1. 2.07 kHz, 4.05 kHz.
2. 5.72 kHz, 11.44 kHz
3. See Fig. 9.14
4. Yes - longitudinal modes, No - transverse modes. Difference is due to restoring forces in bar.
5. The marimba has wooden bars, rather than metal, and they are not perfectly rectangular because the underbelly has been carved to bring several modes into a nearly harmonic relationship.
6. See Fig. 9.16
7. No - the mode frequencies shift because the enclosed air is "springy."
8. Several tests can be used: the finger can lightly touch the bar to feel its movement, or a small microphone may be moved around above the bar to sense where the amplitude is greatest. Mounting the bar in different ways enables one to find nodal lines, or salt or sand may be sprinkled on the bar if the amplitude of vibration is great enough.
9. Chladni patterns are 2-dimensional vibration patterns in which nodal lines and circles are indicated by sprinkling sand or salt on the surface while vibrating it at a resonant frequency.
10. Lightly touching the string at position L/n while striking it gives a spectrum of harmonics nf, 2nf, 3nf, 4nf, 5nf, etc., where n = 1,2,3,4..., corresponding to a pitch frequency nf. In these expressions the fundamental of the original string at its full length is f.
11. L/3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... Striking at L/5 means every fifth is missing, i.e. 5, 10, 15, 20, 25, 30... Thus the spectrum includes 3, 6, 6, 12, 18, 21, 24, 27, 33...
12. Quadruple tension to double frequency.
13. See Sect. 10.3
14. More strings creates more sound and slight mistunings causes longer decays.
15. Piano tuning is stretched as a result of inharmonicity, in order to decrease annoying beats between different strings due to frequency mismatch of fundamentals and overtones.
16. See Sect. 11.2 for a detailed description.
17. Sawtooth harmonic amplitudes are proportional to 1/n, so n=2 is small than n=1 by a factor (20 dB)log(1/2) = -6 dB. However, response at n=2 is greater by 10 dB, so overall n=2 will be stronger in the radiated spectrum, by 4 dB over n=1.
18. Radiation patterns are due to vibrational mode shapes and diffraction.
19. Open: 287, 573, 860 Hz; Closed: 143, 430, 717 Hz.
20. Three mechanisms: cane reed, lip reed, airjet
21. Factor of 2 between registers corresponds to 1 octave.
22. Factor of 3 between clarion and chalumeau corresponds to an octave+P5; factor of 5/3 between chalumeau and altissimo corresponds to a M6.
23. Tone holes enable the acoustic length of the instrument to be changed in order to sound a variety of musical pitches. They also radiate sound.
24. L=v/4f = 0.369 m. The cross section is conical, so the acoustic length is greater than the physical length.
25. Recorder is excited by an airject generated in a fixed windway that strikes an edge to create an edgetone that excites a tube. By overblowing, the edgetone is forced to oscillate more rapidly, driving a higher harmonic which becomes the new pitch frequency due to cooperation between the airjet and the tube resonances.
26. Radiation is very efficient above cutoff. Impedance is small so reflections are not large.
27. Impedance is the ratio of driving force to velocity. For guitars and pianos, the bridge is driven by the strings, and bridge motion is coupled to soundboard motion. Coupling efficiency (energy transfer) depends upon impedance matching between string and bridge, and between bridge and soundboard. Good matching means efficient energy transfer, but also fast string damping, so a tradeoff is made for the best sound.
28. Pitch frequencies, all in Hz: a) 100, b) 100, c) 200, d) 100, e) 300, f) 600