Post on 02-Oct-2014
Zaza Metreveli - experimental particle physicist
Tech F154 (first floor)
847 491 8631 (office phone)
Office hours: Tuesday (11:00-12:00)
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Friday (3:00-5:00)
TA - Boris Harutyunyan (Thursdays)
Required: University Physics with Modern Physics
by Young and Freedman
12th edition, Pearson Addison Wesley
1
12th edition, Pearson Addison Wesley
Lab Manual
Physics 135-3 Problem Manual
by D. Brown (availability TBA)
Final Exam: Monday, March 12, 9-11 am
I wish a great success to all of you.
Mechanical Waves
Chapter 15
University Physics, 12th edition
Hugh D. Young and Roger A. Freedman
Lectures by Zaza Metreveli
2
Goals for Today’s Lecture
� Study waves and their properties
� Define wave functions and wave dynamics
� Calculate the power in a wave� Calculate the power in a wave
� Consider wave superposition
� Study standing waves on a string
3
Mechanical Waves - Definition
A mechanical wave is a disturbance that travels through some material or substance called the medium for the wave.
As the wave travels, the particles that make up the medium
undergo displacements of various kind, depending on the nature of the wave. nature of the wave.
Transverse wave – displacements of the medium are perpendicular to the direction of travel of the wave.
Longitudinal wave – displacements of the medium are parallel to thedirection of travel of the wave.
4
Types of Mechanical Waves
Transverse
Longitudinal
Both Transverse
And Longitudinal5
Transverse wave
Longitudinal wave
Types of Mechanical Waves - Demonstration
Longitudinal wave
Both transverse and
longitudinal
6
General Common Characteristics of Mechanical Waves
Disturbance propagates (travels) with a definite speed through the medium. This speed is called wave speed.
The medium itself does not travel through space. Its individual
particles undergo back-and-forth or up-and-down motions around the equilibrium positions.around the equilibrium positions.
To set any of these systems into motion, we have to put in energy by doing mechanical work on the system.
Wave transports energy, but not matter, from one region to another.
7
Periodic Waves - Parameters
A detailed look at periodic transverse
waves will allow us to extract parameters.
A – amplitude [m, cm]
magnitude of change in the
oscillating variable.
λ – wavelength [m, cm]
distance over with the wave’s shape
repeats.
periodic sinusoidal waves
T – period [s]
length of time taken by one cycle.
– frequency [Hertz, 1/s]
event repeats per second.f
ω – angular frequency [rad/s]
rate of change of angular displacement.
repeats.
fπω 2=ω
π21==
fT
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Periodic Waves - Parameters
• Wave travels with constant speed
• In time interval T wave travels distance λλλλ
υ
T
λυ =
f1
=Since
Tf
1=
fλυ =
This fundamental equation holds for
all types of periodic waves.
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Question:
If you double the wavelength λ of a wave on a string, what happens
to the wave speed v and the wave frequency f?
A. v is doubled and f is doubled.
B. v is doubled and f is unchanged.
C. v is unchanged and f is halved.
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C. v is unchanged and f is halved.
D. v is unchanged and f is doubled.
E. v is halved and f is unchanged.
If you double the wavelength λ of a wave on a string, what happens
to the wave speed v and the wave frequency f?
Answer:
A. v is doubled and f is doubled.
B. v is doubled and f is unchanged.
C. v is unchanged and f is halved.
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C. v is unchanged and f is halved.
D. v is unchanged and f is doubled.
E. v is halved and f is unchanged.
fλυ =
Mathematical Description of a Wave
Wave function – determines the position of any particle of wave at any time.
),( txyy =We consider simple harmonic
motion (SHM) – sinusoidal wavesftAtAtxy πω 2coscos),0( ===
)](cos[),(υ
ωx
tAtxy −=
)(2cos)](cos[),( tx
fAtx
Atxy −=−= πω )(2cos)](cos[),( tx
fAtx
Atxy −=−=υ
πυ
ω
)(2cos),(T
txAtxy −=
λπ
Wave number [rad/m]λ
π2=k
kυω =
Graphing the wave function
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)cos(),( tkxAtxy ω−=
Mathematical Description of a Wave
)cos(),( tkxAtxy ω−= - sinusoidal wave moving in +x direction
)cos(),( tkxAtxy ω+= - sinusoidal wave moving in -x direction
for t=0kxAtxy cos)0,( ==
tAtAtxy ωω cos)cos()0,( =−== for x=0tAtAtxy ωω cos)cos()0,( =−== for x=0
tkx ω± is called the phase.
If phase = 0, 2π, 4π, … y=A
If phase = π, 3π, 5π, … y=-A
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Which of the following wave functions describe a wave that moves
in the –x-direction?
A. y(x,t) = A sin (–kx – ωt)
B. y(x,t) = A sin (kx + ωt)
C. y(x,t) = A cos (kx + ωt)
Question:
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C. y(x,t) = A cos (kx + ωt)
D. both B. and C.
E. all of A., B., and C.
Answer:
Which of the following wave functions describe a wave that moves
in the –x-direction?
A. y(x,t) = A sin (–kx – ωt)
B. y(x,t) = A sin (kx + ωt)
C. y(x,t) = A cos (kx + ωt)
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C. y(x,t) = A cos (kx + ωt)
D. both B. and C.
E. all of A., B., and C.
)cos(),( tkxAtxy ω+=
Particle Velocity and Acceleration in a Sinusoidal Wave
� υy - transverse velocity of any particle in a transverse wave, partial derivative
of with respect to time:
)sin(),(
),( tkxAt
txytxy ωωυ −=
∂
∂=
� ay – acceleration of any particle in a transverse wave, second partial derivative
of with respect to time:
),()cos(),(
),(22
2
2
txytkxAt
txytxa y ωωω −=−−=
∂
∂=
),( txy
),( txy
t∂
x
txy
∂
∂ ),(- slope of the string
2
2),(
x
txy
∂
∂- curvature of the string
),()cos(),( 22
2
2
txyktkxAkx
txy−=−−=
∂
∂ω
2
2
2
22
22
/),(
/),(υ
ω==
∂∂
∂∂
kxtxy
ttxy
2
2
22
2),(1),(
t
txy
x
txy
∂
∂=
∂
∂
υWave Equation – very important
16
Particle Velocity and Acceleration in a Sinusoidal Wave
t=0
17
Speed of a Transverse Wave
F – tension [N]
µ − linear mass density [kg/m]
- Increasing tension increases
the wave speed.
- Increasing mass decreases
the wave speed.F
Fy is constant
1
2
tF υ υ 1⇒=t
t
F
F yy
υ
υtFtF
y
yυ
υ=
m = µ [mass per unit length] υt [length] = µυt
yy mtF υ= (impulse-momentum theorem) y
yttF υµυ
υ
υ=
µυ
F=
equilibrumtoreturntheresistinginertia
equlibrumtosystemreturningforserestoring
_____
_____=υ
18• Wave speed v does not depend on vy, also on amplitude and frequency.