Quick answers to the final sample questions:

1. The carpet is much more absorptive than the tile, and reflections off the floor will be greatly reduced, so that any resonances resulting from standing waves in the vertical direction will be greatly reduced.

2. The flute is open at both ends and the clarinet is closed at the mouthpiece end, so the lowest frequency mode of the clarinet has a wavelength that is one-fourth of the clarinet's length, whereas the flute's lowest mode wavelength is half the flute's length. Thus the flute's lowest note has twice the frequency of the clarinet's, making it an octave higher.

3. The wavelengths of the fundamental modes of these strings are all the same (twice the string length), but the frequencies differ because the speed of waves along the strings are all different. Generally, speed and therefore frequency are increased by increasing the tension in the string. However, we need such a wide range of speeds that one kind of string couldn't support that range of tensions, and so thicker strings are used for the lowest pitches. If two strings have the same length and tension but different masses, the more massive string is lower pitched.

4. If you press down hard at that point, you make the remaining string 4/5 as long as the open string, thus raising its frequency by a factor of 5/4. This corresponds to an interval of a major third, so of course the resulting note is in our equal-tempered scale. This is why the guitar maker placed a fret there.

5. If I pluck the string one-fifth of the way from the other end, I'll be plucking at a node of the fifth harmonic, which will completely suppress that mode. However, if I pluck 1/10th or 3/10th of the way from the other end, I'll be plucking at an antinode, which is a great way to excite that mode.

6. Well, if  I haven't missed a loop, that clarinet is only 3 times as long as the soprano, so I'd say the lowest note is down by a factor of 3 in frequency, which brings us down from D3 to G1.

7. In both cases you're exciting the instrument at or near antinodes of the high-frequency modes and near nodes of the lowest frequency modes, so these phenomena certainly do result from similar causes.

8. That hole is located at or near a pressure node for the third harmonic, making that mode easy to excite (and making the fundamental hard to excite), and it's the third harmonic that you want to excite by overblowing.

9. If the speakers are perfectly out of phase with each other, their sound waves will perfectly cancel and you'll have zero intensity at that point. If they're perfectly in phase with each other, their sound waves will perfectly reinforce each other and you'll have twice the amplitude and four times the intensity, or 4 microwatts per meter squared, at that point. You can switch from one of these situation to the other by reversing the leads on one speaker. The average intensity in the neighborhood of the speakers will be 2 microwatts per meter squared, with the local intensity varying continuously between 0 and 4 microwatts per meter squared.

10. Any strings that have modes whose frequencies match the frequencies of the modes of the middle C string will be excited. The C below middle C has many modes that match those of middle C, because every harmonic of middle C  matches an even harmonic of this lower C. The second harmonic of middle C matches the third harmonic of the note whose fundamental frequency is 2/3 that of middle C, which would be F3. The third harmonic of  C4 matches the fourth harmonic of the note whose fundamental frequency is 3/4 that of middle C, which would be G3. Similarly you'd weakly excite (because the higher harmonics are weaker in general) A3 flat and A3. Going up, the second harmonic of C4 matches the fundamental of C5. The third harmonic of C4 matches the second harmonic of the note whose fundamental is 3/2 of C4, which is G4, and the fourth harmonic matches the third harmonic of the note whose fundamental is 4/3 that of C4, which would be F4. We'd get E4. E4 flat is not a good frequency match, and would require using the 6th harmonic of C4 to excite the 5th harmonic of E4 flat, which is asking a lot of the higher harmonics.

11. We did this estimate in class.

12. Lower voice: 440, 880, 1320, 1760, 2200, 2640 Hz. Higher voice: 550, 1100, 1650, 2200, 2750, 3300 Hz. Both voices produce 2200 Hz. If the equal-tempered interval is used instead, then the frequencies for the higher voice will be 554.4, 1108.8, 1663.2, 2217.6, etc. We see that the 2200 Hz from the lower voice and the 2217.6 Hz will produce very rapid beating, which might sound somewhat harsh.

13. With long cylinders of air, or with guitar strings, cutting the resonating length (whether it's string or air vibrating)  in half doubles the frequency in obeisance to our favorite formula of lambda times f equals v. But if the vibrating object isn't one-dimensional, then halving one dimension doesn't generally halve the wavelength. So while our formula remains true, because we don't know what happened to lambda, we can't say what happened to f.