Wavelength λ: the distance from a crest to the next crest
• Amplitude A: the maximum displacement of a single point from its
equilibrium position
Lecture 23
Purdue University, Physics 149
7
Example: Speed on a String
• A 2 m long string has a mass of m = 6.40 g.
What is the speed of transverse waves on this
string when its tension is 90.0 N?
Linear mass density:
μ = m / L = (6.40 × 10-3 kg) / (2 m)
= 3.20 × 10-3 kg/m
Speed of transverse waves:
v = sqrt(F/μ) = sqrt[ (90.0 N) / (3.20 × 10-3 kg/m) ]
= 168 m/s
Lecture 23
Purdue University, Physics 149
8
Harmonic Waves
• Harmonic waves: a special kind of periodic wave in which
the disturbance is sinusoidal (either a sine or cosine
function).
• In a harmonic transverse wave on a string, every point on
the string moves (in the direction perpendicular to the
direction of propagation of the wave) in simple harmonic
motion with the same amplitude and frequency, although
different points reach their maximum displacement at
different times.
Lecture 23
Purdue University, Physics 149
9
Harmonic Waves
y(x,t) = A sin(ωt – kx)
A = amplitude
ω = angular frequency
k = wave number = 2π/λ
Lecture 23
Purdue University, Physics 149
10
Amplitude and Wavelength
y(x,t) = A cos(ωt – kx)
Wavelength: The distance λ between identical points on the wave.
Amplitude: The maximum displacement A of a point on the wave.
Angular Frequency ω: ω = 2 π f
Wave Number k: k = 2 π / λ
Recall: f = v / λ
y
Wavelength
λ
Amplitude A
A
Lecture 23
Purdue University, Physics 149
x
11
Period and Velocity
l
l
Period: The time T for a point on the wave to undergo one
complete oscillation.
Speed: The wave moves one wavelength λ in one period T
so its speed is v = λ / T.
Lecture 23
Purdue University, Physics 149
12
Period and Frequency
•
Period T
– The time for a single point to repeat itself, or the time for a pulse to move from a
crest to the next crest
– Units: s, min, and so on
•
Frequency f
– The number of cycles (for a single point) per unit time
– Unit: Hz (note that 1 Hz = 1 cycle/s)
•
Wavelength λ
– The distance from a crest to the next crest
– Units: m, cm, and so on
•
Amplitude A
– The maximum displacement (distance) of a single point from its equilibrium
position
– Units: m, cm, and so on
•
v=fλ
Relationships
– The intensity of a wave is proportional to the square of its amplitude (I ∝ A2).
Lecture 23
Purdue University, Physics 149
13
Example: Periodic Waves
• What is the speed of a wave whose frequency
and wavelength are 500.0 Hz and 0.500 m,
respectively? And, what is the period of the
wave?
f = 500.0 Hz and λ = 0.500 m
v = f⋅λ = (500.0 Hz)×(0.500 m)
= 250 m/s
T = 1/f = 1 / (500 Hz)
= 2×10-3 s
Lecture 23
Purdue University, Physics 149
14
Harmonic Waves Exercise
y(x,t) = A cos(ωt – kx)
Label axis and tic marks if the graph
shows a snapshot of the wave
y(x,t) = 2 cos(4t – 2x) at x=0.
Recall: T = 2 π /ω
What is the period of this wave?
T = 2 π / ω = 2 π/ 4 = 1.58 s
+2
What is the velocity
of this wave?
π/4
-2
Lecture 23
π/2
3π/4
t
v = ω/k= 4/2 m/s=2m/s
Purdue University, Physics 149
15
Harmonic Waves
• Direction of Travel:
– Wave travels in the +x-direction, if ωt and kx terms
have the “different” signs.
– Wave travels in the –x-direction, if ωt and kx terms
have the “same” sign (that is, both positive or both
negative).
Lecture 23
Purdue University, Physics 149
16
Direction
A wave y = A cos(ωt - kx) travels in +x direction
A wave y = A cos(ωt + kx) travels in -x direction
Δt
Lecture 23
Purdue University, Physics 149
17
ILQ
•
What is the equation of a harmonic wave
traveling to the left (–x direction) with amplitude
5.0 mm, angular frequency 65 rad/s, and wave
number 4 rad/m ?
a)
b)
c)
d)
y(x,t) = (5.0 mm)⋅sin[(65 rad/s)t + (4 rad/m)x]
y(x,t) = (65 rad/s)⋅sin[(5.0 mm)t + (4 rad/m)x]
y(x,t) = (5.0 mm)⋅sin[(65 rad/s)t – (4 rad/m)x]
y(x,t) =(4 rad/m)⋅sin[(65 rad/s)t +(0.0050 m)x]
Lecture 23
Purdue University, Physics 149
18
Graphing Waves
• If t is held constant, the graph shows an instantaneous
picture of what the wave looks like at that particular
instant (that is, a snapshot).
• If x is held constant, the graph shows the motion of the
(single) point x as a function of time (t).
Lecture 23
Purdue University, Physics 149
19
Wave Properties
• The speed of a wave is a constant that depends only on the
medium, not on amplitude, wavelength or period (similar to
SHM)
λ and T are related !
λ = v T or λ = 2π v / ω
or λ = v / f
•
v=λ/T
(since T = 2π / ω )
(since T = 1/ f )
Recall f = cycles/sec or revolutions/sec
ω = 2πf
Lecture 23
Purdue University, Physics 149
20
ILQ
•
A boat is moored in a fixed location, and waves
make it move up and down. If the spacing
between wave crests is 20 m and the speed of
the waves is 5 m/s, how long does it take the
boat to go from the top of a crest to the bottom
of a trough?
a)
b)
c)
d)
1 second
2 seconds
4 seconds
8 seconds
Lecture 23
Purdue University, Physics 149
21
Wave Description
λ – wavelength: distance between crests (meters)
T – period: the time between crests passing fixed location (seconds)
v – speed: the distance one crest moves in a second (m/s)
f – frequency: the number of crests passing fixed location in one
second (1/s or Hz)
ω – angular frequency: 2πf: (rad/s)
Lecture 23
Purdue University, Physics 149
22
ILQ 1
A rope of length L and mass M hangs from a
ceiling. The rope is not an ideal rope. If the
bottom of the rope is pulsed to cause a wave to
travel up the rope, the speed of the wave as it
travels up
A) increases.
B) stays the same.
C) decreases.
Lecture 23
Purdue University, Physics 149
23
ILQ 2
Suppose a periodic wave moves through some medium. If the
period of the wave is increased, what happens to the
wavelength of the wave assuming the speed of the wave
remains the same?
A) The wavelength increases
B) The wavelength remains the same
C) The wavelength decreases
Lecture 23
Purdue University, Physics 149
λ=vT
24
Principle of Superposition
• When two or more waves overlaps, the net
disturbance at any point is the sum of the
individual disturbances due to each wave.
(ex. this can be seen by dropping two pebbles into a pond.)
Lecture 23
Purdue University, Physics 149
25
Interference and Superposition
• When two waves overlap, the
amplitudes add.
• y(x,t) = y1(x,t) + y2(x,t)
– Constructive: increases
amplitude
– Destructive: decreases
amplitude
Lecture 23
Purdue University, Physics 149
26
ILQ
•
Two waves travel towards each other with v =
10 cm/s. What will be the amplitude of the
resultant wave 1 second from now?
a)
b)
c)
d)
0.30 cm
0.50 cm
0.80 cm
1.30 cm
Lecture 23
Purdue University, Physics 149
27
Reflection
• Reflection occurs at a boundary between different
wave media.
• Some energy may be transmitted into the new
medium and the rest is reflected (that is, some or
all of the energy will travel back into the first
medium).
• How much of the energy is reflected depends on
how different the properties of the two media are
(in particular, on the wave speed in the two
media); the more different, the more reflection
takes place.
Lecture 23
Purdue University, Physics 149
28
Reflection
• When a wave travels from one
boundary to another, reflection occurs.
Some of the wave travels backwards
from the boundary
– Traveling from fast to slow inverted
– Traveling slow to fast upright
Lecture 23
Purdue University, Physics 149
29
Reflection Act
• A slinky is connected to a wall at one
end. A pulse travels to the right, hits the
wall and is reflected back to the left. The
reflected wave is
A) Inverted
B) Upright
– Fixed boundary reflected wave inverted
– Free boundary reflected wave upright
Lecture 23
Purdue University, Physics 149
30
Reflected Wave: Inverted or Not?
• The reflected wave will be inverted if it reflects from a
medium with a higher mass density (that is, with a lower
wave speed; recall
).
• The reflected wave will not be inverted if it reflects from a
medium with a lower mass density (that is, with a higher
wave speed).
Å This phenomenon can be explained by
(1) The principle of superposition
at the fixed point at the end, or
(2) Newton’s third law
for the forces between the string and the wall.
Lecture 23
Purdue University, Physics 149
31