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Chapter 2 Wave motionAugust 25,27 Harmonic waves

2.1 One-dimensional wavesWave: A disturbance of the medium, which propagates through the space, transporting energy and momentum.Types of waves: Mechanical waves, electromagnetic (EM) waves.Longitudinal waves, Transverse waves. Question: The type of wave in the corn field in Macomb, IL. Suppose the wind is weak.

Mathematical description of a wave: For a wave that does not change its shape:

x0

t = 0

x0

tvt

Disturbance is a function of position and time: (x, t) =f (x, t)Example: E(x, t) and B(x, t) of light

(x, 0) =f (x, 0)=f (x) (wave profile, snapshot)(x, t) =f (x-vt) (General form of a wave)Example: (x, t) = exp[-a(x-vt)2]

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2.1.1 The differential wave equation

2

2

22

2 111

1

)(

)(),(

tvxtvtxvx

tvxvtx

vtxftx

2

2

22

2 1

tvx

2

2

22

2

2

2

22

2

1

1

wavesEM :

t

B

cx

B

t

E

cx

E

Example

zz

yy

*A partial, linear, second order, homogeneous differential equation

Specifying a wave: Amplitude and wavelength Second order differential equation

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2.2 Harmonic wavesHarmonic waves:(x, t) = f (x-vt) =A sin k(x-vt)

vkv

k

txtx

txtx

vtxkAtx

2

2

),(),(

),(),(

)(sin),(

Parameters: • Amplitude: A• Wavelength: • Wave vector (propagation number): k = 2/• Period: = /v• Frequency: = 1/• Speed of wave: v=• Angular frequency: =22/• Wave number : =1/

)sin(

)(sin),(

tkxA

vtxkAtx

Real waves:Monochromatic waves Band of frequencies:Quasi-monochromatic wavesRemember all of them by heart.

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2.3 Phase and phase velocityGeneral harmonic wave functions:(x, t) =A sin(kx-t+)Phase: (x, t)= kx – t + Initial phase: (x, t)|x=0, t=0=

Rate-of-change of phase with time:

xt

Rate-of-change of phase with space: kx t

Phase velocity: The speed of propagation of the condition of constant phase.

kt

xvt

kxtkx

t

xv

)(1

tx xtv

In general

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Read: Ch2: 1-3Homework: Ch2: 4,17,18,29,34,35Due: September 5

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August 29 Addition of waves

2.4 The superposition principleSuperposition principle:The total disturbance from two waves at each point is the algebraic sum of the individual waves at that point.

221

2

2221

2

22

2

222

2

21

2

221

2

)(1)(1

1

tvxtvx

tvx

Superposition of harmonic waves Interference: in-phase, out-of-phase

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2.5 The complex representation

Real harmonic wave: ]Re[)cos(),( )( tkxiAetkxAtx

Complex representation: )(),( tkxiAetx

• The actual wave is the real part.•Easy to manipulate mathematically, especially in the addition of waves.•Use with care when perform multiplication of waves.

2.6 Phasors and the addition of waves

itkxi AeAetx )(),(Harmonic wave:

Phasor: A rotating arrow (vector) that represents the waveThe addition of waves = the addition of vectors.

AAA

AeeAeA iii

21

2121

A1

A2

A

2

1

Re()

Im()

A

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2.7 Plane waves

constant : ˆˆˆ rkkjik zyx kkk

Equation for a plane perpendicular to

Wavefront: The surface composed by the points of equal phase at a given time.Plane wave: Waves whose wavefronts are planes.

x

y

z

kr

Description of a plane wave:

rkrrkr iAeA )(or ),cos()(

2

2)(

kk

k

krr

k: propagation vector (wave vector).

Including time variable: )(),( tiAet rkr

In Cartesian coordinates:)()(),,,( tzyxiktzkykxki AeAetzyx zyx

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Significance of plane waves:• Easy to generate (harmonic oscillator).• Any 3-dimensional wave can be expressed as a combination of plane waves

(Fourier analysis).

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Read: Ch2: 4-7No homework

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September 3 Spherical waves

2.8 The three-dimensional differential wave equation)()(),,,( tzyxiktzkykxki AeAetzyx zyx Plane wave:

kv

t

kzyx

kz

ky

kx

z

y

x

,22

2

22

2

2

2

2

2

22

2

22

2

22

2

2

2

22

2

2

2

2

2 1

tvzyx

Laplacian operator: 2

2

2

2

2

22

zyx

2

2

22 1

tv

General solution: )()(),,,( 21 vtzyxgCvtzyxfCtzyx

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2.9 Spherical waves

Spherical waves: Waves whose wavefronts are spheres.Spherical coordinates: (r, , )

x

z

y

r

cos

sinsin

cossin

rz

ry

rx

Laplacian operator in spherical coordinates:

2

2

2222

22

sin

1sin

sin

11

rrr

rrr

Spherical symmetry: )(),,( rr

)(11

2

22

22 r

rrrr

rr

Differential wave equation:

2

2

22

2 1)(

1

tvr

rr

)(1

)(2

2

22

2

rtv

rr

Solution:r

vtrftr

)(),(

General solution:r

vtrgC

r

vtrfCtr

)()(),( 21

The inverse square law: Intensity of a spherical wave 1/r2.

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Harmonic spherical wave:

or ),cos(),( tkrr

Atr )(),( tkrie

r

Atr

2.10 Cylindrical waves

Cylindrical waves: Waves whose wavefronts are cylinders.Cylindrical coordinates: (r, , z)Laplacian operator in cylindrical coordinates:

2

2

2

2

22 11

zrrr

rr

Cylindrical symmetry: )(),,( rzr

x

z

y

r

sin

cos

zz

ry

rx

z

Differential wave equation: 2

2

2

11

tvrr

rr

Solution: When r is sufficiently large, )(),( tkrier

Atr

A is the source strength.

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Read: Ch2: 8-10Homework: Ch2: 40,43Due: September 12