Wave Motion - SKKUicc.skku.ac.kr/~yeonlee/Display Optics/Plane Wave.pdf · Hecht by YHLEE;090901; 0-1 Wave Motion 1. One Dimensional Waves A wave: A self-sustaining disturbance of

Hecht by YHLEE;090901; 0-1

Wave Motion 1. One Dimensional Waves A wave : A self-sustaining disturbance of the medium

Waves in a spring

A longitudinal wave : Medium displacement // Direction of the wave A transverse wave : Medium displacement ⊥ Direction of the wave The disturbance advances, but not the medium (This is why waves can propagate at great speeds) A wave is given by a function of position and time: ψ = f x t,b g The shape of the disturbance at a certain time → The profile of the wave ψ x f x f x, ,0 0b g b g b g= ≡ : Taking a “photograph” at t = 0

A wave on a string


Propagation of a pulse

Moving reference frame

A new coordinate system, ′S , moving with the wave → ψ = ′f xb g : Indep. of time. The profile is measured along x ' In this case x x vt'= − → ψ x t f x vt,b g b g= − : 1-D wavefunction • At a later time t t+ Δ f x v t t f x vt− + ⇒ −Δb g b g : Same profile

↑ change position x x v t→ + Δ • A wave in -x direction ψ = +f x vtb g


2cosA xπψλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

cosAψ ϕ=

3 5 7 0 2 3 2 2 2 2 2π π π π πϕ π π π= −

3 5 3 7 0 4 4 2 4 4 2 4

x λ λ λ λ λ π πλ= −

x

The Differential Wave Eq. Start with ψ = f x vt∓b g and x x vt'= ∓

∂ψ∂

∂∂

∂∂

∂∂x

fx

xx

fx

= ='

'', → ∂ ψ

∂

∂

∂

2

2

2

2xf

x=

'

Similarly

∂ψ∂

∂∂

∂∂

∂∂t

fx

xt

v fx

= ='

''

∓ , → ∂ ψ

∂

∂

∂

2

22

2

2tv f

x=

′

Combining the eqs. : ∂ ψ

∂

∂ ψ

∂

2

2 2

2

21

x v t= , 1-D differential wave eq.

Its solution : ψ = − + +C f x vt C g x vt1 2b g b g There are two kinds of 1-D wave(+ and – waves) → 1-D differential wave eq. should be of second-order. 2. Harmonic waves A wave with sine or cosine profile ( ) ( ),0 cosx A kx f xψ = = k : propagation number A : amplitude A traveling wave by replacing x by ( )x vt− ( ) [ ] ( ), cos ( )x t A k x vt f x vtψ = − = − : Periodic in both space and time • The spatial period : Wavelength, λ [m]

ψ λ ψx t x t± =, ,b g b g [ ]cos[ ( )] cos ( ) cos[ ( ) ]k x vt k x vt k x vt kλ λ− ⇒ ± − ⇒ − ±

↑ ↑ position change by ±λ ↑ =2π → k = 2π λ/ phase, ϕ • The relation between ϕ and x

• The temporal period : Frequency, ν τ= 1/ [Hertz] ψ τ ψx t x t, ,± =b g b g [ ] ( ){ }cos[ ( )] cos ( ) cosk x vt k x v t k x vt vτ τ⎡ ⎤− ⇒ − ± ⇒ −⎣ ⎦∓

↑ time change by ±τ , ↑ = 2π /k → τ λ= /v The temporal frequency : ν τ=1/ → v = νλ Angular frequency : ω πν= 2


3. The Superposition Principle Two waves ψ ψ1 2 and both satisfy the differential wave eq.

2 2

1 12 2 2

1x v tψ ψ∂ ∂

=∂ ∂

, 2 2

2 22 2 2

1x v tψ ψ∂ ∂

=∂ ∂

Add two eqs.

( ) ( )2 2

1 2 1 22 2 2

1x v t

ψ ψ ψ ψ∂ ∂+ = +

∂ ∂

→ ( )1 2ψ ψ+ is also a solution of the diff. wave eq. → Two waves can add algebraically in the overlap region. Outside the overlap region, each propagates unaffected. Superposition of two waves

Waves that are out-of-phase diminish each other → Interference

(a) Two waves are in-phase (b) Two waves are π /3 out-of-phase (c) “ 2 3π / “ (d) “ π “


4. The Complex Representation The complex number ~z x iy= + : i = −1 ↑ ↑ real, imaginary parts In polar coords. x r y r= =cos , sinθ θ

→ ~ cos sinz r ir rei= + ⇒θ θ θ r : magnitude, modulus, absolute value ↑ θ : phase angle Euler formula

e iiθ θ θ= +cos sin The complex conjugate : ~* *z x iy x iy re i= + = − = −b g b g θ A complex exponential

e e e ez x iy x iy~= =+ → Modulus e ez x~

=

It is periodic e e e ez i z i z~ ~ ~+ = =2 2π π Any complex number

~ Re ~ Im ~ cos sinz z i z r ir= + = +b g b g θ θ : Either real or imaginary part can represent ↑ ↑ a harmonic wave 1

2~ ~*z z+e j , 1

2i z z~ ~*−e j A harmonic wave is usually represented by the real part of ~z

ψ ω εω εx t Ae A kx ti kx t, Re cosb g b gb g= LNMOQP⇒ − +− +

The wave function, in general

ψ ω εx t Aei kx t,b g b g= − +


5. Plane Waves The surface of a constant phase form a plane. The phase plane is perpendicular to the propagation direction. • A plane wave ψ r Aeik rb g = •

A constant phase requires k r a• = ↑ constant Assume oa k r= • ↑ constant vector → ( )r r ko− • = 0

→ r forms a plane that is perpendicular to k

• The periodic nature of a plane wave ( ) ( )ψ λ ψr k r+ = k : unit vector of k

↑ ↑

( )Aeik r k• +λ

, Aeik r•

→ k k=2πλ

: Propagation vector

Surfaces joining points of equal phase → wavefronts


• A plane wave with time dependence

( )ψ ω∓r t Aeik r i t, = • The plane wave in Cartesian coord.

ψωx y z t Aei k x k y k z tx y z, , ,b g d i=

+ + ∓ : k k k kx y z= + +2 2 2

The phase velocity (Velocity of the wavefront) ψ ψr rk t t r t+ + ⇒Δ Δ, , e j b g

↑ ↑ eik r rk i t t∓• + +( ) ( )Δ Δω eik r i t∓• ω → k r tΔ = ±ωΔ

→ drdt k

v= ± = ±ω

• Two plane waves with the same wavelength → k k1 2 2= = π λ/

Wave 1 : A e A eik r i t ik z i t1 1

1 1• − −=ω ω ,

Wave 2 : A e A eik r i t ik y z i t2 2

2 2• − + −=ω θ θ ωsin cos b g

In general

ψα β γ ωx y z t Aei k x y z t, , ,b g b g=

+ + ∓ (1)

where k k k k kx y z= = + +2 2 2

→ 2 2 2 1α + β + γ = : Direction cosines ↑ Cosine of the angle subtended by k x and


6. The Three-Dimensional Differential Wave Equation From (1)

∂ ψ

∂α ψ

2

22 2

xk= − , ∂ ψ

∂β ψ

2

22 2

yk= − , ∂ ψ

∂γ ψ

2

22 2

zk= −

→ ∂ ψ

∂

∂ ψ

∂

∂ ψ

∂ψ

2

2

2

2

2

22

x y zk+ + = −

Similarly

→ ∂ ψ

∂ω ψ

2

22

t= −

Combine the two eqs.

∇ =22

2

21

ψ∂ ψ

∂v t : Three dimensional differential wave eq.

where ∇ = + +22

2

2

2

2

2∂

∂

∂

∂

∂

∂x y z : Laplacian operator

One of the solution is a plane wave

ψω α β γx y z t Ae Ae

i k x k y k z i t ik x y z vtx y z, , ,b g d i b g= =+ + + +∓ ∓

Note that the following is also a solution ψ x y z t C f k r vt C g k r vt, , ,b g e j e j= • − + • +1 2

7. Spherical Waves A spherical wave is spherically symmetrical, independent of θ φ and ( ) ( ) ( ), ,r r rψ ψ θ φ ψ= = , The differential wave eq. in this case

∂

∂ψ

∂

∂ψ

2

2 2

2

21

rr

v trb g b g=

The solution

r r t f r vtψ ,b g b g= ∓ → ψ r tr

f r vt,b g b g=1 ∓

The harmonic spherical wave

ψ r t A er

ik r vt,b g

b g=

∓

The wavefront is obtained from kr = constant → Concentric spheres


• The spherical wave decreases in amplitude as it propagates.

A part of a spherical wave resembles a part of a plane wave at far distance

8. Cylindrical Waves The cylindrical symmetry requires ψ ψ θ ψr r z rb g b g b g= =, , The differential wave eq.

1 12

2

2r rr

r v t∂∂

∂ψ∂

∂ ψ

∂FHGIKJ =

The solution is given by Bessel functions. As r → ∞

ψ r t Ar

eik r vt,b g b g≈ ∓


Electromagnetic Theory, Photons, and Light Classical Electrodynamics : Energy transfer by electromagnetic waves Quantum Electrodynamics : Energy transfer by massless particles(photons) 1. Basic Laws of Electromagnetic Theory Electric charges, time varying magnetic fields → Electric field Electric currents, time varying electric fields → Magnetic field The force on a charge q F qE qv B= + × : Lorentz force A. Faraday’s Induction Law A time-varying magnetic flux through a loop → Induced Electromotive Force(emf) or voltage

emf ddt B= − Φ

↑ ↑ E dl

C•z . B dS

A•zz

→ E dl ddt

B dS dBdt

dSC A A

• = − • ⇒ − •z zz zz : Faraday’s law

(Wrong direction of dl in the book)


B. Gauss’s Law-Electric Electric flux : E A A

D dS D dS⊥Φ = = •∫∫ ∫∫

The total electric flux from a closed surface = The total enclosed charge

A VD dS dVρ• =∫∫ ∫∫∫ ,

where D Eε= , ε ε ε= r o , electric permittivity ↑ ↑ Permittivity of free space Relative permittivity, Dielectric constant

A point charge at the origin → D is constant over a sphere, 24E D rπΦ = . → E qΦ =

Coulomb’s law, 24 o

qErπε

=

C. Gauss’s Law-Magnetic No isolated magnetic charge B dS

A• =zz 0

D. Ampere’s Law A current penetrating a closed loop induces magnetic field around the loop

Using current density 2[ / ]J A m

C A

H dl J dS• = •∫ ∫∫ : Ampere’s law

where B Hμ= with μ μ μ= r o , permeability ↑ ↑ Permeability of free space Relative permeability

• A capacitor Between the capacitor plates

E QA

=ε

→ ε∂∂Et

iA

J D= ≡ ,

↑ ↑ Differentiate both sides ↑ Displacement current density The generalized Ampere’s law

C A

DH dl J dSt

∂∂

⎛ ⎞• = + •⎜ ⎟

⎝ ⎠∫ ∫∫ : Maxwell


E. Maxwell’s Equations

In free space : , , 0o o Jε ε μ μ ρ= = = =

C A

BE dl dSt

∂∂

• = − •∫ ∫∫ → BEt

∂∂

∇ × = − (1)

C A

DH dl dSt

∂∂

• = •∫ ∫∫ → DHt

∂∂

∇ × = (2)

0AB dS• =∫∫ → 0B∇ • = (3)

0AD dS• =∫∫ → 0D∇ • = (4)

↑ Differential form

2. Electromagnetic Waves Three basic properties

(1) Perpendicularity of the fields A time-varying D generates H that is perpendicular to D “ B “ E “ B → Transverse nature of E and H (2) Interdependence of and E H → Time varying E B and regenerate each other endlessly (3) Symmetry of the equations → The propagation direction will be symmetrical to both and E H Differential wave equation From (1)

∇ × ∇ × = − ∇ ×Et

Be j e j∂∂

Using (2)

∇ ∇ • − ∇ = −E E Eto oe j 22

2ε μ∂

∂ → ∇ =2

2

2E Eto oε μ

∂

∂

↑ =0 • In Cartesian coord. by separation of variables

∂

∂

∂

∂

∂

∂ε μ

∂

∂

2

2

2

2

2

2

2

2Ex

Ey

Ez

Et

x x xo o

x+ + = , : o

Wave equation known long before Maxwell

with 1/ ov ε μ=

Maxwell calculated

v m so o

= =× • ×

≈ ×− −

1 1

8 854 10 4 103 10

12 78

ε μ π./ :

Fizeau's measurement, c=315,300 /Km s

→ Maxwell concluded that light is an electromagnetic wave.


A. Transverse Waves

A plane wave propagating along x-axis

→ ∂

∂

∂

∂

E x y z ty

E x y z tz

, , , , , ,b g b g= = 0

→ ( ), ( , ) ( , ) ( , )x y zE E x t E x t x E x t y E x t z= ⇒ + +

From divergence eq. (4)

∇ • =E 0 → ∂∂

∂

∂∂∂

Ex

Ey

Ez

x y z+ + = 0 → ∂∂Ex

x = 0 ,

↑ ↑ E constx = (not a traveling wave) = 0, = 0

→ 0xE =

( , ) ( , )y zE E x t y E x t z= + : Transverse wave

• Assume E yE x ty= ,b g From (1)

x y z

x y zE

Bt

y

∂∂

∂∂

∂∂

∂∂

0 0

= − → zEx

xBt

yBt

zBt

y x y z∂

∂∂∂

∂

∂∂∂

= − − − (5)

↑ ↑ = 0, = 0, to match z → Only E By z and exist : Transverse wave

• A plane harmonic wave

( ) [ ], cosy oE x t E kx t= − ω + ε The B- field from (5)

[ ]1 cosyz o

EB dt E kx t

x c∂

= − ⇒ − ω + ε∂∫

→ E cBy z= : E B and are in phase

E B× // Beam propagation direction


3 Light in Matter In a homogeneous, isotropic dielectric (nonconducting material) ε ε μ μo o→ →, → v = 1/ εμ The absolute index of refraction

n cv o o

r r≡ = ⇒εμε μ

ε μ → n r= ε , Maxwell’s relation

↑ μr ≈ 1, except for ferromagnetic materials (μr = − × −1 22 10 5. for diamond)

The index of refraction depends on frequency

→ Dispersion

A. Dispersion Macroscopic view: A matter responses to the electric or magnetic field via ε or μ Microscopic view: The atom interacts with electric field via electric dipole Applied electric field → Distorted charge distribution → Internal field (External) (Electric dipole moment) The electric polarization P : Dipole moment per unit volume oD E P Eε ε= + ⇒ NaCl,… The eq. of motion

Restoring force of the electron for small x : F kx= − From Newton’s second law

2

22

i to o

d x dxqE e m x m mdtdt

− ω = ω + + γ : ωo k m2 = /

↑ ↑ ↑ ↑ Damping effect (Energy loss during oscillation due to ↑ ↑ Mass X Acceleration interaction between neighboring atoms)

↑ Restoring force Driving force from the incident wave The solution

( ) ( )2 2

/ i to

o

q mx t E ei

− ω⎡ ⎤⎢ ⎥=⎢ ⎥ω − ω − γω⎣ ⎦

(1) Without the driving force (no incident wave) → The oscillator vibrates at the resonance freq. ωo (2) For oω ω : E tb g and x tb g are in phase

(3) For oω ω : x tb g is 180o out of phase with E tb g


• The electric polarization, The permittivity

( )

2

2 2

/oo

o

q NE mP qx N

i= ⇒

ω − ω − γω,

( )( ) ( )

2

2 2

/o o

o

P t q N mE t i

ε = ε + ⇒ ε +ω − ω − γω

The dispersion relation

2

22 2

11o o

Nqnm i

⎛ ⎞= + ⎜ ⎟⎜ ⎟ε ω − ω − γω⎝ ⎠

> <1 for ω ωo

< >1 for ω ωo For oω << ω : ω can be neglected in the eq. → constant n For oω→ ω : n gradually increases with frequency (normal dispersion) For oω ≈ ω : The damping term becomes dominant (strong absorption)

dndω

< 0 (anomalous dispersion)

Shaded regions: absorption bands Shaded region: visible band (Note rise in UV and fall in IR) A material opaque near the resonance frequency can be transparent at other frequencies.


The Properties of Light 1. Reflection A beam of light in a dense medium → Scattering mostly in the forward direction A beam of light across an interface → Some backward scattering. Reflection The change of n over a distance > λ → Little reflection The change of n over a distance < λ /4 → Abrupt interface Internal and External Reflection Unpaired atomic oscillators → Reflection Indep. of glass thickness

Beam I : External reflection (n ni t< )

Beam II : Internal reflection (n ni t> ), 180o phase shift Huygens’s Principle Every point on a primary wavefront behaves as a point source of spherical secondary wavelet. The secondary wavelets propagate with the same speed and frequency with the primary wave. The wave at a later time is the superposition of these wavelets. Rays A ray is a line drawn in the direction of light propagation. In most cases, ray is straight and perpendicular to the wavefront A plane wave is represented by a single ray. A. The Law of Reflection A plane wave into a flat medium ( λ >> atomic spacing) → Spherical wavelets from the atoms.

→ Constructive interference only in one direction.


• Derivation of the law At t=0, the wavefront is AB At t= t1 , the wavefront is CD Note v t BD ADi i1 = = sinθ ,

v t AC ADr r1 = = sinθ

→ sin sinθ θi

i

r

rv v=

Since v vi r=

→ θ θi r= : Law of reflection (Part I) 2. Refraction The incident rays are bent at an interface → Refraction A. The Law of Refraction At t=0 the wavefront is AB At t t= Δ the wavefront is ED v t BD ADi iΔ = = sinθ

v t AE ADt tΔ = = sinθ

→ sin sinθ θi

i

t

tv v=

Since v cni

i= , v c

ntt

=

→ n ni i t tsin sinθ θ= : Law of refraction, Snell’s law A weak electric field

→ A linear response of the atom → A simple harmonic vibration of the atom

→ The frequencies of the incident, reflected and refracted waves are equal.


3. The Electromagnetic Approach A. Waves at an Interface An incident plane wave ( )cosi oi i iE E k r t= • − ω

The reflected and transmitted waves ( )cosr or r r rE E k r t= • − ω + ε

( )cost ot t t tE E k r t= • − ω + ε

, ,i r tε ε ε are constant phases

The boundary conditions ( ) ( ) ( )tangential tangential tangentiali r tE E E+ =

↑ ↑ ↑ u En i× u En r× u En t× This relation should be satisfied regardless of r and t → ω ω ωi r t= =

k r k r k ri r r t t• = • + = • +ε ε (1) From the first two of (1)

k ki i r rsin sinθ θ= → θ θi r=

From the first and last of (1)

k ki i t tsin sinθ θ= → n ni i t tsin sinθ θ=


B. The Fresnel Eqs. Case 1. E ⊥ The plane of incidence The relation among E H k, , and

( ) ˆ//E H k× , ( )ˆ //k E H×

At the interface E E Eoi or ot+ = (1)

( ) ( ) ( )tangential tangential tangentialoi or otH H H+ =

↑ ↑ ↑ −H xoi icosθ H xor rcosθ −H xot tcosθ Since H E v= /μ

( )1 1cos cosoi or i ot ti i t t

E E Ev v

− θ = θμ μ

(2)

From (1) and (2) with μ μ μ μi r t o= = = , v c n= / Amplitude reflection coefficient

cos coscos cos

or t t i i

oi t t i i

E n n rE n n ⊥

⊥

⎛ ⎞ θ − θ= − ≡⎜ ⎟ θ + θ⎝ ⎠

Amplitude transmission coefficient

2 cos

cos cosot i i

oi t t i i

E n tE n n ⊥

⊥

⎛ ⎞ θ= ≡⎜ ⎟ θ + θ⎝ ⎠


Case 2. E // The plane of incidence

E tangential should be continuous across the interface

→ ( ) ( ) ( )tangential tangential tangentialoi or otE E E+ = (3)

↑ ↑ ↑ E xoi icosθ , −E xor rcosθ , E xot tcosθ , : E is such that B points outward H tangential should be continuous across the interface

→ ( ) ( ) ( )tangential tangential tangentialoi or otH H H+ = (4)

↑ ↑ ↑

1μi i

oivE z 1

μr rorv

E z 1μt t

otvE z

From (3) and (4) with θ θi r= , v vi r= , μ μ μ μi r t o= = = , v c n= / Amplitude reflection coefficient

////

cos coscos cos

or t i i t

oi t i i t

E n n rE n n

⎛ ⎞ θ − θ= ≡⎜ ⎟ θ + θ⎝ ⎠

Amplitude transmission coefficient

////

2 coscos cos

ot i i

oi t i i t

E n tE n n

⎛ ⎞ θ= ≡⎜ ⎟ θ + θ⎝ ⎠

]

• Applying Snell’s law assuming θi ≠ 0 , Fresnel Eqs. become ↑ n ni i t tsin sinθ θ=

( )( )

sinsin

t i

t ir⊥

θ − θ=

θ + θ ( )( )//

tantan

t i

t ir

θ − θ= −

θ + θ

( )2sin cossin

t i

t it⊥

θ θ=

θ + θ ( ) ( )//

2sin cossin cos

t i

t i t it θ θ

=θ + θ θ − θ


C. Interpretation of the Fresnel Eqs.

Amplitude Coefficients

At normal incidence, θi = 0

t i

t i

n nr r

n n⊥

−= =

+

The external reflection ( ), t i i tn n> θ > θ

→ r⊥ < 0 .

// 0r = when ( ) 90ot iθ + θ = : Polarization angle of i pθ = θ .

The internal reflection ( ), i t t in n> θ > θ

→ 1r⊥ = when 90otn = : Critical angle of i cθ = θ in sini i tn nθ =

// 0r = when ( ) 90ot iθ + θ = : Polarization angle of 'i pθ = θ .

( 90op p′θ + θ = )

n nt i> , nt = 15. n n ni t i> =, . 15 Stronger reflection at glacing angle Reflectance and Transmittance The power per unit area : S = ×b e , poynting vector In phasor form : ( )*1

2S E H= ×

The intensity ( )2/W m : Irradiance

→ 212 o r o

cI S En

= = ε ε : Average energy per unit time per unit area


The cross sectional area of the incident beam = A icosθ “ “ reflected beam = A rcosθ “ “ transmitted beam = A tcosθ The reflectance

R I AI A

II

EE

rr r

i i

r

i

or

oi≡ ⇒ ⇒ = =

Reflected powerIncident power

coscos

θθ

22

The transmittance

⎛ ⎞θ θ θ

≡ ⇒ ⇒ = ⎜ ⎟θ θ θ⎝ ⎠

22cos cos cosTransmitted power

Incident power cos cos cost t ot t t t t

i i oi i i i i

I A E n nT t

I A E n n

• Energy conservation I A I A I Ai i r r t tcos cos cosθ θ θ= +

→ n E n E n Ei oi i i or i t ot t2 2 2cos cos cosθ θ θ= +

→ 2 2

cos1cos

or t t ot

oi i i oi

E n EE n E

⎛ ⎞ ⎛ ⎞ ⎛ ⎞θ= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟θ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

↑ ↑ R T


4. Total Internal reflection The Snell’s law for n ni t>

sin sinθ θit

it

nn

= : θ θi t<

At the critical angle, θt = 90o

sin θct

i

nn

=

For θ θi c>

→ All the incoming energy is reflected back into the incident medium

Total Internal Reflection

5. The Interaction of Light and Matter Reflection of all visible frequency → White color 70%~80% reflection → Shiny gray of metal Thomas Young : Colors can be generated by mixing three beams of light well separated in frequency Three primary colors combine to produce white light : No unique set The common primary colors : R, G, B

• Two complementary colors combine to produce white color

M G WC R WY B W

+ =+ =+ =

,,

• A saturated color contains no white light (deep and intense) An example of an unsaturated color ( ) ( )M Y R B R G W R+ = + + + = + : Pink


• The characteristic color comes from selective absorption

Example: (1) Yellow stained glass White light → Resonance in blue → Yellow is seen at the opposite side ↑ ↑ Red + Green Strong absorption in blue

(2) H O2 has resonance in IR and red → No red at ~30m underwater (3) Blue ink looks blue in either reflection or transmission Dried blue ink on a glass slide looks red. → Very strong absorption of red. Strong absorber is a strong reflector due to large nI . Resonance of materials Most atoms and molecules → Resonances in UV and IR Pigment molecules. → Resonances in VIS Organic dye molecules → Resonance in VIS • Subtractive coloration Blue light → Yellow filter → Black at the other side ↑ It removes blue


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Wave Motion - SKKUicc.skku.ac.kr/~yeonlee/Display Optics/Plane Wave.pdf · Hecht by YHLEE;090901; 0-1 Wave Motion 1. One Dimensional Waves A wave: A self-sustaining disturbance of