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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]
2
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
3
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.
4
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
6
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
7
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
8
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
9
Significance of plane waves:• Easy to generate (harmonic oscillator).• Any 3-dimensional wave can be expressed as a combination of plane waves
(Fourier analysis).
11
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
12
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.
13
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.