Exam 1 Solutions - Winter, 2000
1. The order of connecting components is: CD player and phonograph are both connected to the preamplifier as inputs; the preamplifier output is input to the amplifier, and the speakers are connected to the amplifier outputs.
2. Sound waves are longitudinal, as are compressional waves in metal and wooden bars. Waves on a string are transverse, as are electromagnetic waves and water waves. Longitudinal waves require a medium to propagate, while transverse waves do not, e.g., light is a transverse wave and propagates in a vacuum. The oscillation of a transverse wave is perpendicular to the propagation direction, while that of a longitudinal wave is along the direction of propagation.
3. Changing the pressure by a factor of five requires increasing the power by a factor of 25, since power is proportional to pressure squared.
4. The reverberation time of a space is defined as the time required for the sound pressure at a given location to drop by a factor of 1000, or as the time required for the sound power at a given location to drop by a factor of 1,000,000, or 60 dB in sound power level. It depends on the ratio V/A, where V is the room volume, and A is the total absorption of the room, equal to the sum of many terms, each of which is the product of a surface area and the absorption coefficient for the material of that surface.
5. The frequency is constant as the wave moves from water to air. Underwater, f = speed/wavelength = 1432 /0.10 Hz = 14,320 Hz. In the air, wavelength = speed/frequency = 344/14,320 m = 0.024 m, or 2.4 cm.
6. Standing waves are oscillatory patterns that arise when the wavelength matches the geometrical dimensions of the vibrating medium. When this happens, interference between waves reflecting around on or in the medium causes a regular pattern of vibration to build up such that the waves don't appear to be traveling, hence the name "standing" waves. The frequencies of standing waves are characteristic of the vibrating medium and are called "resonant" frequencies, or "resonances." Musical instruments generate sound by resonating in standing wave patterns. Standing waves are important in room acoustics, where the should be avoided. Standing waves may build up on speaker cones, affecting the frequency response of the speaker.
7. The fundamental of the tube is 344/(4*0.75) Hz = 114.7 Hz. A tube closed at one end resonates only in odd harmonics, so the next two resonances are n = 3, 5 at frequencies of 344 Hz and 573 Hz.
8. A tube that is closed at both ends will have pressure antinodes at both ends. The fundamental would have 1 half wavelength inside the tube, so the pressure at the two ends will be out of phase, and there will be a node in the center. The second harmonic will have two nodes, each L/4 from the tube ends, where L is the tube length, and the pressure antinodes at the two ends will be in phase. The third harmonic will have nodes at L/6 from each end, as well as a node in the middle, and the pressure antinodes at the two ends will be out of phase.
9. A vocal formant is a resonance of the vocal tract. The tract can be modeled as a tube that is closed at one end, where the larynx is located. Adjustment of the tract by moving the lips, soft palate and tongue can change its shape and shift the formant peak frequencies in order to generate different vowel sounds.
10. Dispersion refers to the spreading of sound waves as they are emitted by a speaker. Large dispersion means the sound waves spread out a great deal. The relevant wave property that causes dispersion is diffraction, which can be thought of as an interference effect due to the interaction of many wavelets generated by the different surface elements of the speaker. Dispersion begins to be significant when the wavelength of sound is equal to the speaker diameter. For a given wavelength, dispersion is increased as the speaker size is decreased.
11. Since 9 meters is the longest dimension and is more than twice the length of the other two dimensions, the second lowest frequency will correspond to a mode number of 2 along this longest direction, or [344*2]/[2*9] Hz = 38 Hz.
12. The increase in decibel level is: (10 dB)*log[4/0.5] = (10 dB)*log[8] = 9.0 dB
13. Sound pressure is related to sound pressure level by: SPL = (20 dB)log(P/P0), where P0 is the reference pressure, 0.00002 Pa. A range of 120 dB in hearing thus corresponds to a log of 6, i.e., a pressure ratio of 106, or one million-fold change in pressure.
14. Compliance is a measure of how far a speaker moves for a given applied force, and is measured in meters per Newton. It is analogous to the inverse spring constant of a spring. Compliance of a cone speaker depends upon the springiness of the speaker suspension. The important quantity referred to is the main speaker resonance, which is proportional to the inverse square root of the compliance.
15. For each 10 dB of dynamic range, the apparent loudness doubles, so 50 dB would correspond to 25, or 32 times louder.