Phys 12a Assignment 5

Physics 12a
Assignment 5
Caltech, 11/6/14
Web Page:
http://www.ligo.caltech.edu/∼ajw/ph12a
Check it for late-breaking announcements.
Problems:
Due in class or the Ph12 IN box in 1st floor East Bridge, 10:30 AM, Thursday,
11/13/14.
Note that this assignment is 4 pages long. The problems are actually very fundamental and
important, but hopefully straightforward, so I hope the assignment itself is not too long.
V.1 Reflections in transmission lines (Crawford 5.5, 5.6).
Suppose a coaxial transmission line having 50 ohms characteristic impedance is joined to one
having 100 ohms characteristic impedance.
(a) A voltage pulse of +10 volts (maximum value) is incident from the 50 ohmline to the
100 ohm line. What is the “height” (in volts, including the sign) of the reflected pulse?
Of the transmitted pulse?
(b) A + 10-volt pulse is incident from the 100 ohm to the 50 ohm line. What are the reflected
and transmitted pulse heights?
(c) How can you insert an ordinary resistor so that an incident pulse traveling from the 50
ohm to the 100 ohm line is transmitted without generating any reflected pulse? We want
to know how many ohms the resistance has, and we want a schematic sketch showing the
center conductor and outer conductor of each of the lines at the place where they join
and showing the resistor connected. (Do not worry about “distributing” the resistor. If
the wavelengths are long compared with the diameter of the cable, there is no need to
distribute the resistance.)
(d) Suppose a + 10-volt pulse is incident on this modified circuit. What is the size of the
transmitted pulse?
(e) Now suppose a +10-volt pulse is sent down this line in the “wrong” direction, i.e., from
the 100 ohm line to the 50 ohm line. What happens? Find the reflected and transmitted
pulse heights.
(f) Next consider the problem of transmitting a pulse from the 100 ohm line to the 50 ohm
line without generating any reflection. What should be the resistance value and how
should it be connected at the place where the lines are joined? What is the pulse height
transmitted if +10 volts is incident? What happens now when a + 10-volt pulse is
incident from the 50 ohm to the 100 ohm line, i.e., in the “wrong” direction?
V.2 Multiple reflections: (Crawford 5.26).
In the following derivations you are to use complex numbers. Suppose ψinc is the real part of
Aei(ωt−kz) , where A is real. At z = 0 the impedance suffers a sudden change from Z1 to Z2 .
At z = L the impedance changes again from Z2 to Z3 . Let R12 = (Z1 −Z2 )/(Z1 +Z2 ) = −R21 ,
and similarly for R23 = −R32 . Assume that in medium 1 there is a reflected wave that is the
real part of RAei(ωt+kz) , where R is complex, and may be written R = |R|e−iδ .
(a) Show that if we neglect all contributions except the reflection from z = 0 and the first
reflection from z = L, we obtain
R = R12 + T12 R23 T21 e−2ik2 L
, where T12 = 1 + R12 and T21 = 1 + R21 = 1 − R12 .
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(b) Show by explicit summation of the infinite series corresponding to an infinite number of
multiple reflections that the exact solution for R is
2 R e−2ik2 L
1 − R12
23
R = R12 +
.
1 − R23 R21 e−2ik2 L
where the first term, R12 , is due to the “prompt” reflection at the first discontinuity
at z = 0, and the rest is due to one or more reflections at z = L. Show that in the
small-reflection approximation this result reduces to that of part (a).
(c) Show that the exact result can be written in the form
R=
R12 + R23 e−2ik2 L
.
1 − R23 R21 e−2ik2 L
Show that this exact expression for R vanishes for the same combinations of R23 /R12
and k2 L as does the approximate expression for R obtained in the small-reflection approximation. Thus the approximate expression gives the zeroes correctly, but is inexact
as to the intensity at the maxima.
V.3 Boundary-condition method for reflection and transmission coefficients. (Crawford 5.27).
The physical situation is exactly as in the previous problem. The method of solution will
be completely different. Instead of summing an infinite series of multiply-reflected rays we
take the following approach: Each “ray” of the superposition of multiply-reflected rays is
continuous. Therefore the superposition itself is continuous. That is the basis of the method
(also known as the method of “self-consistent fields”. Thus we do not bother with summing
over multiple reflections. Instead we write ψ(z, t) in three regions: region 1 (z < 0), region 2
(0 < z < L), and region 3 (z > L), as the real part of
ψ1 (z, t) = ei(ωt−k1 z) + Rei(ωt+k1 z)
ψ2 (z, t) = F ei(ωt−k2 z) + Bei(ωt+k2 z)
ψ3 (z, t) = T ei[ωt−k3 (z−L)]
where R (reflected), F (forward), R (backward), and T (transmitted) are unknown complex
numbers to be determined. (For simplicity we have taken the amplitude of the incident wave
to be unity.) Notice that the term with complex amplitude F is the superposition of all of the
multiply-reflected rays between z = 0 and L that are going in the forward direction at time
t. Similarly the term with complex amplitude R is the superposition of all the backwardgoing rays. At the two discontinuities, z = 0 and z = L, you are to apply the boundary
conditions of continuity. Assume ψ(z, t) is continuous, and assume dψ(z, t)/dz is continuous.
(This means that the string tension is constant, if we have a string; or that the equilibrium
pressure p0 times the factor γ is constant, for sound waves; or that the magnetic permeability
µ is constant, for electromagnetic waves.) These two boundary conditions at each of the
two places give four linear equations in the four complex numbers T , F , B, and R. That is
sufficient to determine T , F , B, and R uniquely. Justify that statement. Find T , F , B, and
R. Show that your result for R is identical with that obtained by the method of multiple
reflection, in the previous problem.
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V.4 Transmission resonance: (Crawford 5.28).
(a) Show that for reflection due to two discontinuities (Multiple Reflections problem above)
the fractional time-averaged energy flux that is not reflected (and hence by energy conservation must be transmitted) is given by
|T |2 = 1 − |R|2 =
2 − R2 + R2 R2
1 − R12
23
12 23
2 R2 .
1 + 2R12 R23 cos 2k2 L + R12
23
(b) Show that if medium 3 has the same impedance as medium 1 this becomes
|T |2 = 1 − |R|2 =
2
2
1 − R12
2
4 .
1 − 2R12 cos 2k2 L + R12
(c) Show (assuming the conditions of part (b)) that at certain values of k2 L the fractional
time-averaged energy flux not reflected is unity, i.e., for those values, all the energy
is transmitted and none reflected. Call any one of these “resonance values” of k2 by
the name k0 . Show that the resonance values are given by k0 L = nπ, where n is any
positive integer. Plot |T |2 as a function of k2 L to show several resonance peaks; and
superimpose plots of |T |2 for several different values of R12 (properly labeled) to show
how the resonances depend on R12 .
(d) Show that for k2 sufficiently near a resonance value k0 the transmitted (time-averaged)
energy flux is given by
2
2
1 − R12
2
|T | = 1 − |R| =
2
1 − R12
2
2
2 [2L(k − k )]2
+ R12
2
0
.
Plot it, and show that this form is that of a “Breit-Wigner resonance shape”, with a full
width at half-maximum transmitted intensity ∆k2 , given by
2
1 − R12
(∆k2 )L ≈
,
|R12 |
provided that |R12 | is not too much less than unity. Show that for |R12 | 1, the BreitWigner approximation is useless, because it does not hold except very near k0 , i.e., it
doesn’t hold out even to the “half-maximum transmitted power” points. In that case
one should use the exact result.)
(e) Show that for —|R12 | ≈ 1, when the Breit-Wigner shape holds for many resonance
widths away from k0 , the resonance full width is given by
(∆k2 )L ≈ 2 (1 − |R12 |) .
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V.5 Termination of waves on a string: (Crawford 5.32).
(a) Suppose you have a massless dashpot having two moving parts 1 and 2 that can move
relative to one another along the x direction, which is transverse to the string direction
z. Friction is provided by a fluid that retards the relative motion of the two moving
parts. The friction is such that the force needed to maintain relative velocity x˙ 1 − x˙ 2
between the two moving parts is Zd (x˙ 1 − x˙ 2 ), where Zd is the impedance of the dashpot.
The input (part 1) is connected to the end of a string of impedance Z1 stretching from
z = −∞ to z = 0. The output (part 2) is connected to a string of impedance Z2 that
extends to z = +∞. A picture might help:
Show that a wave incident from the left experiences an impedance at z = 0 which is the
same as that it would experience if connected to a “load” consisting of a string stretching
from z = 0 to +∞ and having impedance ZL given by
ZL =
Zd Zs
,
Zd + Z2
that is,
1
1
1
=
+
.
ZL
Zd Z2
Thus it is as if the dashpot and string 2 were impedances connected “in parallel” and
driven by the incident wave.
(b) Show that if string Z2 extends only to z = λ2 /4, where λ2 is the wavelength in medium
2 (assuming we have a harmonic wave with a single frequency), and there is terminated
by a dashpot of zero impedance (frictionless), the wave incident at z = 0 is perfectly
terminated (ie, Zd = Z1 ). Show that the output connection of the dashpot at z = 0
cannot tell whether it is connected to a string of infinite impedance or is instead connected to a quarter-wavelength string that is “short-circuited” by a frictionless dashpot
at z = λ2 /4. In either case the output connection remains at rest.
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