10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 1
PHY 113 C General Physics I11 AM – 12:15 PM TR Olin 101
Plan for Lecture 16:Chapter 16 – Physics of wave motion
1. Review of SHM2. Examples of wave motion3. What determines the wave velocity4. Properties of periodic waves
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 2
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 3
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is k ?k = m(2p)2
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 4
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the frequency of oscillations ?w=2pf=2p f=1 Hz
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 5
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is amplitude of the displacement ?xmax=2m
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 6
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the maximum velocity ?v(t)=-2(2p) cos(2pt+3p)vmax=4p
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 7
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the maximum acceleration ?a(t)=-2(2p)2 cos(2pt+3p)amax=8p2
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 8
Some comments on Simple Harmonic Motion
k
mktAtx
xmk
dtxddtxdmmakxF
2
2
2
2
2
ωthatprovided
)φω(cos)(:solutiongeneral
:equationaldifferenti
Suppose that you know: (in standard units)
x(t)=2 cos(2pt+3p)
What is the displacement at t=0.3 s ?x(0.3)=2 cos(2p(0.3)+3p)
=2(0.309)=0.618 m
m=1 kg
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 9
Some comments on driven Simple Harmonic Motion
k
)
)
)
mk
tmFtAtx
tFxmk
dtxd
dtxdmmatFkxF
2
220
02
2
2
2
0
ωwhere
sin/)φω(cos)(:solutiongeneral
sin:equationaldifferenti
sin
w
F(t)=F0sin(t)
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 10
Webassign question (Assignment 14)
Damping is negligible for a 0.175-kg object hanging from a light, 6.30-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.430 m?
)mktmFtAtx
2
220 ωsin/)φω(cos)(
w
) 00 amplitudesin)( XtXtx
0
022
00 /
/mXF
mk
mkmFX
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 11
The wave equation
Wave variable
What does the wave equation mean?
Examples
Mathematical solutions of wave equation and descriptions of waves
2
22
2
2
xyc
ty
),( txyposition time
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 12
Source: http://www.eng.vt.edu/fluids/msc/gallery/gall.htm
Example: Water waves
Needs more sophistocatedanalysis:
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 13
Mechanical waves occur in continuous media. They are described by a value (y) which changes in both time (t) and position (x) and are characterized by a wave velocity c: y=f(x-ct) or y=f(x+ct)
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 14
Waves on a string:
Typical values for c:
3x108 m/s light waves
~1000 m/s wave on a string
343 m/s sound in air
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 15
Transverse wave:
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 16
Longitudinal wave:
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 17
General traveling wave –
t = 0
t > 0
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 18
iclicker question:
t=0t=1 s t=2s
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 19
Basic physics behind wave motion --
example: transverse wave on a string with tension T and mass per unit length m
y
ABT
dtydx
xm
TTdtydm
xy
xyμ
μ
θsinθsin
2
2
AB2
2
qBy
x
BBB x
y
θtanθsin
2
2
0 xy
xy1
xy
xLimABx
2
2
2
2
μ xyT
ty
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 20
The wave equation:
Solutions: y(x,t) = f (x ct)
2
22
2
2
xyc
ty
string)a(for
μwhere Tc
function of any shape
22
22
2
2
2
2
2
22
2
2
2
2
Let :Note
cuuf
tu
uuf
tuf
uuf
xu
uuf
xuf
xu
uuf
xuf
ctxu
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 21
iclicker question
Is it significant to write the wave equation with the special symbols?
A. YesB. No
2
22
2
2
xyc
ty
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 22
Examples of solutions to the wave equation:
Moving “pulse”:
Periodic wave:
20),( ctxeytxy
φsin),( 0 ctxkytxy
“wave vector”not spring constant!!!
ωπ2π2λπ2
fT
kc
k
cTT
txytxy
λφλ
π2sin),( 0
phase factor
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 23
cTT
txytxy
λφλ
π2sin),( 0
Periodic traveling waves:
Amplitude
wave length (m)
period (s); T = 1/f
phase (radians)
velocity (m/s)
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 24
Snapshot of periodic wave at t=t0
Time plot of periodic wave at x=x0
l
1/f
f l = c
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 25
φ
λπ2sin),(φ
λπ2sin),( 00 T
txytxyTtxytxy leftright
Ttxytxtxy leftright
π2cosφλπ2sin2),(y),( 0
“Standing” wave:
Combinations of waves (“superposition”)
BABABA 21
21 cossin2sinsin
: thatNote
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 26
Summary of wave properties:
constant)al(fundament/102.9979:fieldsmagneticandelectriccoupled todueLight wave:Example
:density andpressureair with in waveSound:Example
:h mass/lengtandion with tensstringaon Wave:Example
occurs.avein which wmediumand/or processon thedependsspeedWave
8 smc
pcp
TcT
c
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 27
Example from webassign:
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 28
cTT
txytxy
λφλ
π2sin),( 0
Periodic traveling waves:
Amplitude
wave length (m)
period (s); T = 1/f
phase (radians)
velocity (m/s)
10/22/2013 PHY 113 C Fall 2013 -- Lecture 16 29
Example from webassign:
0
0
0
2:speede transversmaximum
φλ
π2cos2),(
λφλ
π2sin),(
yT
Ttxy
Tttxy
cTT
txytxy
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