5. Superposition of Waves5. Superposition of Waves
Last Lecture• Wave Equations• Harmonic Waves• Plane Waves• EM Waves
This Lecture• Superposition of Waves
Superposition of Waves Superposition of Waves 1 2
2 2 2 22
2 2 2 2 2
1 2
1 2
:
1v
. ,,
Suppose that and are both solutionsof the wave equation
x y z t
Then any linear combination of and isalso a solution of the wave equation For exampleif a and b are constants then
a b
is a
ψ ψ
ψ ψ ψ ψψ
ψ ψ
ψ ψ ψ
∂ ∂ ∂ ∂+ + = ∇ =
∂ ∂ ∂ ∂
= +
.lso a solution of the wave equation
E. Hecht, Optics, Chapter 7.
5-2. Superposition of Waves of the Same Frequency 5-2. Superposition of Waves of the Same Frequency
1 2
01 1 02 2cos( ) cos( )RE E E
E t E tα ω α ω= += − + −
Constructive interference and Destructive interference Constructive interference
and Destructive interference
Constructive interference :
( )
( )
2 1
1 2 01 1 02 1
01 02 1
2cos( ) cos( 2 )
cos( )
mE E E t E m t
E E t
α α πα ω α π ω
α ω
− =
+ = − + + −
= + −
Destructive interference :
( )( )
( )
2 1
1 2 01 1 02 1
01 02 1
2 1
cos( ) cos( 2 1 )
cos( )
m
E E E t E m t
E E t
α α π
α ω α π ω
α ω
− = +
+ = − + + + −
= − −
General superposition of the Same Frequency General superposition of the Same Frequency
( )
( )
1 2
1 2
( ) ( )01 02
0 01 02
( )0 0
Re
Defining
Re cos( )
i t i tR
i ii
i tR
E E e E e
E e E e E e
E E e E t
α ω α ω
α αα
α ω α ω
− −
−
= +
≡ +
⇒ = = −
2 20 01 02 01 02 2 1
01 1 02 2
01 1 02 2
2 cos( )sin sintancos cos
E E E E EE EE E
α αα ααα α
= + + −
+=
+
Superposition of Multiple Waves of the Same Frequency Superposition of Multiple Waves of the Same Frequency
( ) ( ) ( )
( )
0 01
2 20 0 0 0
1 1
01
01
:
, cos cos
2 cos
sin tan
cos
N
i ii
N N N
i i j j ii j i i
N
i iiN
i ii
The relations just developed can be extended to the addition ofan arbitrary number N of waves
E x t E t E t
E E E E
E
E
α ω α ω
α α
αα
α
=
= > =
=
=
= − = −
= + −
=
∑
∑ ∑∑
∑
∑
5-3. Random and coherent sources 5-3. Random and coherent sources
( )2 20 0 0 0
1 12 cos
N N N
i i j j ii j i i
E E E E α α= > =
= + −∑ ∑∑
Randomly phased sources :
( )2 2 20 0 0 0 0
1 1 12 cos
N N N N
i i j j i ii j i i i
E E E E Eα α= > = =
= + − =∑ ∑∑ ∑ 0 01
N
ii
I I=
= ∑
The resultant irradiance (intensity) of the randomly phased sources is the sum of the individual irradiances.
Coherent and in-phase sources :
( )2 2 20 0 0 0 0 0 0
1 1 1 12 cos 2
N N N N N N
i i j j i i i ji j i i i j i i
E E E E E E Eα α= > = = > =
= + − = +∑ ∑∑ ∑ ∑∑2
0 01
N
ii
I E=
⎛ ⎞= ⎜ ⎟⎝ ⎠∑
The resultant irradiance (intensity) of the coherent sources is N2 times the individual irradiances.
0 01I N I= ×
20 01I N I= ×
5-4. Standing waves 5-4. Standing waves
1 2 0 0sin( ) sin( )R RE E E E t kx E t kxω ω ϕ= + = + + − −to the left to the right
02 cos( )sin( )2 2R R
RE E kx tϕ ϕω= + −
{ } ( )02 sin cosR RE E kx tϕ π ω= → =
Mirror reflection
2 xkx mπ πλ
= =
2x m λ⎛ ⎞= ⎜ ⎟
⎝ ⎠Nodes at
In a laser cavity In a laser cavity
2md m λ⎛ ⎞= ⎜ ⎟
⎝ ⎠
2 2m m
m
d c cmm d
λ νλ
⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
5-5. The beat phenomenon 5-5. The beat phenomenon
1 2
2pk kk +
=
1 2
2gk kk −
=
( )( )( ) 2 cos cosg px A k x k xψ =
Superposition of Waves with Different Frequency Superposition of Waves with Different Frequency
: beat wavelength
kp =
kg =
If the 2 waves have approximately equal wavelengths,
Phase velocity and Group velocity Phase velocity and Group velocity
When the waves have also a time dependence,
1 2
1 2
2
2
p
g
ω ωω
ω ωω
+=
−=
1 2
1 2
2
2
p
g
k kk
k kk
+=
−=
1 1 1
2 2 2
( , ) cos( )( , ) cos( )x t A k x tx t A k x t
ψ ωψ ω
= −= −
higher frequency wave
lower frequency wave (envelope)
phase velocity : 1 2
1 2
pp
p
vk k k kω ω ω ω+
= = ≈+
1 2
1 2
gg
g
dvk k k dkω ω ω ω−
= = ≈−
group velocity :
( )
( )
2 1
1 2 /
pg p p
p p p
p
dvd dv kv v kdk dk dk
d c c dn k dnv k v k vdk n n dk n dk
dnv kn d
ω
λ π λλ
⎛ ⎞= = = + ⎜ ⎟
⎝ ⎠− ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + = + = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞= + ← =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Phase velocity and Group velocity Phase velocity and Group velocity
phase velocity : 1 2
1 2
pp
p
vk k k kω ω ω ω+
= = ≈+
1 2
1 2
gg
g
dvk k k dkω ω ω ω−
= = ≈−
group velocity :