Assignment 1 (due Feb. 4):
Read Chapters 1 and 2. Chapter 2 Exercises 1, 2, 4, 5
Assignment 2 (due Feb. 11):
Read Chapter 3.
Chapter 3 Exercises 1, 3, 5, 6, 7, 10, 11, and
(i) A wire is under a tension T. If the speed of a wave on the wire is to be doubled, the tension must be changed to what value?
(ii) Check if the following functions are solutions of the wave equation. Show work.
y(x,t) = exp[ - (x - vt)^2]
y(x,t) = exp[- x^2 - 2xvt -(vt)^2]
y(x,t) = exp[-x^2 - (vt)^2]
Assignment 3 (due Feb. 18):
Read Chapter 4
Problems:
1. (a) A 1000 Hz sound wave in oxygen gas (O2) at temperature T and pressure p travels at 350 m/s. At what speed will a 5000 Hz sound wave travel in the same gas?
(b) The pressure of the oxygen gas is doubled to 2p, with no change in temperature. What is the sound velocity now?
(c) The temperature is doubled from part (a) to 2T. What is the sound velocity now?
(d) What is the temperature T in part (a)?
(d) Consider a hydrogen gas (H2) at the same temperature T. What is the sound velocity in this gas?
(e) Consider a helium gas (He) at the same temperature T. What is the sound velocity in this gas?
2. Consider a spherical sound wave at a frequency f emanating from the origin.
(a) Write down an expression for the pressure variations Δp(r,t) as a function of distance r from the origin and time t.
(b) Show explicitly that this wave solves the 3D wave equation. This is tedious in Cartesian coordinates, but quite straightforward if you use spherical coordinates (r, theta, phi). Note that in spherical coordinates, the Laplacian operator reads ∂2/∂r2 + (2/r)(∂/∂r) (+ angular derivative terms that are zero when operated on a function that does not depend on angles theta and phi).
(c) How does the sound intensity in such a spherical wave vary as a function of distance from the source?
(d) How does the total radiated sound power, integrated over all directions, vary as a function of distance from the source?
No homework assignment due Feb. 25!
Assignment 4 (due March 3):
Read the tutorial on complex number representation if you are not familiar with this.
Problems:
1. An ultrasound wave is beamed from air into soft human tissue (sound velocity 1540 m/s). In order for the sound beam inside the tissue to propagate at an angle of 45 degrees with respect to the surface normal, at what angle (with respect to the surface normal) does the sound beam have to be incident? (Draw a diagram first.)
2. When a sound source moves at Mach m, i.e., at a speed m times the speed of sound (where m > 1), a shock wave forms in the form of a cone around the propagation axis (see here). Considering that the shock wave moves at the speed of sound in a direction perpendicular to the shock front, determine the opening angle of the cone as a function of the Mach number m. (Define the angle in a drawing!)
3. A plane sound wave of frequency f is passed through a slit. Diffraction causes the wave to spread out in a wedge of 10 degree opening angle. What is the opening angle of the corresponding wedge for a sound wave of frequency 2f ?
4. A pipe closed at one end and open at the other, is filled with air at room temperature. A standing wave is observed at a frequency of 1000 Hz. As the frequency is raised, the next resonance frequency is observed at 1200 Hz (this should have been 1400 Hz - sorry) Hz. Determine the length of the pipe and the fundamental (lowest) resonance frequency.
Assignment 5 (due March 10):
1. A train approaches a steep cliff while sounding its whistle at a frequency 1000 Hz. A person on the train hears a beating sound due to refection of the whistle sound from the cliff. The beating has a frequency of 8 Hz. What is the speed of the train?
2. A damped harmonic oscillator has spring constant of 10 N/m, a mass of 0.1 kg, and a damping constant of 0.001/s. A sinusoidal driving force of amplitude 0.1 N is acting on the mass. What is the steady-state amplitude of the oscillation, (a) in the limit of zero driving frequency, (b) in the limit of infinite driving frequency, and (c) for a driving force oscillating at the resonant frequency. (d) What is the Q-factor of this oscillator?
3. By changing the parameter a from 0 to 0.25, the following function morphs from a square wave (for a = 0) to a triangle wave (for a = .25):
The "cosine" Fourier coefficients (called Bn in the lecture) of this function (for a > 0) are given by:
Calculate the Fourier coefficients for a = 0.001 (almost square), a = 0.125 (intermediate), and a = 0.25 (triangle wave), up to n = 10. Then plot the Fourier sums for all three waves up to n = 10 (or higher if you want) using your favorite graphing software (Excel, Matlab, Maple, Mathematica, etc.), or using this applet, and print the resulting waveforms. Check that the obtained waveforms approximate a square wave, a "flattened" triangle wave, and a triangle wave, respectively. If they don't, you have probably made a mistake! What happens for a > 0.25, e.g. a = .375? Hand in your calculation and the printouts.
4. The Dirac delta function δ(t) is a sharp impulse at the time t = 0, in such a way that when integrated over, the integral equals one. If you are not familiar with the concept, educate yourself by googling "Dirac delta function", or check out the Wikipedia entry. Consider a train of delta functions, f(t) = Σ δ(t-nT), where the sum is over all whole numbers n from n = - infinity to + infinity. (This function is also called the "Dirac comb".) The Dirac comb is clearly periodic with period T. What are the Fourier coefficients Bn and Cn of this periodic function?
Assignment 6 (due March 24):
Read textbook chapters 5 and 6. No problems due.
Assignment 7 (due April 7):
1. A sound source emits spherical sound waves in all directions at a power of 10 mW. Calculate the Sound Intensity Level (SIL) in dB at a distance of 2 m.
2. Given a sound pressure level of 40 dB at a distance of 3 m from a point source, what is the sound pressure level at a distance of 6 m?
3. A Harley Davidson motorcycle driving by your house creates engine noise at a SPL of 80 dB. What is the SPL of the local Hell's Angels (20 such motorcycles) driving by your house?
4. What is the loudness in phons of the above 1 and 20 motorcycles, respectively, assuming that the ear's sensitivity for motorcycle engine noise is 10 dB less than for a pure tone at 1000 Hz?
5. What is the loudness in sones (Stevens' definition) in the above example?
6. Assume you are serving on the city council, devising an ordinance limiting sound pollution. In order to limit traffic noise in your town, you propose a cap of 80 dB. Would it make more sense to use a limit of 80 dBA or 80 dBC, if the objective is that the limit on the perceived loudness is approximately the same at all frequencies, taking into account the ear's frequency response? Explain your choice.
Assignment 8 (due April 14):
Read Chapters 7 and 8 (through section 8.7).
Last Assignment (9) due May 5:
1 Consider two pure tones at frequencies 1000 Hz and 1200 Hz sounded together. What combination tone frequencies might be heard (a) in the presence of second order nonlinearities, and (b) in the presence of third order nonlinearities?
2. A line microphone consists of a tube with small holes attached to a microphone. Phase matching of the guided waves inside the tube with free sound waves outside ensures that the microphone is primarily sensitive to sound traveling in the direction of the axis of the tube. How is this modified if the tube is filled with (a) He, and (b) CO2?
3. A 5m x 5m room, with a 3 m high ceiling, has perfectly reflecting concrete walls. The floor is covered by carpet (sound absorption coefficient 0.4), and the ceiling is covered with sound absorbing tiles (sound absorption coefficient 0.7). What is the reverberation time in this room?