Physics 2111
Unit 23
Today’s Concept:
Waves
Wave Equation
Resonance
Energy
Mechanics Lecture 23, Slide 1
What is a Wave?
A wave is a traveling disturbance that transports energy but not matter.
Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Light waves (what moves?)
Mechanics Lecture 23, Slide 2
Types of Waves
Longitudinal: The medium oscillates in the same direction as the wave is moving.
Mechanics Lecture 23, Slide 3
Transverse: The medium oscillates perpendicular to the direction the wave is moving.
• Waves on a string
• Water waves
• Light Waves
• Sound Waves
How to make a Function Move
Suppose we have some function y = f (x):
x
y
0
Mechanics Lecture 23, Slide 4
f (x - a) is just the same shape moveda distance a to the right:
x = ax
y
0
Let a = vt Then
f (x - vt) will describe the same shape moving to the right with speed v.
x = vt
v
x
y
0
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?
v
x
y
0
y = f (x - vt)
v A) y = - f (x - vt)
B) y = f (x + vt)
C) y = f (-x + vt)
Question
Mechanics Lecture 23, Slide 5
x
y
0x
y
0x
y
Harmonic Wave
Consider a wave that is harmonic in xand has a wavelength of .
Mechanics Lecture 23, Slide 6
Has the functional form:
xAxy
2cos)(
Harmonic Wave
Mechanics Lecture 23, Slide 7
2
( ) cosy x A x vt
-
cos( )A kx t -
Give it speed v
=Acos(2𝜋
𝜆𝑥 −
2𝜋
𝜆𝑣𝑡)
v=/P
Amplitude:
The maximum displacement A of a point on the wave.
A
Wavelength: The distance between identical points on the wave.
Period: The time P it takes for an element of the medium to make one complete oscillation.
cos( )y A kx t -
Pv
k
2k
f2
P
2
Wave Properties
Mechanics Lecture 23, Slide 8
Wavelength
Not the spring constant!
position(x)
NOTE:
Wave Properties
Mechanics Lecture 23, Slide 9
PeriodP
Same plot but vs time
time(t)
Question
Mechanics Lecture 23, Slide 10
A boat is moored in a fixed location and waves make it move up
and down. If the spacing between wave crests is L and the
speed of the waves is v, how much time Δt does it take the boat
to go from the top of one wave to the top of the next?
A. Δt = L/v
B. Δt = Lv
C. Δt = v/L
You shake a rope up and down a distance of 40cm with a frequency of 10Hz. The wave formed has a velocity of 5m/sec. What is the equation for this wave?
Example 23.1 (Traveling Wave)
Mechanics Lecture 23, Slide 11
Example 23.2: Wave Graph
Mechanics Lecture 11, Slide 12
The figure to the left shows
height vs. displacement plot for a
string which has a wave traveling
in the positive x direction at time
t=5.0 sec with a velocity of 12.0
m/sec.
a) What is the amplitude of this wave?
b) What is the wavelength of this wave?
c) What is the frequency of this wave?
d) What is the equation of motion of this string? (i.e. y(x,t)=?)
e) What is the vertical (y) velocity of a piece of string at the
point labeled 1?
f) What is the vertical (y) acceleration of a piece of string at
the point labeled 1?
Mechanics Lecture 23, Slide 13
You have a 2m long piece of string that has a mass of 0.6 grams. If you loop it over a pulley and hang a 200gram mass from one end, what is the velocity of wave in the string?
Example 23.2 (Wave Speed)
Mechanics Lecture 23, Slide 14
Linear Superposition
What happens when two waves “collide”?
x
Mechanics Lecture 23, Slide 15
x
Constructive Interference
Or……
Destructive Interference
x
Mechanics Lecture 23, Slide 16
x
Demo
http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html
Y(x,t) = 2*cos(kx)*cos(t)
In the equation we just derived, when is the value of Y always equal to zero?
A. When kx = 0
B. When kx = /2
C. When kx =
D. (A) and (C)
E. Y varies with time and isn’t always zero anywhere
Question
Mechanics Lecture 23, Slide 17
Resonance
Mechanics Lecture 23, Slide 18
Fundamental
1st harmonic
1st Overtone
2nd harmonic
2nd Overtone
3rd harmonic
L
L = n* /2Resonant frequencies for strings always node at both ends
CheckPoint
Mechanics Lecture 23, Slide 19
L
The wave length of the above wave is:
A. 2 L
B. 1.5 L
C.1 L
D.2/3 L
E. 0.5 L
L = n* /2
Resonant frequencies
Mechanics Lecture 23, Slide 20
L = n* (v/f)/2
f= v*n/(L 2)
Velocity -
determined
by tension
and density
of string
(v2 = T/m
Length of
string
But what is n?
Resonance
Mechanics Lecture 23, Slide 21
Fundamental
1st harmonic
1st Overtone
2nd harmonic
2nd Overtone
3rd harmonic
L
f= v*n/(L 2)
n = 1
n = 2
n = 3
When you tune a guitar, you are adjusting what resonant frequency at which each string will vibrate. You do this by:
A. Adjusting the length of the string
B. Adjusting the density of the string
C. Adjusting the tension in the string
D. Adjusting the speed of sound near the string
Question
Mechanics Lecture 23, Slide 22
When you play a guitar, you are also adjusting what resonant frequency at which each string will vibrate. You do this by:
A. Adjusting the length of the string
B. Adjusting the density of the string
C. Adjusting the tension in the string
D. Adjusting the speed of sound near the string
Question
Mechanics Lecture 23, Slide 23
In our previous example problem, we had a 2m long piece of string that has a mass of 0.6 grams. We looped it over a pulley and hung a 200gram mass from one end. What is fundamental frequency of the string?
Example 23.3 (fundamental frequency)
Mechanics Lecture 23, Slide 24
5m
The velocity we find using the formula
is the velocity of the wave.
Velocity
Mechanics Lecture 23, Slide 25
2 Tvm
The velocity of any little piece of the string varies with time. We get it by taking the derivative of y(x,t) = A*cos(kx-t)
Energy
Mechanics Lecture 23, Slide 26
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