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