reading:
HRK vol. 2 Ch. 46: Diffraction
HRK vol. 2 Ch. 47: Gratings and Spectra
much of the same material is covered in French, pp. 280-298

problems:

 1. HRK Ch. 45, Q2 ( Why screen $A$ in Young's double slit experiment?)
 2. HRK Ch. 45, Q27 ( soap film vs. oil film)
 3. HRK Ch. 45, Q28 ( angle dependence of thin film interference)
 4. HRK Ch. 45, P10 ( mica flake in front of one slit) (Ans.: 6.64 microns.)
 5. HRK Ch. 45, P18 ( coherence of atomic radiation) (Ans.: 3 m.)
 6. HRK Ch. 45, P26 ( interference intensity) (Ans.: I_0(5 + 4 cos phi)/9.)
 7. HRK Ch. 45, P38 ( oil drop) (Ans. (a) 594 nm.)
 8. HRK Ch. 45, P51 ( Michelson interferometer measurement of index of refraction of air)
 9. HRK Ch. 46, P10 ( diffracting wire) (Ans. 2.45 cm.)
10. HRK Ch. 46, P19 ( diffraction by telescope aperture)
11. HRK Ch. 46, P35 ( double slit interference and single slit diffraction combined)
12. HRK Ch. 47, P19 ( overlapping orders of grating spectra)
13. HRK Ch. 47, P25 ( resolution of sodium doublet)
14. HRK Ch. 47, P39 ( Bragg scattering)
15. 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?

honors problem:
due ***Wednesday***, 12/12 by 5pm
turn in at Dr. J's office, place under door if I'm not there.

LIGO: Laser interferometer Gravitational WaveObservatory.Visit the website "About LIGO":  http://www.ligo.caltech.edu/LIGO_web/about/. View  the slide show and read the fact sheet. 1) Based on the information there, calculate how many times the laser light will bounce back and forth between the interferometer mirrors during one cycle of the typical gravitational wave LIGO is designed to detect.  2) Suppose the souce of a gravity wave is a coalescing pair of black holes and  the (dimensionless) gravitational wave amplitude is 0.01 at a distance of 100 kilometers from the center of the binary orbit. If LIGO can detect waves of amplitude 10^-21, from how  far away could waves from this system be detected? (Assume the usual 1/r falloff of wave amplitude. Give your answer in light years, and also as a fraction of the size of the observable universe, that being around 10^10 light years.) 3) By what distance would the separation of the LIGO mirrors  be changed when such a gravitational wave passes?

reading:
HRK vol. 2 Ch. 48:Polarization (except 48-4).
Circular polarization supplement (note the definition of left and right circular polarization int he supplement is opposite to that in the textbook!)
HRK vol. 2 Ch. 45: Interference (except 45-5)

problems:

1. 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.

2.  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.)

3. 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.)

4. 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?

5. HRK Ch. 42, P8 (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.

6. HRK Ch. 42, P14 (red light, green light)

7.  (a) Superpose right and left circularly polarized plane electromagnetic waves with the same amplitude and propagation direction, and with the x-component of the electric field for the left circularly polarized wave a phase phi ahead of that of the right 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 counterclockwise with the x-direction (viewed head-on with the approaching 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 counterclockwise through an angle Pi (n_L - n_R) L/lambda_0. (c) If  the linear polarization of blue light (475 nm) rotates more than that of red (650 nm) by 90 degrees counterclockwise after propagating though 5 cm of a sugar syrup, what is the difference [(n_R - n_L)/lambda_0]_blue -  [(n_R - n_L)/lambda_0]_red in the syrup?

8. HRK Ch. 48, P3 (polarizing filter and electric field component)

9. HRK Ch. 48, P7 (stacked polarizer sheets)

10. HRK Ch. 48, P10 (mix of unpolarized and polarized light)

11. HRK Ch. 48, P13 (polarization by reflection from water)

12. If you face south at sunset, with the sun directly to your right, how will the sky light in the direction you are facing be polarized? Explain.

honors problem: TBA

reading:
week of  11/19:
RHK vol. 1 pages 457-460 on Doppler effect (handed out on paper, available on reserve at EPSL, scans linked here:
p457, p458, p459, p460) ;
spacetime diagram derivation of Doppler effect;
supplement on Doppler effect for electromagnetic waves;
Also note old reading assignment Ch. 42-3 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) [available on reserve, some copies
for checkout at EPSL].

problems:
all from HRK Ch. 43:

1. Q6 (color dependence of speed of light in quartz}

2. Q14 (interaction between light and transparent medium?)

3. Q25 (change of wavelength but not frequency for light entering glass from air)

4. P7 (refraction fromwater to air)

5. P8 (ocean wave refraction at shore)

6. P17 (apparent depth) Note: The pupil of the eye with which you view the coin subtends only a very small angle from the vertical. Use the small angle approximation. Also: Put in n for water at the end.

7. P20 (layered indices of refraction)

8. P44  (fish and campfire)

9. P48 (internal reflection in optical fiber)

honors problem:
due Wednesday, 11/28

1. Read the sections of the Feynman lectures listed as "optional" above.

2. The index of refraction of the ionosphere for radio waves of frequency 100 MHz is n=0.90.
(a) Find the density of electrons per cubic centimeter in the ionosphere.
(b) Evidently the phase velocity is greater than c! Find the group velocity and show that it is less than c.

Hint 1: I think 100MHz is much greater than the natural frequency of the electrons in the ionosphere, called the "plasma frequency". Look into this a bit and see if that's right.

Hint 2: The answer to part (a) is approximately 10^7.

Hint 3: In doing part (b), the calculation seems simple if you find a polynomial equation satisfied by the frequency, and take it's derivative with respect to wavenumber, without first solving for frequency in terms of wavenumber.

reading:
week of  11/12: HRK: Ch. 43 (Reflection and Refraction) ; French pp. 265-274

problems:

1. HRK Ch. 41, P14 (electric & magnetic fields in a plane wave)

2. HRK Ch. 41, P16 (intensity and magnetic field of EM waves)  You must assume a harmonic wave to get this result. Add part (c): What is the maximum value of the electric field in the wave if the magnetic field is one gauss?

3. HRK Ch. 41, P19 (fields and intensity of spherical waves)

4. HRK Ch. 41, P30 (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.

5. HRK Ch. 41, P34 (force and pressure of radiation with given intensity)

6. HRK Ch. 41, P35 (laser pressure)

7. HRK Ch. 41, P44  (laser rocket)

8. HRK Ch. 42, Q13 (speed of light in matter)

9. HRK CH. 42, P4    (light travel times)

10. HRK Ch. 42, P6 (Ole Romer's measurement of the speed of light using Jupiter's satellites)

honors problem:

none

reading:
week of  11/5: HRK: Ch. 41 (Electromagnetic waves), except material from eqn. (3) through eqn. (10); Ch. 42 Sections 1,2,3. Also Supplement: Differential form of Maxwell's equations, if you haven't yet read it.

problems:

1. F7-19 (Dispersion relation graph)

2. Electromagnetic waves in an optiical fiber satisfy a wave equation of the form E_tt - v^2 div grad E + mu^2 E = 0. A similar equation describes electromagnetic waves in the ionosphere and certain
elementary particles, such as pions. (a) Find the dispersion relation w(k) corresponding to this equation. (b) Find the corresponding phase and group velocities in the limits of "small"and "large" k. (c) To
what is k being compared in part (b)? That is, what dimensionless quantity is small or large in the two limits?

3. HRK Ch. 40, Q26  (Faraday's and Ampere's laws in a cavity)

4. HRK Ch. 40, P7  (displacement current in a square capacitor)

5. HRK Ch. 40, P8  (magnetic field outside a capacitor)

6. HRK Ch. 41, Q4  (characterizing radiation of different wavelengths and frequencies)

7. HRK Ch. 41, Q9  (deflection of em waves by em fields?)  What is the relevance of TEACHING PHOTONS TO TANGO to this question?

8. HRK Ch. 41, P6  (radiation pattern of magnetic dipole antenna)

9. Quantized energy levels as normal modes of the Schrodinger wave equation [worth 30 points]

As discussed in the class notes for Friday Nov. 2, energy levels of a quantum system correspond to the normal modes of the Schrodinger equation. To see how  this works in a simple example, consider a particle in one dimension confined to the interior of a "box" running from x = 0 to x = L. The particle feels no potential except for an infinite one that keeps it inside the box. This means that the wave function satisfies the free particle Schrodinger equation, but the wave function must vanish outside the box. Looking more closely at the Schrodinger equation at the box walls, it can be shown that in fact the wave function must actually vanish at the box walls, psi(0,t) = 0 = psi(L,t).
(a) Solve the Schrodinger equation for states of definite energy in the box and by doing so determine the quantized energy levels. That is, assume psi(x,t) = exp(-iEt/hbar) f(x), and find the allowed values of E and the corresponding spatial wave functions f(x).  Is the lowest energy zero? If not, what is it?
(b) Sketch the wavefunctions f(x) corresponding to the first three energy levels.
(c) For a given allowed energy E, what are the specific momentum components exp(ipx/hbar)  that enter the wave function? Explain this result in relation to the corresponding classical situation, by referring to the physical interpretation of the wave function.
(d) For each energy level, find the normalization for the wave function required by the probability interpretation.
(e)  If the particle is in the lowest energy level, what is the probability of finding it in the middle third of the box, i.e. between x = L/3 and x = 2L/3?
(f) If L is one Angstrom  (around the size of an atom) and m is the electron mass, what is the lowest  energy level in electron-volts?

honors problem:

1. Derive eqn. (14) of the Maxwell supplement.

2.  Read "Understanding Why Sound Waves Travel Faster along Earth's Axis in the Inner Core" from the November 2001 issue of Physics Today, pp. 17-18, and write a brief summary of the main ideas,  about 300 words long.

reading:
week of  10/29: HRK: Ch. 40 (Maxwell's equations), Supplement: Differential form of Maxwell's equations. (See also last week's reading.)

problems:

1.  Intensity of spherical waves:
You showed in hw8 that the wave intensity I is the product of the energy density u (energy per unit volume) and the speed of propagation v of a wave disturbance. In this problem let us apply that to 3d spherical outgoing waves. (a) Assuming an outgoing spherical wave satisfies the wave equation (7-43), show that the intensity falls as 1/r^2. (Hint: The energy density is proportional to the square of the amplitude.) (b) Show that the energy in a pulse that flows through a spherical surface of radius r _1 is the same as the energy that flows through a larger surface of radius r_2.

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.  Limits on the frequency dependence of the speed of light [worth 20 points]

5. Exercise 6.1, Blandford and Thorne  Practice with group and phase velocities

Honors problem:

Exercise 6.2, Blandford and Thorne  Wave packet size and spreading
Actually, don't do part (a). Instead, look at the wavepacket Mathematica notebook and investigate quantitatively whether or not L is indeed approximately Pi/Delta k. Do part (b), assuming part (a) holds. Add a part (c): Use the same analysis as in part (b) to predict the amount of spreading of the wave packets in the mathematica notebook after 1 and 5 units of time, and compare quantitatively with the plots in the notebook. You'll have to come up with some working definition of the "width" of the wave packets.

reading:
week of  10/22: French, Ch. 7, pp. 230-234; Blandford & Thorne, Chapter 6, sections 6.1,2 (I extracted the
relevant 5 pages as a crummy looking pdf file here.), group velocity Mathematica notebook or output in html.

problems:
1. F7-24  (energy density and intensity of compressional waves)
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 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.

2. 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 7.

3. F8-1 (reflection amplitudes)

4. 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. (Hint for part (b): See problem F7-24.)

5. Show that if Y (r,t) = u(r,t)/r satisfies the 3d wave equation (7-44), then u satisfies the 1-d wave equation u_tt = v^2 u_rr. Use this to write the general  spherically symmetric solution to the 3d wave equation in terms of two free functions representing incoming and outgoing spherical waves.

6. (a) Show that the complex function s(x,t) = A exp(-iwt) exp(ikx) satisfies the wave equation with wave speed v if and only if w^2 = v^2 k^2. (b) 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. Write down Newton's law for the nth mass point in terms of the longitudinal positions (absolute positions, not displacements from equilibrium) x_{n-1}, x_n, and x_{n+1} of that mass point and its nearest neighbors. (Note that the equilibrium position of the nth mass is na.) (c) 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 is w^2=(4h/m)sin^2(ka/2). This is INDEPENDENT OF n, so you really have a solution! (d) Show that the solution with k + 2p/a is indistinguishable from the solution with k. (e) Plot the dispersion relation w(k) and indicate the range of distinct values of k. (f) Since w(k)/k is not the same for all k, the different k's propagate with different speeds. Take the limit as k goes to zero to find the speed of long wavelength waves. Show that it is aSqrt[h/m].

honors problem: TBA.

reading:
week of  10/15: French, Ch. 7, pp. 237-246; Chapter 8, pp. 253-259

problems:
1. F7-2  (wave parameters from y(x,t))

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

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

4. F7-12 (understanding transverse velocity, and y(x,t) from wave parameters)

5. F7-13 (transverse velocity of a pulse and longitudinal motion)

6. F7-15 (superposition of pulses)

honors problem: F7-14  waves on a rotating loop of string

Another way to state the result of part (b) is that the wave speed is equal  to the tangential speed of the loop. This is a striking result, as it is independent of the mass and radius of the loop. There must be some simple way to see the result, without doing the calculation...

reading:
week of  10/8: French, Ch. 6,  pp. 189-191 (Fourier analysis), Ch. 7  pp. 201-212, 223-225, 228-230.

problems:

1. F6-1 (string harmonics)

2. 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 amplitude of vibration if the transverse speed is not to exceed the speed of sound?

3. 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?

4. 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. Do (c) before (b), and include illustrations for n = 1,2,3.

5. F6-10 (laser modes)

6. F6-11 (energy of string modes and superpositions) In part (b), verify the statement that 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.

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

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

9. If the wavelength of sound is a factor of 10 larger than the mean free path of the molecules, then sound waves can 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.

10. 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). )

honors problem: Normal modes of a loaded guitar string

reading:
week of  10/1: French, Ch. 6, through p. 180. (Last week's assignment included 161-170.)

problems:
1.  (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.

2. HRK Ch. 39, problem 26 (air conditioner)

3. HRK Ch. 39, problem 29 (max 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.

4. HRK Ch. 39, problem 30 (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.

5. HRK, Ch. 39, problem 36 (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.

6. 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/w²L, C_1/(1 - wRC_1), Sqrt[2].)

7. French, 5-1 (make your own coupled oscillator) Alternatively, find a coupled oscillator system. Make the suggested measurements and compare the observed transfer frequency with the computed one.

8. French 5-2 (spring coupled pendula)

9. 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.

10. 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. In the other normal mode 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.
In part (a), take "solve the equations for the two normal modes" to mean simply find the frequencies of these modes in terms of m_O and m_C. Add the following part: (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?

honors problem:
French 5-10 (hanging pair of masses on springs) Hint: Let y_1 and y_2 denote the displacements of the two masses. Find the normal modes and their frequencies 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.)

reading:
week of  9/24: French, Ch. 5, pp. 119-131; Ch. 6 pp. 161-170;
supplement on complex impedance for AC circuits (.dvi, .ps, .pdf) (I'm not sure any of these work well yet. Let me know of any problems.)

problems:
1. F4-7 (Energy in a damped, driven oscillator)

2. F4-8 (Transients) For clarity make a small change of notation: In part (a), show that the displacement takes the form x(t) = C + Dexp(-gt) and
find C & D in terms of the initial position and velocity. In part (c) find the vavlues of C & D for the stated initial conditions. Note that the presence
of the steady state term in the solution changes the relation between C & D  and the initial conditions. Also, be careful  to correctly resolve the
ambiguity of the arctangent.

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

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

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

6. 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.

reading:
week of  9/17:  finish reading Chapter 4 of French

problems:

1. To find the power absorbed or output by an oscillating system, it is often necessary to compute the time average of cos²(wt + f)  and sin²(wt + f) . Compute these time averages over the time interval [0,T], check that their sum is precisely unity, and show that  each approaches 1/2 in the limit as T goes to infinity.

2. Another "oscillations around a local minimum" problem: A particle of mass m moves in one dimension in
a potential V(x) = (1/4) b (x² - a²)².
(a) Sketch the potential V(x), and indicate the location of any stable or unstable equilibrium points.
(b) Find quantitatively the value of x at the stable equilibrium points.
(c) Find the frequency of small oscillations about these points.

3. A particle of mass m and electric charge q is situated in an alternating electric field along the x-axis, E = E₀cos(wt) . The particle also experiences a force proportional to the third derivative of its x-position: F_a= a d³x/dt³ (which is due to the recoil of the particle from the radiation it emits when accelerated). Find the amplitude and phase (relative to the electric field) of the charged particle in the steady state. This model gives an approximate description of a charged particle that scatters radiation. Do this problem using the complex exponential method. (Note: Don't leave any ambiguous arctangents lying around. Be specific about the phase.)

4. 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. 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 exceot to eventually drive the system to a steady state, and use the small amplitude approximation. Then check that your result is consistent with this approximation. (Hint: The restoring force is proportional to the displacement from the equilibrium position. In this problem, the equilibrium position is oscillating with time in a prescribed fashion.)

5. Suppose a damped oscillator at rest at its equilibrium position is suddenly struck at t = 0 so that its velocity at t = 0 is non-zero but its position is zero. Find the particular solution x(t)  describing the subsequent motion.

6. Critical damping: (a) Verify that the general solution for critical damping is x(t) = (A + Bt) exp(-gt/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.

7.  Play with the following applets for harmonic oscillator simulations, and answer the questions below.

A. For a damped, undriven oscillator go to http://thorin.adnc.com/~topquark/fun/JAVA/dho/dho.html. You can plot curves simultaneously for different values of the damping coefficient.
(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 damped case. Why? Account for the difference quantitatively.
(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.

B. For the driven oscillator, go to http://webphysics.ph.msstate.edu/javamirror/explrsci/dswmedia/drivosc.htm. Using the fact that below resonance the force and displacement are in phase, and above resonance they are out of phase, find a nice resonance for some setting of the parameters. Write down these parameters, and compare with the theoretical predictions for the resonant frequency.

honors problem: In hw1 you determined the first order correction to the period of an oscillator of mass m with anharmonic potential  V(x) = 1/2 kx² + ax⁴, by expanding the integral to first order in the dimensionless quantity ax_m²/k, where x_m is the amplitude of the oscillation. This week let's use numerical integration to check this two ways. Use Mathematica or whatever program you wish, and hand in a printout of your code, plot, and calculations. If you don't  yet  know how to use any computer program to do this, try to find a fellow student who can show you, or ask Dr. J. or Luis. Once again, let's adopt units with m = k = x_m = 1.

(1) Numerically evaluate the integral giving the period as a function of a. Plot this function and verify that it is linear for small a. Find the slope for small a and compare with the result T = 2p - 3pa + O(a²)  from hw1. By what percent do they deviate if a= 0.1? How about a =0.2?  (Hint: Since the integrand diverges at an endpoint the computer may squeal. You may have to keep slightly away from the endpoint.)

(2) Assume a = 0.1 and integrate the equation of motion with the initial condition that the particle starts out at rest at the maximum displacement. Using your solution find the period and compare with the results of part (1).

reading:
week of  9/10:  damped and driven oscillators, resonance, HRK38:7, F3:62-69, F4:77-101

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^iq, 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*)

5. The natural logarithm ln z of a complex number z is defined by the equation exp(ln z) = z. This equation does not determine ln z uniquely, so ln z is said to be a multi-valued function of z.

    (a) Find all the values of ln 1.
    (b) Show that for any z there are infinitely many values of ln z. What do they differ by?
    (c) Express ln (r expiq) in terms of r and q.

6. Using the logarithm we can define z^w for any two complex numbers z and w by z^w = exp(w lnz).
Find all the values of 1², 1ⁱ , iⁱ.

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

8. F2-1 (practice with polar form)

9. F2-3 (beats)  In addition, explain what is wrong with Fig 2-6 on page F26.

10. F3-13 (verify damped oscillator solution)

11. F3-14 (damped oscillator example)

12. 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?

13 . 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,
show that there are solutions of this form only if w = 0. Find these solutions, 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:

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?

reading:
week of 8/27:  harmonic oscillators,  F1:3-7, F3:41-43, F3:45-62, HRK38:5-6
week of  9/3:  harmonic oscillators,   complex numbers,  F1:7-15,  F2:19-26, F3:43-45

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 m₁ and m₂ moving in one dimension and interacting with a potential energy V(r),  where r is the distance between the two masses. Let r₁ and r₂ be the distances of the masses from their common center of mass.
(a) Show that r₁ = [m₂ /(m₁+m₂)] r and  r₂ = [m₁/(m₁+m₂)] r.
(b) Show that the kinetic energy of the motion about the center of mass is 1/2 m (dr/dt)²,
where m = m₁m₂/(m₁+m₂) is the so-called ``reduced mass".
(c) Suppose that V(r) = -a/r + b/r², with a and b positive.
(i)  Sketch a graph of V(r).
(ii) What is the equilibrium separation r₀?
(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 k₁ and k₂ 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 = k₁ + k₂, while if they are connected in series then 1/k = 1/k₁ + 1/k₂.  (Hint: What quantity is additive in each case?)

9.  Microscopic basis of elasticity: As discussed by French on page 49, the elastic moduli of a solid can be related to the interatomic forces in the solid. Let us make a rough analysis of this relationship. Let k₀ be the effective spring constant for an adjacent pair of atoms, and suppose the atoms are arranged in a cubic lattice with interatomic spacing a₀ .
(a) Ignoring all but the nearest neighbor interactions, show that the Young's  modulus for stretching is given by Y =  k₀/a₀. (Hint: Use the results of problem 8.)
(b) Using the Young's modulus for Aluminum  from Table 3-1 on p. 47 of  French, and the interatomic
spacing 3 Angstroms (1 A = 10^-10 m), compute k₀.
(c) If your value of k₀ is reasonable, the energy required to stretch one of these atomic "springs" to a distance of 1A should be of the order of 1 eV, which is a typical chemical covalent binding energy.
Compute this stretching energy and compare with 1 eV.

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