(VI) Reflection and Transmission of Waves
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Characteristic Impedance
This is a property of the medium in which the wave
propagates, which describes how hard it is to set up a
wave in the medium.
For all mechanical waves this takes the form;
i.e.
1
Characteristic impedance (2)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
e.g. for transverse waves on a string;
2
Consider the wave: y = y 0 exp[i(ωt − kx )]
z=
transverse driving force
transverse velocity
=
=
−T
∂y
∂x
∂y
∂t
T
ω k
since the phase velocity v p =
T
ρ
Characteristic impedance (3)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Writing energy flow in terms of z:
1
Rate of energy flow = Tωky 0 2
2
3
Reflection and Transmission at a boundary
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Consider this string that has a density junction in it at x = 0
4
y
x
(z turns out to come in useful)
string
(Tension in this string is
the same everywhere)
The density junction affects the travelling wave in the following way...
Reflection and Transmission (2)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we calculate the R and T coefficients.
5
ρ1
y
ρ2
x=0
string
x
Incident:
→
A1 exp[i(ω1t − k1 x )]
Reflected:
Bexp[i(ω1t + k1 x )]
→
Transmitted:
A2 exp[i(ω 2 t − k 2 x )]
←
Amplitude Reflection Coefficient:
Amplitude Transmission Coefficient:
Reflection and Transmission (3)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we begin to work out the R and T in terms of z.
6
Use the following boundary conditions at x = 0:
1. ………………………………………………………. –
otherwise the string would break
2. The …………………………………………………. must
be ………………...
• Remember the transverse force is proportional to this
∂y
derivative.
F = −T
∂x
• If the force were not continuous, there would be a finite
net force on an infinitesimal segment of the string at
x = 0, which would cause an infinite acceleration.
Reflection and Transmission (4)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we’re still working out the R and T in terms of z...
Using condition 1 at x = 0:
A1 exp(iω1t ) + Bexp(iω1t ) = A2 exp(iω 2 t )
This can only be true for all times if ω1 = ω 2 = ω , in
which case:
7
Reflection and Transmission (5)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we’re still working out the R and T in terms of z…
8
Using condition 2:
∂yi ∂y r ∂y t
+
=
∂x ∂x ∂x
at x = 0, at all times:
Tk
Now remember that z =
so for fixed ω and fixed T,
ω
then z is proportional to k
Thus the second boundary condition becomes:
Reflection and Transmission (6)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we calculate the R in terms of z.
A1 + B = A2 (A1 − B)z1 = A2 z2
Eliminate A2 by dividing these two equations:
9
Reflection and Transmission (7)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Here we calculate the T in terms of z.
10
Now from the first condition,
A1 + B = A2
Therefore
Hence
R=
(z1 − z2 )
(z1 + z2 )
Reflection and Transmission (8)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
R=
11
(z 1 − z 2 )
(z 1 + z 2 )
T =
2 z1
(z 1 + z 2 )
• Wave goes from less dense to
more dense region:
z1
z2
R is NEGATIVE:
transmitted amplitude < incident amplitude (T < 1)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Reflection and Transmission (9)
2z1
(z − z )
R= 1 2 T=
(z1 + z 2 )
(z1 + z 2 )
• Wave goes from more dense to
less dense region:
z1
z2
R is POSITIVE
transmitted amplitude > incident amplitude (T > 1)
12
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
Reflection and Transmission (10)
2 z1
(z − z )
R= 1 2 T=
(z1 + z 2 )
(z1 + z 2 )
• String fixed rigidly at x = 0:
z2
z1
13
Reflection and Transmission (11)
Dr. Pete Vukusic, Exeter University:
PHY 1106: Waves and Oscillators (Lecture 18)
R=
14
2 z1
(z1 − z 2 )
T=
(z1 + z 2 )
(z1 + z 2 )
• String has free end at x = 0:
z1
z2