Boundary effects
• What happens to a traveling wave
when it encounters a barrier, or discontinuity, or obstacle of
some kind
• Example: inhomogeneity in mass density
Incident
Reflected
Transmitted
! incident = Asin (" t # k1 x )
Im{Aei{k1 x ! wt } } (A, A', A'' real here)
! reflected = A'sin (" t + k1 x )
Im{A'e!i{k1 x + wt } }
! transmitted = A''sin (" t # k2 x )
Im{A''ei{k2 x ! wt } }
• All waves have same ! - necessary to satisfy boundary condition at x=0
• Right moving waves ! k1 x
• Left moving waves + k2 x
Incident
Reflected
Transmitted
! incident = Asin (" t # k1 x )
! reflected = A'sin (" t + k1 x )
! transmitted = A''sin (" t # k2 x )
Boundary conditions
y(x = 0!,t) = y(x = 0+,t)
!y
!y
(x = 0",t) = (x = 0+,t)
!x
!x
String continuous
T
Forces continuous
T
Incident
Reflected
Transmitted
! incident = Asin (" t # k1 x )
! reflected = A'sin (" t + k1 x )
! transmitted = A''sin (" t # k2 x )
Boundary conditions
y(x = 0!,t) = y(x = 0+,t)
!y
!y
(x = 0",t) = (x = 0+,t)
!x
!x
! incident (x = 0,t) + ! reflected (0,t) = ! transmitted (0,t)
!" incident (x = 0,t) !" reflected (0,t) !" transmitted (0,t)
+
=
!x
!x
!x
Incident
Reflected
Transmitted
! incident = Asin (" t # k1 x )
! reflected = A'sin (" t + k1 x )
! transmitted = A''sin (" t # k2 x )
Boundary conditions
y(x = 0!,t) = y(x = 0+,t)
!y
!y
(x = 0",t) = (x = 0+,t)
!x
!x
A + A! = A!!
(
)
k1 A ! A" = k2 A""
! incident (x = 0,t) + ! reflected (0,t) = ! transmitted (0,t)
!" incident (x = 0,t) !" reflected (0,t) !" transmitted (0,t)
+
=
!x
!x
!x
Asin (! t " k1 x )
A''sin (! t " k2 x )
A'sin (! t + k1 x )
A + A! = A!!
(
)
k1 A ! A" = k2 A""
r=
A! k1 " k2
=
A k1 + k2
t=
2k1
A!!
=
A k1 + k2
reflection amplitude
transmission amplitude
Special cases:
1) k1 = k2 !
2) k1 < k2
A""
= 1 No reflection
A
! A" is negative
Reflected wave = $ A" sin (# t + k1 x ) = A" sin (# t + k1 x + % )
A" = 0, t =
i.e. PHASE CHANGE at rare-dense boundary
3) k1 > k2
4)
2
'( !
(k1 < k2 !
1
<
2)
[v =# / k = T / & ]
! A" is positive
k2'(
r=
A"
' –1
A
T ' 0 No wave in very heavy string
Energy flux at boundaries
Power flux
1
P = T ! kA2
2
(Energy flow per unit time)
1
PI = T ! k1 A2
2
1
PR = T ! k1 A"2
2
1
PT = T ! k2 A""2
2
Incident power flux
Reflected power flux
Transmitted power flux
#k "k &
PR A! 2
RP =
= 2 = r2 = % 1 2 (
PI
A
$ k1 + k2 '
2
Coefficient of reflection
2
k2 " 2k1 %
4k1k2
2
TP =
=
=
t
=
=
PI
k1
k1 $# k1 + k2 '&
k1 A2
k1 + k2
PT
k2 A!! 2
k2
(
2
" k1 ! k2 %
4k1k2
RP + TP = $
+
'
# k1 + k2 &
k1 + k2
(
)
2
=
k12 + 2k1k2 + k22
(k + k )
1
2
2
)
2
Coefficient of transmission
= 1 Conservation of energy
Summary
♠ Continuity of y and ∂y/∂x at the boundary x = 0 determines the
amplitudes of the reflected and transmitted waves
in terms of the amplitude of the incident wave
as a function of the wave numbers k1 and k2 :
k1 − k2
2k1
r=
; t=
k1 + k2
k1 + k2
♠ Energy transport across the boundary:
ref lected f lux
= r2
R≡
reflection coefficient
incident f lux
k2 2
transmitted f lux
t
=
T ≡
incident f lux
k1
1=R+T
transmission coefficient