i. Do problems with symbols first (introduce them if you have to). Only put in numbers at the end.

ii. Check your answers for dimensional correctness.

iii. Examine limiting cases to check your answers (masses to zero or infinity, angles to zero or pi/2, etc.)

iv.

due Thurs, 12/12

reading:

12/6-12/12

HRK vol. 2, Ch. 43: Gratings and Spectra (section 43-5 on Holography is optional)

problems:

1. HRK Ch. 41, Q2 ( *Why
screen $A$ in Young's double slit experiment?*)

2. HRK Ch. 41, Q28 ( *angle
dependence of thin film interference*)

3. HRK Ch. 41, E14 ( *coherence
of atomic radiation*) (Ans.: 3 m.)

4. HRK Ch. 41, E30 ( *oil
drop*) (Ans. (b) 594 nm.)

5. HRK Ch. 41, E40 ( *Michelson
interferometer measurement of index of refraction of air*)
(Ans.: 1.0003)

6. HRK Ch. 41, P6 ( *interference
intensity*) (Ans.: I_0(5 + 4 cos phi)/9.)

7. HRK Ch. 42, E17 ( *diffraction by telescope aperture)*

8. HRK Ch. 42, P1 ( *diffracting wire*) (Ans. 2.45 cm.)

9. HRK Ch. 42, P7 ( *double
slit interference and single slit diffraction combined*)

10. HRK Ch. 43, E11 ( *overlapping orders of grating spectra*)

11. HRK Ch. 43, E21 ( *resolution
of sodium doublet*)

12. HRK Ch. 43, E33 ( *Bragg
scattering*)

13. Two vertical dipole radio antennas are separated by a distance lambda/4
in the north-south direction, where lamba is the wavelength being broadcast.
The same signal is fed to both antennae, but with a phase lag of pi/2 for
the northernmost one relative to the other one. Suppose the intensity of
the radiation from each antenna alone is I_0 at some large distance d. When
both antennae broadcast together what is the intensity, at the distance d,
towards (a) the north, (b) the south, (c) the east, (d) the west?

14. Two vertical dipole radio
antennas are separated by a distance 2 lambda in the north-south direction, where lamba
is the wavelength being broadcast. The same signal is fed in phase to both
antennae. Suppose the intensity of the radiation from each antenna alone
is I_0 at some large distance d. When both antennae broadcast together
and are viewed from the distance d, (a) how many interference maxima are
there around a complete circle encompassing the two antennae? (b) At what
angles are the maxima located? (c) What is the intensity of each of these
maxima? (d) What is the intensity in the SW direction (45 degrees south of
west)?

due Thurs, 12/5

**reading:**

12/2-12/5

HRK vol. 2, Ch. 41: Interference

HRK vol. 2, Ch, 42: Diffraction

12/6-12/12

HRK vol. 2, Ch. 43: Gratings and Spectra (section 43-5 on Holography is
optional)

For the week ending
12/1 the reading was:

HRK Ch. 44: Polarization (section 44-4 on Double Refraction is optional,
since we won't have time to go into it, but the subject is very interesting,
so I invite you to read it and ask me outside of class if you have questions).

*Supplement* (1 page) on circular polarization under the supplements
link at the course web page.

**problems**:

1. HRK Ch. 39, E40 (*approximate
Doppler shift for electromagnetic waves*) Add part (b): The mass of a
giant black hole in the center of a galaxy can be inferred from the orbital
radius and orbital velocity of gas orbiting the hole. The velocity of the
gas is measured by the Doppler shift of observed spectral lines. Suppose
we are looking edge on at a disk of such gas, and a spectral line of natural
wavelength lambda_0 is received as lambda_a from the approaching side and
lambda_r from the receding side of the disk at a given radius. What is the
orbital speed v of the gas at this radius? Assume v/c << 1.

2. HRK Ch. 39, E 43 (*cosmologiocal
redshifts) *Give your answers in terms of a factor times the speed of light.
(*Note*: the correct interpretation of this cosmological redshift is
not that it is due to recession but rather that it is due to the expansion
of the universe.)

3. HRK Ch. 39, E46 (*red
light, green light*)

4. HRK Ch. 39, P 15 (*microwave
radar beats) *(*Note: *The speed of the plane is much less than c,
so use the small v/c approximation!)

5. HRK Ch. 44, E 2 (*no
interference from orthogonal polarizations)
*

6. HRK CH. 44, E7 (*stacked polarizer sheets*)
Add part (b): If the third
of the four sheets is removed, what fraction of the incident intensity is
then transmitted?

7. HRK Ch. 44, E11 (*polarization
by reflection from water*)

8. HRK Ch. 44, P1 (*polarizing
filter and electric field component at the beach*)

9. HRK Ch. 44, P2 (*mix
of unpolarized and polarized light*)

10. (a) If you face south
at sunset, with the sun directly to your right, how will the sky light coming
from the direction you are facing be polarized? (b) Same question is you look
up directly overhead. *Explain your answers with diagrams and words.*

11. **[20 points]** (a)
Superpose right and left circularly polarized plane harmonic electromagnetic
waves with the same amplitude and propagation direction, and with the vertical
component of the electric field for the right circularly polarized wave a
phase phi ahead of that of the left circularly polarized wave. Show, by drawing
the two rotating electric field vectors and their resultant at one point
in space at several times, that the resultant wave is linearly polarized
in the direction making an angle phi/2 clockwise with the vertical direction.
(The polarization directions and the the final angle are all defined here
as viewed along the propagation direction of the wave). (b) The optical
activity (polarization rotating property) of substances like sugar syrup
can be understood as a consequence of the fact that the index of refraction
n_R for right circularly polarized light is different from the index n_L for
left circularly polarized light. Show that the angle of rotation of the polarization
direction for linearly polarized light of vacuum wavelength lambda_0 passing
through a length L of such a substance is rotated clockwise through an angle
Pi (n_R - n_L) L/lambda_0. (c) If the linear polarization of blue light
(475 nm) rotates by 90 degrees clockwise after propagating though some sugar
syrup, by how much does the polarization of red light rotate? (d) If the
sample of syrup in part (c) is 5 cm thick, what is the difference (n_R -
n_L) for this syrup? (*Hint: *For part (b), to determine the relative
phase of the two polarizations count the number of wavelength cycles that
``fit" in the length L.) (*Notes:* 1) In parts (c) and (d), neglect the
variation of the indices of refraction between the two frequencies of light.
2) If the light is not composed of infinite harmonic wavetrains but is rather
a wavepacket of finite size, the right and left circular polarized wavepackets
will eventually separate if they keep propagating at different speeds through
enough syrup. At that stage, they will cease to recombine coherently, and
the color effect will disappear, I think.)

Read the first 9 pages of the article by Scott Hughes, Listening to the Universe with Gravitational Wave Astronomy, (http://arxiv.org/abs/astro-ph/0210481), and peruse the website "About LIGO" (http://www.ligo.caltech.edu/LIGO_web/about/). Ask Dr. J. if you have any questions about this. Referring to the article by Hughes, answer the following questions:

1. Derive equation (4) using dimensional analysis. That is, assume h is proportional to GQ/r on the general physical grounds discussed in the article, and deduce the missing power of the speed of light c and the number of time derivatives of the qudrupole moment Q.

2. Verify equation (5).

3. Derive equation (6), and verify the numerical value. (Here Delta L is the same as delta L in the previous discussion.)

4. Find the laser wavelength implicitly assumed in arriving at the numerical value of 100 Watts for the laser power discussed in the first paragraph of page 9. I imagine the wavelength should close to visible if not visible. (

5. Derive equation (7) from the equipartition theorem (1/2 kT energy per degree of freedom), and verify the numerical value.

due Wed, 11/27

**reading:**

11/14-11/25: All available
at the supplements link: RHK vol. 1 (4th ed.) pages 457-460 on Doppler effect (p457, p458, p459, p460) ;

spacetime diagram derivation of Doppler effect;
supplement on
Doppler effect for electromagnetic waves;

Also note old reading assignment
Ch. 39-6 of HRK deals with Doppler effect for electromagnetic waves.

*Optional: *Feynman
Lectures vol. 1, Ch. 31-1,2,3,4 (*The Origin of the Refractive Index*)

**problems**:
Ch. 39 HRK unless otherwise noted.

1. Q22 (*speed of light
in quartz*)

2. Q27 (*interaction
between light and medium*?) Explain your answer.

3. Q35 (*change of
wavelength but not frequency*) Explain your answer.

4. E25 (*apparent depth*)

5. E35 (*viewpoint
of a fish*)

6. P2 (*Roemer's measurement
of the speed of light*)

7. P3 (*ocean wave
refraction)
*

8. P6 (*layered indices
of refraction*)

9. P12 (*optical fiber*)

10. An electron and a
positron at rest annihilate and produce two photons each of energy equal
to the rest mass of an electron times c^2. (i) Using the quantum relation
E = hf determine the frequency and wavelength of such photons. (This wavelength
is called the Compton wavelength of the electron. It is the smallest length
scale over which an electron can be localized without creating electron-positron
pairs.) (ii) What is the ratio of the Compton wavelength of the electron
to the radius of a hydrogen atom (the "Bohr radius")? (Since the former is
much smaller than the latter one can think of the electron in the atom as
a single particle rather than a particle accompanied by an indeterminate number
of electron-positron pairs!)

11. Refer to the supplement giving a spacetime diagram derivation of the Doppler effect for approaching source and observer. Make your own spacetime diagram derivation of the Doppler effect for the case of a stationary observer and an approaching source.

12. A police car (P) with
a siren and a speeder (S) move along a long straight road. The frequency of
the sound emitted by the siren in its own rest frame is f_source=1125 Hz.
Find the frequency and wavelength observed by the speeder in the following
five situations. In the first four the relative speed of the police car and
speeder is always 29 m/s (65 mph). For the speed of sound use 340 m/s:

i) v_P = 29 m/s, v_S = 0

ii) v_P = 0, v_s = - 29 m/s

iii) v_P = 14.5 m/s, v_S = -14.5 m/s

iv) v_P = 39 m/s, v_S = 10 m/s

v) v_P = v_S = 0, wind blowing at 29 m/s from speeder to police car

13. In a discussion
of Doppler shifts of ultrasonic (high-frequency) waves used in medical diagnosis,
the authors remark: "For every millimeter per second that a structure in
the body moves, the frequency of the incident ultrasonic wave is shifted
approximately 1.3 Hz/MHz." What speed of the ultrasonic waves in tissue
do you deduce from this statement? (*Hint*: There are two Doppler shifts.
First treat the reflecting structure as a moving observer and then as a
moving source emitting at the frequency it "observed".) (*Note*: After
deriving the exact relation between the incident and reflected frequencies,
use the fact that the speed of the structure is much less than the speed
of the ultrasound to find an approximate expression, and use this to answer
the question.)

14. A bat is flittering about
in a cave, navigating by the use of ultrasonic bleeps (short emissions of
high-frequency sound lasting a millisecond or less and repeated several
times a second). Assume that the sound emission frequency of the bat is
39.2 kHz. During one fast swoop directly toward a flat wall surface, the
bat is moving at 8.58 m/s. Calculate the frequency of the sound the bat
hears reflected off the wall. (*Hint*: See the previous problem. *Answer*:
41.2 kHz.)

15. A beam of high speed electrons in water is moving faster than the speed of light in water and so emits Cerenkov radiation. If the wavefront is a cone of angle 58 degrees, what is the speed of the electrons?

**honors problem **(due Wed, 11/27):
The microwave anisotropy
probe (MAP) is a satellite that was placed at the L2 Lagrange point
to make measurements of the cosmic microwave background to learn about the
early history of the universe and the origin of the structure that eventually
formed stars and galaxies and all of us. The mission has finished taking
a year of data and the first results are set to be announced in a month or
two. The scientists lips are sealed until the announcement, but there are
hints that they may have found some surprises. For this assignment, you are
to visit the MAP website
(http://map.gsfc.nasa.gov/), read this Introduction
to the Cosmic Microwave Background (http://background.uchicago.edu/~whu/beginners/introduction.html),
and see what it's all about. Write a one (or more) page summary of some physics
of the MAP mission, emphasizing any concepts discussed in Physics 273.

**Assignment 10:**

due Thur, 11/14

**reading:**

11/8-11/14: HRK: Ch.
39 (Light waves), omitting section 6.

**problems:**

(Q = question, E = exercise,
P = problem in HRK.)

1. A conducting sphere
of radius R is being charged by a current I flowing through a thin wire
connected to the sphere at one point. What is the displacement current through
a spherical surface enclosing the conducting sphere?

2. HRK Ch. 38, E5 (displacement
current in a square capacitor) (*Hint for part (a)*: It is simplest
to use the fact that i + i_d is the same through any surface spanning a
loop.)

3. We showed in class that
the electric and magnetic fields in vacuum satisfy the 3d wave equation
with a wave speed equal to c = (epsilon_0 mu_0)^(-1/2). For fields that
are plane symmetric, which only depend on z and t, each field component is
therefore a superposition of a function of z - ct and a function of
z + ct. Assuming a purely unidirectional wave, **E**=**E**(z-ct) and
**B**=**B**(z-ct), show that Maxwell's equations imply that, apart
from constant fields, **E** and **B **are perpendicular to each other
and to the z-direction, that E = cB, and that **E**x**B **is in the
+z direction. (*Hint: *If you need some help see the discussion in the
supplement on Maxwell's equations.)

4. HRK Ch. 38, Q21 (*deflection
of em waves by em fields?*) Explain why or why not.

5. HRK Ch. 38, E14 (*radiation
pattern of a magnetic dipole antenna*)

6. HRK Ch. 38, P13 (*electric & magnetic fields, intensity,
momentum and pressure of a plane wave*)

7. A WMUC radio transmitting
antenna is a vertical electric dipole. In your dorm room you have a magnetic
dipole receiving antenna in the shape of a circular loop, which works
by Faraday's law: a changing magnetic flux through the loop generates an
emf in the loop. If the vertical direction is +z, and the direction to
to the transmitter is +x, your receiving antenna will pick up a signal if
it is in (choose one): (a) the x-y plane; (b) the y-z plane; (c) the
x-z plane; (d) [(a) & (b)]; (e) [(b) & (c)]; (f) [(a) &
(c)]; (g) all of the above; (h) none of the above. **Explain!**

8. The antenna of problem
7 operates at a (very low) power of 10 W. (a) If you listen to the station
at a distance of 1 km from the antenna,

what is the direction and average magnitude of the Poynting vector
at your location? (Treat the waves as spherical for this part.)

(b) What is the rms electric field in the waves where you are listening,
in volts per meter, and the rms magnetic field in Tesla? (Treat the waves
as planar for this part.) (c) What is the pressure exerted by the waves
on a reflecting surface?

9. In the supplement equation (14) expresses conservation of electromagnetic field energy plus particle energy. Derive this equation from Maxwell's equations.

10. HRK Ch. 38, P12 (*power
flow into a resistor)* Neglect end effects. *Hint:* Note
that the potential difference across the resistor can be expressed as the
line intergral of the electric field over a curve running along the
resistor from one end to the other.

11. HRK Ch. 38, P14 (*power flow
into a capacitor)
*

12. HRK Ch.
38, P15 (*laser pressure on a tiny sphere*)

13. HRK Ch. 38, E42
(*laser rocket*)

**honors
problem: **TBA

reading:

11/1-11/7:

HRK: Ch. 38 (Maxwell's equations and electromagnetic waves)

Supplement: Differential form of Maxwell's equations and electromagnetic waves.

problems:

*1. Transverse
waves on a stiff string* : The wave equation for transverse waves
on a string with some stiffness (resistance to bending) has the form

y_tt = a y_xx - b y_xxxx

where the subscripts
denote partial derivatives with respect to the indicated variable, and a
and b are both positive constants. The term with coefficient a is the tension
term, while that with coefficient b is the stiffness term. (a) What are the
dimensions of $a$ and $b$? (i.e. what powers of length, time and mass do
they contain?) (b) Find the dispersion relation w(k) for waves satisfying
this equation. (c) Sketch the dispersion relation. (d) Does the stiffness
term make the phase and group velocities smaller, larger, or the same as
without this term? Give a physical reason for this. (e) What *dimensionless*
quantity involving the wavenumber k should be much less than unity if (i)
the stiffness term is negligible compared with the tension term in the dispersion
relation, or (ii) the tension term is negligible compared with the stiffness
term? (f) What are the phase and group velocities under the conditions (i)
and (ii) of the previous part?

2. * Longitudinal waves in a 1d crystal :*
Consider
an infinite linear chain of masses m, connected by springs with force
constant h, with a nearest neighbor separation a in the equilibrium
situation. We are going to study longitudinal vibrations of this chain.
(a) Write down Newton's law for the nth mass point in terms of the longitudinal
displacements from equilibrium x_{n-1}, x_n, and x_{n+1} of that mass
point and its nearest neighbors. (The equilibrium position of the nth
mass is na.) (b) Assume a solution of the form x_n(t) = A exp(-iwt) exp(ikna),
and show that the relation between w and k that is required by the equation
of motion for the nth mass is w^2=(4h/m)sin^2(ka/2). This is independent
of n, so you really have a solution for the whole chain! (c) Show that
the solution with k + 2 pi/a represents exactly the same physical situation
as the solution with k. (d) Plot the dispersion relation w(k) and indicate
the range of distinct values of k. (e) Since w(k)/k is not the same for
all k, the different k's propagate with different speeds. Find the phase
and group velocities for lambda = 2 pi a. (f) Take the limit as k goes to
zero to find the speed of long wavelength waves. Show that it is aSqrt[h/m].
(g) Find a value of k for which the group velocity vanishes, and show
that for this k the motion of adjacent masses is 180 degrees out of phase,
so it is just a standing wave on the infinite chain.

3. * Capillary waves
on water *: For sufficiently short wavelengths, surface waves
on water are governed by the restoring force of surface tension sigma
rather than by gravity. The surface tension is defined as the energy per
unit area required to increase the surface area, i.e. the energy to increase
the area by dA is sigma dA. (a) Show that no quantity with dimensions of
velocity can be made from just the surface tension and the mass density
rho of the water. Therefore the wave speed cannot be the same for all wavelengths.
(b) Show using dimensional analysis that the dispersion relation must take
the form w^2 = B k^3 for these waves, and find B in terms of sigma
and rho up to an unknown numerical constant (which I think is in fact equal
to one!). (c) If capillary waves with a wavelength of 5 mm have a phase
velocity of 5 cm/s, what is the group velocity of a wavepacket consisting
of wavelengths centered on 5 mm? (Do not evaluate B for this part. It
drops out of the result.) (d) Now evaluate B (i) using the surface tension
and mass density of water (look them up!), and (ii) using the wave parameters
given in part (c), and compare. Are the given parameters consistent with
capillary waves on water?

4. *Limits on the frequency
dependence of the speed of light*

5. ** Wavepacket motion
and spreading for a quantum particle** : The dispersion relation
for the quantum wave corresponding to a (non-relativistic) particle of
mass m is w(k) = hbar k^2/2m. (Using energy E = hbar w and momentum
p = hbar k this corresponds to E = p^2/2m, the kinetic energy expressed
in terms of momentum.) (a) Show that the group velocity corresponds to the
classical velocity p/m, while the phase velocity is half of that. (b) Now
adopt a system of units for which hbar = m = 1, and adopt an arbitrary unit
of length a and use units where a = 1. The unit of time hbar/ma^2 is then
also unity. Construct a wavepacket

Y(x,t) = ∫ dk A(k) exp[i(kx - w(k)t)],

using an amplitude A(k)
that is zero if k is not between 9 and 11, and A(k) = exp(i 7 k) for
9< k <11. (i) Numerically evaluate Y(x,0) and plot the
real part of the resulting function. Where is it centered? (This integral
is simple to do by hand.) (ii) Numerically evaluate Y(x,3) and plot the
real part of the resulting function on the same graph. Where is it centered?
(This integral cannot be usefully done by hand.) (Use a computer program such as Matlab,
Mathematica, Maple, or IDL to do the integrals and plots.) (c) (i) How far did the center of the
wavepacket travel in the one unit of time? (ii) Evaluate the group velocity
for this wavepacket and compare with the speed you found directly from
the wavepacket. Do they agree? (d) (i) What is the difference between
the width of the wavepacket at time t=3 and the width at time t=0? (ii)
Evaluate the difference between the group velocities at k = 9 and at k
= 11. (iii) If you imagine that the spreading of the wavepacket can be
attributed to this difference in group velocities, by how much would it
spread over one unit of time? (iv) Do your answers to parts (i) and (iii)
agree?

honors problem:

due Tuesday 11/12

You may have noticed sometimes a standing pattern of ripples upstream from a rock in a stream. The same effect can be created by moving a submerged object with constant velocity relative to standing water, since all that matters is the relative motion of the object and the water. This phenomenon only occurs once the relative speed reaches a particular critical value, and when that happens the pattern appears with a definite wavelength. It can be understood in reference to the dispersion realtion for waves on water, and involves the balance between surface tension and gravity as the restoring force.

a) Observe this phenomenon in your bathtub or in the fountain on McKeldin mall or in some other body of water. Try to estimate the critical flow speed and the wavelength. What happens to the wavelength as you increase the speed?

b) First neglect surface tension, and assume the wavelength is much shorter than the water's depth. (i) Find the wave number for which the phase velocity of a gravity surface wave to be equal to the flow speed v of the water. (ii) Show that at this wave number the group velocity is *lower* than v, so wave energy would be swept downstream with the flow. I think this means that no standing pattern would form.

c) Now include surface tension. In class we discussed the dispersion relation for gravity surface waves on deep water, and in the homework you discovered the dispersion relation for capillary surface waves. (i) Combine these into a single dispersion relation that takes both effects into account, and sketch the dispersion relation. To make an educated guess at how to combine them, think about the fact that the gravitational and surface tension restoring forces add to give the total restoring force, to justify a dispersion relation of the form

w^2 = a k + b k^3where a and b are some positive constants. (ii) Find the minimal flow speed v for which the phase velocity can be equal to v, and find the wavenumber for which this occurs. Show that v^2 = 2 Sqrt[ab] and k = Sqrt[a/b]. Then put in the numerical values for the acceleration of gravity g, the surface tension sigma, and the mass density rho, and find the critical v in cm/s and the corresponding wavelength in cm, and compare with your estimates from part a). (

**reading:**

10/24-31:

1. French, Ch. 8, pp. 253-265;

2. French Ch. 7 pp. 230-234 (whatever you can get out of it);

3. an applet and a bit of text about group velocity and dispersion:

www.phys.virginia.edu/classes/109N/more_stuff/Applets/sines/GroupVelocity.html

4. Mathematica
notebook output (html) on group velocity

**problems:**

*2. Intensities and
energy densities of some waves:*

Consider two
very different types of waves, (a) Sound waves 4.82 km from a 47.5
kW siren, assuming the waves to be spherical, the propagation isotropic
with no atmospheric absorption, and the speed of sound to be 343 m/s.
(b) Gravitational waves satisfy the wave equation (7-43). Consider
gravitational waves produced by the merger of two neutron stars
spiraling in and crashing into each other 10^9 light years away. Assume
the power output in gravitational waves is spherical and equal to 1%
of the rest energy (E = mc^2) of the stars, divided by the time it takes
one of the stars to travel a distance equal to its own radius. The stars
each have a mass equal to 1.4 times the mass of the sun and a radius of
about 20 kilometers, and they are moving at around 10% of the speed of
light when they collide. Compute the intensities and energy densities
of the waves (a) and (b) and compare to the intensity and energy density
of sound at the threshold for human hearing, 10^-12 W/m^2.

3. A large parabolic reflector having a circular opening of radius 0.50 m is used to focus sound. If all the energy falling on the reflector is delivered tfrom the focus to the ear of a listening detective though a tube of diameter 1.0 cm with 12% efficiency, how far away can a whispered conversation be understood? Assume that the intensity of a whisper is 10^-10 W/m^2 at 1.0 m from the source, considered to be a point.

4. Sound intensity is sometimes
measured in *decibels*. The sound level (SL) in decibels (dB)
is defined by SL = 10 log(I/I_0), where I_0 = 10^-12 W/m^2, (the threshold for human hearing
at some frequency) is chosen as the reference value. (a) What is the
sound level in decibels of the whisper in problem 3? (b) Find the intensity
ratio of two sounds whose sound levels differ by 1 dB. (c) How many
persons in a room speaking at the same time each at the average human
speech level of 65 dB are needed to produce a sound level of 80 dB?

5. F8-1 *(reflection amplitudes)*

6. F8-5 *(penetration of
sound from air into water)* For part (a) show that
the amplitude ratio is 2v_1r_1/(v_1r_1 + v_2r_2) which is approximately 2v_1r_1/v_2r_2, where v_1,2 are the
sound speeds in air and water and r_1,2 are the densities of air and water. Then evaluate
this numerically. For part (b) show that the energy flux ratio is approximately
4v_1r_1/v_2r_2, and evaluate this. Add a third part, (c):
Suppose the incident wave has displacement s_in(x,t) = f(x-vt) forsome
function f, where x is the coordinate normal to the interface at x=0.
Neglecting the weak transmitted wave, find the reflected wave displacement
s_ref(x,t) in terms of the function f. (*Hint for part (b): *See
problem F7-24.)

7. F7-19 (Dispersion relation graph)

**honors problem: **none

due
Tuesday 11/5

**reading:**

10/17-24:

French, Ch. 7, This chapter is a mine field. My recommendation
is that, having already read 201-212 last week, you should skip everything
other than what is in the following list: pp. 215 *(waveforms of
musical instruments)*, 223-224 *(motion of pulses)*, 228-230
*(superposition of pulses)*, 237-240 *(energy in a mechanical
wave)*, 244-246* (waves in 2 & 3 dimensions).*

**problems:**

2. F7-13 (

3. F7-15 (*superposition
of pulses)*

Add part (b): The intensity of a wave is defined as the energy per unit time flowing through a unit area perpendicular to the direction of wave propagation. Show (as discussed in class) that this is equal to the energy density times the wave speed, and use this to obtain an expression for the average intensity using your result from part (a).

*Hint for part (a):*
For the potential energy, first think about a simpler problem. Show that
the work required to stretch a rod of length L by an amount dL is (YSL/2)(dL/L)^2.
Note: As shown in the text on p. 239 in the case of transverse waves
on a string, the kinetic and potential energy densities are precisely
equal for ANY purely right-going or purely left-going wave.

5. *Displacement amplitude
and intensity in sound waves*

a) Show that
in a sound wave, the pressure variation amplitude q_m is related to
the displacement amplitude s_m by q_m = w v r s_m, where w is the
angular frequency, v is the wave speed, and r is the mass density of air.

b) The maximum pressure variation that the human ear can tolerate in loud sounds is about 28 Pa at 1000 Hz. The faintest sound that can be heard at 1000 Hz has a pressure amplitude of about 2.8 E-5 Pa. Find (i) the corresponding displacement amplitudes and (ii) the wave intensities in W/cm^2.

*Hint for part (a)*:
Remember that (stress) = (elastic modulus) (strain). For a compressional
wave in air, the stress is the pressure p, the relevant elastic modulus
is the bulk compressibility K, and the local strain at a point x is
ds/dx, where s(x,t) is the displacement of the air at x. *Hint for
part (b):* See problem 5.

due Tuesday 10/29

Try your hand at some relativistic string theory. Consider a string with the property that its tension T is constant no matter what its length is, and suppose that when the length is L the rest energy of the string Mc^2 is equal to the work TL it costs to stretch the string from zero length to length L (c here is the speed of light). That is, the rest energy is entirely due to the work done to stretch the string.

1) What is the speed of transverse waves on a straight string of this type?

2) Consider a circular loop of this string. If initially at rest, it will contract and then reexpand, followed by an identical cycle and so on. Show that the equation that governs the radius r(t) as a function of time is a harmonic oscillator equation with a frequency w determined by the total energy E of the motion. What is the speed of the string as it goes through the origin?

3) Now let's quantize the string: suppose the only allowed energies are the ones for which E = n hbar w, for some nonnegative integer n. What are the allowed energies, and what are the corresponding rest masses?

4) Express the tension as a dimensionless number g times a quantity built entirely from Planck's constant, the speed of light, and Newton's constant.

5) Express the masses you found in part 3) in terms of the dimensionless number g from part 4), and find the numerical values for these masses in grams. (They are huge compared to elementary particle masses. If an elementary particle were to correspond to one of these strings it must be the n=0 case, which is massless. The massive elementary particles would then have to get their tiny masses from some other small effect.)

6) Ignoring quantum mechanics, what would be the total mass in grams and maximum radius in centimeters of an oscillating string with a period of one second?

**due Thurs, 10/17 (may
also be turned in Mon 10/21 but I don't recommend it)**

**reading:**

week
of 10/14:

French, Ch. 7, pp. 201-212 *(Progressive waves)*

**problems:**

1. F6-6 (*longitudinal
harmonics of a rod clamped at center) *In part (a) express your answer
for the nth mode in terms of the the fundamental frequency, and give the
fundamental frequency in terms of the length, density, and Young's modulus
of the rod. In parts (b) and (c) give explicit answers for n = 1,2,3, and
then a general formula valid for any n. Include illustrations for n = 1,2,3.
(*Note*s: (i) Your answer may look different from the answer in the
book but may nevertheless be equivalent. (ii) Since the rod is clamped in
the center the midpoint is required to be a node. You also have the bc that
the endpoints are anti-nodes.)

2. The pitch of the wind instruments rises and that of the string instruments falls as an orchestra warms up. Explain why.

3. If the wavelength of sound is less than a factor of about 10 larger than the mean free path of the molecules, then sound waves cannot propagate through a gas. For air at room temperature the mean free path is about 0.1 micron and the sound speed is 343 m/s. Calculate the frequency above which sound waves cannot propagate.

4. The wave equation for
waves on a string is y_tt = v^2 y_xx (where y_tt and y_xx are the second
partial derivatives of y w.r.t. t and x respectively.) (a) Confirm that y(x,t)
= f(x - vt) + g(x + vt) is a solution to this wave equation for any pair
of functions f and g. (b) Actually, this is the form of the general solution.
Find the expressions for the functions f and g in terms of the general
initial conditions y(x,0) and y_t(x,0). (*Hint for part (b)*:
First find f' and g', the derivatives of f and g with respect to their
argument, and then integrate to find f and g. The answer will involve
an integration over x of y_t(x,0).)

6. F7-3 (*y(x,t)
from wave parameters)*

7. F7-5 (*y(x,t)
and parameters from other data)*

**Assignment 5:**

**due
Thurs, 10/10**

**reading:**

week
of 10/7:

French, Ch. 3, pp. 45-49 (*elasticity and Young's
modulus*), pp. 57-59 (*spring of air, bulk modulus*), 168-181
(*longitudinal vibrations: rod and sound*; *generalities*)*,
*pp. 189-191* (Fourier analysis)*

**problems:**

1. French 5-6 (*spring
coupled masses between two walls*) Do only parts (a) and (b). For part
(b) note that the normal mode coordinates are q_1 = x_A + x_B and q_2 = x_B
- x_A. Find the equations for q_1 and q_2 by adding and subtracting the equations
for x_A and x_B. Add the following part: (c) Describe the motion of the normal
mode oscillations, and explain why one of them has the same frequency
as would a single mass on a spring with force constant k_0, and why
the other has a higher frequency.

2.** [20 points]** French
5-9 (CO_2 *molecule*) *Hint:* One of the normal modes
is obvious, and can be completely described by the position of one
of the oxygen atoms. To simplify finding the other normal mode use the
information that the distance between the oxygen atoms is constant and
the center of mass remains at rest. Thus the position of the carbon atom
determines all three positions. To implement this let x_1,2,3 denote
the displacements from equilibrium of m_1,2,3. Set x_3 = x_1 to eliminate
x_3, and solve for x_1 in terms of x_2 using the fact that the center
of mass remains at rest. In part (a), take "solve the equations
for the two normal modes" to mean find the frequencies and amplitude
ratios in the normal modes in terms of m_O and m_C. Add the following
parts: (c) Since there are three atoms, you'd think there should be three
normal modes. What is the third one and what is its frequency? (d) Look
at this information
on carbon dioxide vibrations, and compare the ratio of your
mode frequencies to the measured value for the corresponding modes
of carbon dioxide. (Note you don't need to know the spring constant
to do this! Also note that frequency is given there in units of cm^-1,
because they are referring to the frequency of the radiation that is
in resonance. What is indicated is the inverse wavelength of the radiation,
which is proportional to the frequency since frequency = 1/period = wavespeed/wavelenth.
The wavespeed here is the speed of light.) Discuss the level of agreement
and try to give reason(s) why the model in the problem may differ from
the experimental result. (One reason is "quantum mechanics versus classical
mechanics", which I think turns out not to really be a reason in this
case. Think of another.)

3. French 5-13 (*bead(s)
on massless stretched string*) *Hint*: The y-component
of the force on the bead from the left hand side in part (a) comes
from the tenstion and is given by T_y = Ty/L.

4. F6-1 *(string harmonics)*

5. A piano wire 1.4 m long
is made of steel with a density 7.8 g/cm^3 and a Young's modulus
2.2 x 10^11 N/m^2. The tension in the wire produces a strain of 1.0%.
(a) Calculate the fundamental frequency of the wire. (b) What is
the maximum transverse speed if the maximum amplitude of vibration
is A = 1 mm? (c) What must be the amplitude is the transverse speed
is to exceed the speed of sound? Would the small displacement approximation
hold for this amplitude? Why or why not? Would the string break?!
(*Answers*: (a) 190 Hz, (b) 1.2 m/s)

6. A guitar string is a bit out of tune. Four beats per second are heard when the string is sounded together with a tuning fork that is vibrating at 440 Hz (an "A" note). What is the period and frequency of the guitar string vibration? Is there more than one possibilty?

7. **[20 points]** F6-11
*(energy of string modes and superpositions)* Add parts (b1)
Show that the increase of potential energy when the string is displaced
from equilibrium is equal to the tension times the increase in length,
and that for small displacements this is approximately equal to
(T/2) integral_0^L dx (y_x)^2; and (b2) Show that for ANY superposition
of two normal modes such as the one in the problem the total energy is just the sum
of the energies of the two modes taken separately. You may find it
helpful to read p. 196.

8. F6-12 (*energy and shape
recurrence of a plucked string)*

**honors problem: **Normal modes
of a loaded guitar string

due
Tuesday 10/22

**Assignment
4:**

**due
Thurs, 10/3**

**reading:**

week
of 9/30: French, Ch. 5, pp. 119-131 (*coupled oscillators*),
135-136; Ch. 6 pp. 161-170 (*vibrating string*)

**problems:**

1. [**25 points] **Consider five compound circuit elements:
(i) an inductor and a resistor in series, (ii) an inductor and
a resistor in parallel, (iii) a capacitor and an inductor in series, (iv) a capacitor
and an inductor in parallel, (v) a resistor, an inductor, and
a capacitor all in parallel. For each of these compund circuit
elements, (a) Find the net complex impedance and its limiting
forms as the frequency goes to infinity or zero. Explain why these limiting
forms are expected from the behavior of the individual circuit
elements. (b) Find the amplitude of the total current. Sketch
this amplitude as a function of frequency and explain the behavior of the
current at high frequency, low frequency, and any interesting frequencies
in between.

2. (a) For a parallel
RC circuit as discussed in class Wednesday, show that no power
is dissipated in the capacitor. Do this by computing the time
average <V(t) I_C(t)>, where V(t)=V_0 cos(wt) is the driving voltage
across the circuit and I_C(t) is the current in the capacitor. (b) Since
I_(total) = I_R + I_C, part (a) means that the net average power
dissipated can be computed either using the TOTAL current as <V(t)
I_(total)(t)> or using just the current in the resistor as <V(t)
I_R(t)>. Find I_total(t) and I_R(t), and confirm that these two
averages give the same result.

3. Consider a series RLC
circuit with a generator supplying an alternating voltage of
fixed frequency w. The capacitor is variable. (a) For a value
C = C_1, the current I_1 is found to be in phase with the applied voltage.
What is C_1, in terms of L
and w?
(b) The capacitance is then changed to C = C_2, so that the voltage
is observed to lead the current I_2 by a phase angle of 45 degrees.
What is C_2 in terms of C_1, R and w? (c) What was the ratio I_1/I_2? (*Answers:
*1/w^{2}L, C_1/(1 - wRC_1), Sqrt[2].)

4. HRK Ch. 37, Exercise 20
(*air conditioner*)

5. HRK Ch. 37, Exercise 22
(*antenna response*) The phrase "alternating emf in the
antenna" means an emf between the antenna and the ground. That
is, you can think of it

as
a closed circuit with the radio wave producing an AC emf.

6. HRK Ch. 37, Exercise 23
(*"impedance matching"---maximum power when source and load
resistance are matched*) Note: The internal resistance
r of the generator is fixed. You are to find the value of the resistance
R that maximizes the power dissipated in R.

8. HRK, Ch. 37, Problem 11
(*light dimmer*) This is my favorite problem of the bunch!
In part (b), compare also compare the heat dissipated in the resistor
to the heat dissipated in the bulb.

**honors problem:**

due Tuesday 10/8

French 5-10 (hanging pair
of masses on springs) *Hint:* Let y_1 and y_2 denote the displacements
of the two masses. One nice method to find the normal modes and
their frequencies is by writing q = y_1 + r y_2 and solving the condition
on r that results when you require this to be a normal mode. (If you
know linear algebra you'll recognize that this is equivalent to writing the coupled
equations as a single equation on a two-component vector with
a matrix for a spring constant, and finding the eigenvalues and
eigenvectors of the matrix. Find these both by hand and with Mathematica
or another computer program, just for practice.)

due Thur, 9/26

**reading:**

week of 9/23: HRK 37 (Alternating current circuits);
supplement on complex impedance for AC
circuits

**problems:**

1. For a ** damped oscillator
applet** go to http://thorin.adnc.com/~topquark/fun/JAVA/dho/dho.html.
You can plot curves simultaneously for different values of
the damping coefficient (which is measured here in units of the
critical damping value 2w_0).

(i) Plot under, over, and critically damped solutions with the same initial conditions on the same graph. Note that the overdamped case damps less quickly than the critically damped case. Why?

(ii) Explore the relation between the period T and the damping coefficient. Does T get longer or shorter as damping increases? What formula governs this relationship?

(iii) Look at the energy vs. time curve in the underdamped case. Identify when the energy is changing most rapidly, and when it is not changing at all, in relation to the position and velocity. Explain what you find.

2. For a * driven damped
oscillator applet*, go to http://webphysics.ph.msstate.edu/javamirror/explrsci/dswmedia/drivosc.htm.
Find
a nice resonance for some setting of the parameters. (It may help to use the fact that
the displacement lags the force by between 0 and (approximately) 90 degrees
below resonance and by between 90 and (approximately)
180 degrees above resonance.) Write down these parameters,
and compare with the theoretical predictions for the resonant
frequency.

3. ** Energy and damping:**
Suppose a damped oscillator at rest at its equilibrium position is
suddenly struck at t = 0 so that its velocity at t = 0 is v_0 and
its position is x = 0. (a) Find the particular solution x(t)
describing the subsequent motion by writing the general solution and
imposing the initial conditions. (b) What is the total energy E(t)
as a function of time? (c) Using a computer (and Mathematica, Matlab,
Maple, or something else) make a plot of E(t), E(0)exp(-gamma t), and
0.1 x(t)

*4. Critical damping:
*(a) Verify that the general solution for critical damping is
x(t) = (A + Bt) exp(-gamma t/2) (eqn (3-43) of French.) (b) Determine
the conditions on the initial position and velocity for which the
coefficient B = 0. (c) Suppose the initial position x(0) is positive.
Find the condition on the initial velocity v(0) if the position
is to swing to negative values before damping out.

5. ** Transients:**
A damped oscillator is at rest at x = 0 until a force F = F_0 cos(wt)
is turned on at t = 0. Find x(t) for t > 0 and sketch the
resulting function. (Hint: The general solution is the sum of
the steady state solution and a free solution.)

6. F4-7 (Energy in a damped, driven oscillator)

7. F4-11 (Damped driven oscillator, numerical example)

8. F4-13 (Extracting information from a resonance curve)

9. F4-17 (Extracting
other information from another resonance curve)

**honors problem:
**due Tuesday
10/1

** Shaken pendulum: **The
pivot point of a simple pendulum having a natural period of
1.00 s is moved laterally in a sinusoidal motion with an amplitude
1.00 cm and a period 1.10 s. (a) With what amplitude should the
pendulum bob swing after a steady motion is achieved? Assume the damping
force is very small and plays no role except to eventually drive the
system to a steady state, and use the small amplitude approximation.
(b) Find the length of the pendulum and check that your result is consistent
with this approximation. (

**reading:**

week of 9/16: damped and driven oscillators,
resonance, transients: HRK36:7, F4:77-111

**problems:**

1. Express the following in
"Cartesian form" x+iy, where x and y are real: 1/(2-3i), (1 +2i)/(3+4i), 5e^{6i.}

2. Express the following in
"polar form" re^{i}^{q}, where r and q are real: -6, -5i, (1+i)/Sqrt[2], 2-3i, (2+i)/(1+2i)

3. (i) Find all the cube roots
of unity 1^{1/3 }and all the cube roots of -1, (-1)^{1/3},
and express them all in both polar form and in Cartesian
form. (ii) Plot and label them in the complex plane.

4. Show that the complex conjugate operation that sends z = x + iy to z* = x - iy enjoys the following properties:

(a)
(z+w)* = z* + w*

(b) (zw)* = z* w*

(c) (z/w)* = z*/w*

(d) (exp z)* = exp(z*)

(e) zz* = |z|^2

5. F1-8 (*trig. identities
using the complex exponential)*

6. F2-1 (*practice with
polar form)*

7. F2-3 (*beats*)
In addition, explain what is wrong with Fig 2-6 on page F26.
(You may like to
look at this visual
applet illustrating beats, and this audio one.)

8. F3-13 (*verify damped
oscillator solution)*

9. F3-14 (*damped oscillator
example)*

10. In a gravitational wave detector mirrors are suspended from a fused silica fiber. The quality factor for the "violin mode" of vibrations of such a fiber was measured to be 57 million for a vibrational motion at 726 Hz (see xxx.lanl.gov/abs/gr-qc/0009035). If the fiber is initially vibrating with an amplitude of 1 mm, how long will it take before the amplitude is damped to 0.1 mm? What fraction of the initial mechanical energy will remain at that point?

11 . Verify the statement after eqn (4-36) on page F111 that no solution of the form x = A cos wt works for the anharmonic oscillator. (Here A and w are assumed to be arbitrary constants.) More precisely, (i) show that there are solutions of this form only if w = 0; (ii) find these solutions with w=0, and interpret them physically by sketching the potential energy curves for the two cases (a) and (b) and referring to the relevant features of these curves.

**honors problem:
**due Tuesday 9/24

F3-18 (*sloshing water in
Lake Geneva---the seiche) *Also find the period for a bathtub of length
1.5 m and water of depth 20 cm. Does your answer seem reasonable?
The derivation in this problem is long and involved, but the final result is
very simple (given the approximations that were made): T =
2L/Sqrt[gh]. You cannot infer the form of the answer by simple dimensional
analysis since there are two lengths in the problem---the length L and
depth h of the lake---and the period might have depended on an arbitrary
function of the
dimensionless ratio h/L. Can you come up with a simple physical
argument that resolves this ambiguity without going through the
whole calculation, so that you might have been able to guess the answer
up to an overall constant?

**reading:
**week of 9/2: harmonic oscillators,
F1:3-9, F3:41-43, F3:49-54, HRK36:5-6

week of 9/9: harmonic oscillators, complex numbers, F1:10-16, F2:19-27, F3:43-45, F3 62-70

**problems:**

1. How are each of the following properties of a simple harmonic oscillator affected by doubling the amplitude: period, force constant, total mechanical energy, maximum velocity, maximum acceleration?

2. A loudspeaker produces
a musical sound by the oscillation of a diaphragm. If the
amplitude of the oscillation is limited to 1.20 x 10^{-3} mm, what
frequencies will result in the acceleration of the
diaphragm exceeding g? (Assume
harmonic oscillation.)

3. A 5.22 kg object is attached to the bottom of a vertical spring and set vibrating. The maximum speed of the object is 15.3 cm/s and the period is 645 ms. Find (a) the force constant of the spring, (b) the amplitude of the motion, and (c) the frequency of oscillation.

4. Two particles execute simple harmonic motion of the same amplitude and frequency along the same straight line. They pass one another going in opposite directions each time their displacement is half their amplitude. Find the phase difference between them.

5. A thin uniform rod of length L hangs from a pivot at a distance d from one end. (a) What is the period of small oscillations of the rod? (To check your answer see problem 3-5 in French.) (b) For what value of d is the period minimized?

6. **Vibrating molecule model:**
Consider two masses m_{1} and m_{2} moving in one dimension
and interacting with a potential energy V(r), where r is the distance
between the two masses. Let r_{1} and r_{2} be the distances
of the masses from their common center of mass.

(a) Show that r_{1} = [m_{2} /(m_{1}+m_{2})]
r and r_{2} = [m_{1}/(m_{1}+m_{2})]
r.

(b) Show that the kinetic energy of the motion about
the center of mass is 1/2 m (dr/dt)^{2},

where m = m_{1}m_{2}/(m_{1}+m_{2})
is the so-called ``reduced mass".

(c) Suppose that V(r) = -a/r + b/r^{2}, with
a and b positive.

(i) Sketch a graph of V(r).

(ii) What is the equilibrium separation r_{0}?

(iii)What is the frequency of small oscillations about
equilibrium? (*Hint: *See hint in problem 8.)

7. French, Problem 3-19, part
(c) only. (*Hint*: Remember that the effective spring constant
is the second derivative of the potential energy evaluated
at the equilibrium point.)

8. **Adding springs:
**If two springs with force constants k_{1} and
k_{2} are connected together the force they exert is
equivalent to that of a single spring with force constant k.
Show that if the springs are connected in parallel then k = k_{1}
+ k_{2}, while if they are connected in series then 1/k =
1/k_{1} + 1/k_{2}. (*Hint:* What quantity
is additive in each case?)

**HONORS HOMEWORK: **Anharmonic
correction to the period of an oscillator