sections 1,2,4,5, i. outlook ii. what is wave? iii...
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Wave Motion
Wave MotionWave MotionWave MotionSections 1,2,4,5,
I. OutlookII. What is wave?III.Kinematics & ExamplesIV. Equation of motion – Wave equationsV. Examples
Wave Motion
OutlookOutlookTranslational and Rotational Motions
with Several physics quantitiesEnergy (E)Momentum (p)Angular momentum (L)
With Conservation lawsConservation of energyConservation of linear momentumConservation of angular momentum
Wave Motion
Wave Motion
What is Wave?What is Wave?VibrationVibration
Wave Motion
What is Wave?What is Wave?InterferenceInterference
Wave Motion
What is Wave?What is Wave?Mechanical vibration
Spring systemString fixed at both endsSound (vibration of air density)
Various Types of Waves Various Types of Waves Movie Wave1Movie Wave1
Water wave Movie Wave2Electromagnetic vibration PHYS208
Light
Wave Motion
Quick Look at KinematicsQuick Look at Kinematics
[Q] How can you describe the shape of the rope?[A] Well, I use period T (because they are SHOs)![Q] Anything else?
Continuum of Continuum of SHOsSHOsMovie Wave3Movie Wave3
KinematicsKinematicsy
x
Wave Motion
A
– A
2 sin )( ⎥⎤
⎢⎣⎡= xAxy
λπ
Wave number is“number of wavesin unit length”:
k = 2π/λSo, how many wavesin this case?
0 1 m
[Q] How can you describe the shape of the rope?[A] T, λ (or k), and A[Q] Anything else?
Wave Motion
x
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ
[Q] When did I take this snap shot?[Q] When did I take this snap shot?
Wave Motion
Pick one SHO and consider its motions.Pick one SHO and consider its motions.
x
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ
1
2 3
4
[Q] Can we find [Q] Can we find TT??[A] [A] ……
Wave Motion
x
y
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ [y] SHM angular frequency
[x] Motion with a constant velocity
Wave Motion
x
y
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ
22 sin ),(⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛−= tvxAtxy
321ω
λπ
λπ
42
2 λπλπ
=⇒= xx
vtx
tvx
+=⇒
=⎟⎠⎞
⎜⎝⎛−
4
2 22
λ
πλπ
λπ
The shape of the rope (wave)moves to +x direction.
Wave Motion
x
y
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ
2
2
2
2
2
4 22
4
⎥⎦⎤
⎢⎣⎡−=
⎥⎦⎤
⎢⎣⎡ −=
⎥⎦⎤
⎢⎣⎡ −=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛−==
xcosA
xsinA
vTxsinA
TvxsinA)Tt,x(y
λπ
πλπ
λπ
λπ
ωλπ
λπ
321
S.H.M.S.H.M.Math and PhysicsMath and Physics
Trig. functions:sin(θ + π/2) = cos(θ)sin(θ + π ) = −sin(θ)cos(θ − π/2) = sin(θ)
Derivative and integralTrig. functions
S.H.O.1) Spring plus block
HorizontalVertical
2) Pendulum
S.H.M.S.H.M.Math and PhysicsMath and Physics
Trig. functions:sin(θ + π/2) = cos(θ)sin(θ + π ) = −sin(θ)cos(θ − π/2) = sin(θ)
Derivative and integralTrig. functions
S.H.O.1) Spring plus block
HorizontalVertical
2) Pendulum
Wave Motion
[19] A transverse traveling wave (amplitude A, wave length λ, and frequency f) on a cord at t = 0 is represented by
y = A sin(2πx/λ + φ). Here φ is a constant phase factor.
(a) What will be the equation for a wave traveling to the left along the x axis as a function of x and t? [Hint] y(x,t) = ?
(b) What is its maximum acceleration of particles on the cord?[Hint] ay(x,t) = ?
Example 1
Wave Motion
Example 2[20] A transverse traveling wave on a cord is
represented by y(x,t) = 0.48 sin(0.56x + 84t)
where y and x are in meters and t in seconds. For this wave determine
(a) the amplitude, (b) wavelength, frequency, velocity (magnitude and
direction), (c) maximum and minimum speeds of particles of the
cord, and (d) maximum acceleration (magnitude) of the particles.
[A] …
Wave Motion
Example 3[23] A transverse wave pulse travels to the right along a
string with speed v = 2.0 m/s. At t = 0, the shape of the pulse is given by the function y = 0.45 cos(3.0x + 1.2) where y and x are in meters and t. For this wave determine
(a) the wavelength, frequency, and amplitude, (b) maximum and minimum speeds of particles of the
string, and(c) maximum and minimum accelerations (magnitudes)
of the particles.
[A] …
Wave Motion
Wave MotionWave MotionWave MotionSections 1,2,4,5,
I. OutlookII. What is wave?III.Kinematics & ExamplesIV. Equation of motion – Wave equationsV. More Examples
Wave Motion
x
y
A
– A
2 sin )0,( ⎥⎦⎤
⎢⎣⎡== xAtxy
λπ
[y] SHM angular frequency
[x] Wave propagation: moving with a constant velocity
22 sin ),(⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎟⎠⎞
⎜⎝⎛−= tvxAtxy
321ω
λπ
λπVisualizationVisualization
Wave Motion
Math & Physics Math & Physics –– Equation of Motions Equation of Motions ––
?) (2 sin )(
dd ) cos( )(
)( 21 )(
22
2
200
⇔⎥⎦⎤
⎢⎣⎡ −=
⇓
⎟⎟⎠
⎞⎜⎜⎝
⎛−=⇔=
⇓
=⇔++=
tvxAt,xy
xtxtAtx
m/Fatatvxtx
λπ
ωω
Wave Motion
Finding Wave EquationsFinding Wave Equations-- Transverse Wave on Rope Transverse Wave on Rope --
Consider a segment (mass m) of the rope under FT.
y
x
Wave Motion
Finding Wave EquationFinding Wave Equation
1
2
FT
FT
m
y
xConsider a segment (mass m)of the rope under FT.
Look at the vertical (y) motion.
Wave Motion
1
2
FT
FT
m
Finding Wave EquationFinding Wave Equationy
x
mF
a yy =
Wave Motion
Finding Wave EquationFinding Wave Equation
1
2
FT
FT
m
:velocity Wave
:Equation Wave
T
2
2T
2
2
µ
µ
Fv
xyF
ty
=
∂∂
=∂∂
y
xConsider a segment (mass m)of the rope under FT.
Look at the vertical (y) motion.
Wave Motion
Example 4[27] Determine if the function y = A sin (k x – ω t)
is a solution of the wave equation.
[A] …
Wave Motion
MathMath Differential Equation
[ ]
[ ]
[ ]
2
2
2
2
2
2
22
2
22
2
2
22
2
2
sin
[RH]
sin
[LH]
2 2 where
cos
) (2 sin )(
xy
kty
tkxAkxy
tkxAty
v,k
,tkxAty
xyv
tytvxAt,xy
∂∂
=∂∂
∴
⎪⎪⎩
⎪⎪⎨
⎧
−−=∂∂
−−=∂∂
==
−−=∂∂
∂∂
=∂∂
⇔⎥⎦⎤
⎢⎣⎡ −=
ω
ω
ωω
λπω
λπ
ωω
λπ
Solution of D.Eq.
Where is Physics?Where is Physics?
Wave Motion
[ ]
[ ]
[ ]
2
2
2
2
2
2
22
2
22
2
2
22
2
2
sin
[RH]
sin
[LH]
2 2 where
cos
) (2 sin )(
xy
kty
tkxAkxy
tkxAty
v,k
,tkxAty
xyv
tytvxAt,xy
∂∂
=∂∂
∴
⎪⎪⎩
⎪⎪⎨
⎧
−−=∂∂
−−=∂∂
==
−−=∂∂
∂∂
=∂∂
⇔⎥⎦⎤
⎢⎣⎡ −=
ω
ω
ωω
λπω
λπ
ωω
λπ
MathMath && Physics Physics –– Equation of Motions Equation of Motions ––
ay = –ω2 y (You have seen this!)
Wave Motion
Example 5[71] The figure shows the wave shape of a
sinusoidal wave traveling to the right at two instants of time. Find the mathematical representation of the wave?
[A] …
Wave Motion
Example 5
4
λ = 6 cm
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