Homework Assignments

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. As the course syllabus states: Homework sets must show reasoning leading to the final answers in a clear and readable fashion to obtain credit.

Notation: F1:3-9 means "French, Chapter 1, pages 3-9, and HRK36:5-6 means Halliday Resnick and Krane, volume 2, 5th edition, Chapter 36, section 5 and 6.

Assignment 13:
due Thurs, 12/12

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


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)?

Assignment 12:
due Thurs, 12/5

HRK vol. 2, Ch. 41: Interference
HRK vol. 2, Ch, 42: Diffraction

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.


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.) 

honors problem (due Tues, 12/10): Gravitational Wave Astronomy

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. (Hint: Use E = hbar w = h f.)
5. Derive equation (7) from the equipartition theorem (1/2 kT energy per degree of freedom), and verify the numerical value.

Assignment 11:
due Wed, 11/27

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

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


(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 ExB 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

Assignment 9:
due Thurs, 11/7


HRK: Ch. 38 (Maxwell's equations and electromagnetic waves)
Supplement: Differential form of Maxwell's equations and electromagnetic waves.

[20 points each]

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:
Stationary ripples on a flowing stream
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^3
where 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). (Hint: It really helps to graphically represent the flow speed by a straight line of slope v on your sketch of the dispersion relation.) (iii) Indicate on your graph what happens when v is greater than the critical value. Show that there is then a wave number  with phase velocity v and group velocity greater than v. Is the corresponding wavelength shorter or longer than the critical one? Compare with your observations from part a).

Assignment 8:
due Thurs, 10/31

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:
4. Mathematica notebook output (html) on group velocity


1. (a) Show that Y(r,t) = u(r,t)/r satisfies the 3d wave equation (7-44) if and only if u satisfies the 1-d wave equation u_tt = v^2 u_rr.

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

Assignment 7:
due Thurs, 10/24

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).


1. (speed, normal modes, energy, tension & force for a vibrating string) A guitar string of length 60 cm is plucked while touching the string at a point one third of the way from one end. The resulting vibration has a frequency of 600 Hz, an amplitude of 1 mm, and is the lowest frequency mode with a node at that point. (a) What is the speed of transverse waves on this string? (b) Write an expression for the transverse displacement y(x,t) in this mode with x in centimeters and t in seconds. (c) If the total energy in this mode is 0.01 J, what are the tension and linear mass density of the string? (d) What is the maximum transverse force exerted by the string on the bridge? (The "bridge" is the support at one end. Assume the string is not moving at either end.)

2. F7-13 (transverse velocity of a pulse and longitudinal motion) For part (c), sketch y(x,t) at t=-epsilon and at t = +epsilon on the same graph, where epsilon is small enough that the pulse does not travel much in that time. Sow how the change of y relates to the velocity you computed. (It should be the velocity times 2 epsilon.) 

3. F7-15 (superposition of pulses)

4. 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 (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.

honors problem:  Relativistic String Theory
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?

Hints for 2): Start by writing the total energy as a function of r and dr/dt, and then use the fact that this energy is conserved to get an equation for the second time derivative of r with respect to t. To find the total energy, use the fact that the total (rest plus kinetic) energy of a mass m moving with speed v is E = gamma mc^2, where gamma = 1/Sqrt[1-(v/c)^2]. There is no separate potential energy---that is included in the rest mass.

Assignment 6:

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

week of  10/14:
French, Ch. 7, pp. 201-212 (Progressive waves)


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. (Notes: (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).)

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

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

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)


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

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


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/w2L, 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.

7. HRK, Ch. 37, Problem 6 (black box & phase shift)                 

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.)

Assignment 3:
due Thur, 9/26

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


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)  on the same graph. Use units with m = v_0 = w_0 = 1, and choose gamma = 0.1. Plot the functions from t = 0 to t = 15. Print out the graph and indicate where the energy is changing most rapidly and where it is constant, and relate that to what the position is doing at those times. Explain.

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. (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.)

Assignment 2:
due Thursday, 9/19

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


1. Express the following in "Cartesian form" x+iy, where x and y are real: 1/(2-3i), (1 +2i)/(3+4i), 5e6i.

2. Express the following in "polar form" reiq, 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 11/3 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?

Assignment 1:
due Thursday, 9/12

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


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?)

HONORS HOMEWORK:  Anharmonic correction to the period of an oscillator