Announcements
§ smartPhysics homework deadlines have been reset to 3:30 PM on December 15 (beginning of final exam). You can get 100% credit if you go back and correct ANY problem on the HW from the beginning of the semester!
§ Last year’s final exam has been posted
§ Final exam is worth 200 points and is 2 hours:
Ø Tuesday December 15, 3:30 – 5:30 pm
Ø Room: JFB 101 & 103 (both)
§ Grade Estimation:
Ø You can do it on your own!
Ø I will upload an Excel Spreadsheet to help.
Mechanics Lecture 23, Slide 1
Lecture 23: Harmonic Waves and
the Wave Equation
Today’s Concept:
Harmonic Waves
Mechanics Lecture 23, Slide 2
What is a Wave?
A wave is a traveling disturbance that transports energy but not matter.
Examples:
Ø Sound waves (air moves back & forth)
Ø Stadium waves (people move up & down)
Largest in history???
Ø Water waves (water moves up & down)
Ø Light waves (what moves?)
Mechanics Lecture 23, Slide 3
Types of Waves
Transverse: The medium oscillates perpendicular to the direction the wave is moving. This is what we study in Physics 2210.
Longitudinal: The medium oscillates in the same direction as the wave is moving.
Mechanics Lecture 23, Slide 4
From sinusoidal function to traveling wave
Start with a drawing of cos(✓)
✓ = 2⇡
−2π
−π
π
2π
θ
Mechanics Lecture 23, Slide 5
From sinusoidal function to traveling wave
Start with a drawing of cos(✓)
Write angle as a function of position: ✓ = kx ! cos(✓) = cos(kx)
Ø Wavelength:
⇣ x⌘
Ø Wave number: k = 2⇡/
! cos(kx) = cos 2⇡
x=
x
Mechanics Lecture 23, Slide 6
From sinusoidal function to traveling wave
Start with a drawing of cos(✓)
Write angle as a function of position: ✓ = kx ! cos(✓) = cos(kx)
Ø Wavelength:
⇣ x⌘
Ø Wave number: k = 2⇡/
! cos(kx) = cos 2⇡
Shift the wave by adding a “phase angle”: cos(kx)
! cos(kx
)
x=
x
Mechanics Lecture 23, Slide 7
From sinusoidal function to traveling wave
Start with a drawing of cos(✓)
Write angle as a function of position: ✓ = kx ! cos(✓) = cos(kx)
Ø Wavelength:
⇣ x⌘
Ø Wave number: k = 2⇡/
! cos(kx) = cos 2⇡
Shift the wave by adding a “phase angle”: cos(kx)
Let the phase angle vary in time:
= !t
! cos(kx
! cos(kx
)
!t)
x=
x
Mechanics Lecture 23, Slide 8
From sinusoidal function to traveling wave
Multiply by an amplitude to give physical dimensions:
cos(kx
!t)
! y(x, t) = A cos(kx
!t)
y
x=
A
x
Mechanics Lecture 23, Slide 9
Wave Properties
Wavelength: The distance λ between identical points on the wave.
Amplitude: The maximum displacement A of a point on the wave.
Period: The time P it takes for an element of the medium to make one complete oscillation.
y = A cos(kx − ωt )
λ
A
2π
= 2π f
ω=
P
k=
2π
λ
Mechanics Lecture 23, Slide 10
Wave speed vs. Rope speed
A
B
C
D
Imagine we have a sinusoidal wave on a rope:
Student question: “what is k doing here?”
y = A cos(kx − ωt )
λ
A
x
The horizontal axis shown is:
A) A time axis.
k=
2⇡
B) A space axis.
In this situation, k is called the wavenumber or wavevector. It has units of inverse length and characterizes how the wave varies in space (i.e., when time is “frozen”).
Mechanics Lecture 23, Slide 11
Wave speed vs. Rope speed
A
B
C
D
Imagine we have a sinusoidal wave on a rope:
y
λ
A
x
If y = A cos ( kx − ω t ) then Aω sin ( kx − ω t ) gives: A) The horizontal speed of the wave.
B) The vertical speed of a segment of rope.
Aω sin ( kx − ω t )
This is the speed of a segment of rope at position x and time t.
Mechanics Lecture 23, Slide 12
Wave speed vs. Rope speed
Imagine we have a sinusoidal wave on a rope:
y
λ
A
x
To find the speed of the wave, first pick out a distinctive feature of the wave (e.g., a peak).
That feature corresponds to a particular value of the cosine (or sine) function, which does not change in time:
y = A cos(kx
!t)
Constant in time
kx
!t = constant
constant + !t
x=
k
dx
vwave =
dt
!
= = f
k
Mechanics Lecture 23, Slide 13
ACT
If a function moving to the right with speed v is described by f (x - vt) then what describes the same function moving to the left with speed v?
A
B
C
D
y = f (x − vt)
y
v
x
0
y
A) y = - f (x - vt)
B) y = f (x + vt)
v
0
x
C) y = f (-x + vt)
Mechanics Lecture 23, Slide 14
Suppose the function has its maximum at f (0). y
y = f (x - vt)
x – vt = 0
x = vt
v
x
0
y
A) y = - f (x - vt) x – vt = 0
x = vt
v
0
v
x
B) y = f (x + vt)
C) y = f (-x + vt)
x + vt = 0
x = -vt
-x + vt = 0
x = vt
v
y
v
v
x
0
Moves to the right when the signs in front of the x and t terms are different
Moves to the left when the signs in front of the x and t terms are the same
Mechanics Lecture 23, Slide 15
CheckPoint: Wave Equation
We have shown that the functional form y(x,t) = Acos(kx − ωt)
represents a wave moving in the +x direction. y
A
x
Which of the following represents a wave moving in the –x direction?
A) y(x,t) = Acos(ωt – kx)
B) y(x,t) = Asin(kx – ωt)
C) y(x,t) = Acos(kx + ωt)
A) y(x,t) = Acos(ωt − kx)
I have no clue but this is the only answer with a “-­‐” in front of the x. Plus, its derivative, velocity, is negative so that helps with my reasoning.
C) y(x,t) = Acos(kx + ωt)
As the wave reaches its amplitude, kx+wt=0; which means if t is increasing constantly at all times then x must increase constantly in the -­‐ direction as well.
Mechanics Lecture 23, Slide 16
How to make a Function Move
y
Suppose we have some function y = f (x):
x
0
y
f (x − a) is just the same shape moved
a distance a to the right:
x
0 x =a
Let a = vt Then f (x − vt) will describe the same shape moving to the right with speed v.
y
v
0
x
x = vt
Mechanics Lecture 23, Slide 17
Harmonic Wave
Consider a wave that is harmonic in x and has a wavelength of λ.
If the amplitude is maximum at
x = 0 it has the functional form:
y
λ
A
x
⎛ 2π ⎞
y ( x) = A cos⎜
x⎟
⎝ λ ⎠
Now, if this is moving to the right with speed v it will be described by:
y
v x
⎛ 2π
(x − vt )⎞⎟ = Acos(kx − ωt )
y ( x) = A cos⎜
⎝ λ
⎠
Mechanics Lecture 23, Slide 18
Transverse Wave on a String
Mechanics Lecture 23, Slide 19
CheckPoint: Wave Frequency 1
y
A
x
y
B
x
Waves A and B shown above are propagating in the same medium. How do their frequencies compare?
A) ωA > ωB
B) ωA < ωB
C) ωA = ωB
Mechanics Lecture 23, Slide 20
CheckPoint : Discuss and Re-vote
A
B
C
D
y
A
x
y
B
x
Waves A and B shown above are propagating in the same medium. How do their frequencies compare?
A) ωA < ωB
B) ωA = ωB
C) ωA > ωB
v=
s
T
!
=
= f=
µ
P
2⇡
Which one has the larger wavelength?
A) As they both have the same velocities, and the wavelength of A is larger, the frequency must be smaller than that of B. B) The tension and mass density are the same in both strings so they have the same frequency. C) It's period is shorter. Mechanics Lecture 23, Slide 21
ACT
A
B
C
D
The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 m/s. Suppose we make a sound wave and a light wave that both have a wavelength of 3 meters. What is the ratio of the frequency of the light wave to that of the sound wave?
A) About 1,000,000
B) About 0.000001
C) About 1000
v = f =) f =
flight =
vlight
light
v
fsound =
vsound
sound
flight
vlight
=
= 106
fsound
vsound
Mechanics Lecture 23, Slide 22
Energy
Mechanics Lecture 23, Slide 23
CheckPoint: Wave Frequency 2
y
A
x
y
B
x
Waves A and B shown above are propagating in the same medium with the same amplitude. Which one carries the most energy per unit length?
A) A
B) B
C) They carry the same energy per unit length
Mechanics Lecture 23, Slide 24
CheckPoint Results: Wave Frequency 2
y
A
x
y
B
x
Which one carries the most energy per unit length?
A) A B) B C) Same
2⇡v
!
=) ! =
vwave = f =
2⇡
s
T
Also, vwave is the same for both: vwave =
µ
2
2
So: !A < !B =) !A
< !B
Mechanics Lecture 23, Slide 25
Homework Problem
y = Asin(kx + ωt )
A
ω
f =
2π
2π
λ=
k
ω
v= fλ =
k
Mechanics Lecture 23, Slide 26
Homework Problem
v=
T
µ
T = v2µ
Mechanics Lecture 23, Slide 27
Homework Problem
ω
f =
2π
y = Asin(kx + ωt )
dy
vy =
= ω A cos(kx + ωt )
dt
dvy
ay =
= −ω 2 Asin ( kx + ω t )
dt
Mechanics Lecture 23, Slide 28
Homework Problem
ω
f =
2π
Average speed = distance traveled / time taken
Distance traveled by a piece of rope during one period is 4A
Mechanics Lecture 23, Slide 29