P1. To examine how sound travels from the source to the observer in an auditorium, consider a person seated at the center of a 40m x 20m hall with a ceiling height of 15m. Using t=d/v with v=343 m/s, the times to reach the observer are: for direct sound to=_s, for first reflection from the side wall tside s, for first reflection from the ceiling tceil _S, giving a first reflected delay t=_ ms. P2. The reverberation time is RT = 0.161 V/A, where V is the room volume in m 3 and A = >(S,xa; ) is the sum for each surface of the room of its area (S ;) times its appropriate absorption coefficient (a;). At f= 2000 Hz, for a bare room 20m x 15m and 8m high with painted concrete block walls (a = 0.09), ceiling plaster on lath (a = 0.04), and floor carpet on concrete (a = 0.60), RT = S P3. Adding 200 unoccupied, upholstered seats (A,=0.43/seat) to this room gives RT =_ s. To see the effect of an audience, having 200 occupied, upholstered seats (As=0.64/seat) gives RT = S. P4. A piano has 88 notes grouped into 12 semitones per octave (see figure below). If it is tuned to the scale of equal temperament, each semitone is given by a factor 2 1/12 = 1.0595. The modern tune standard has the note A4 = 440 Hz. On this piano, middle C or C4 (9 semitones lower) = Hz and the next higher C or Cs (3 semitones above A4) = Hz. [Note: you may check that the octave condition, f(Cs )/ f(C4) = 2.0, is satisfied.] P5. Agreeable consonance occurs for a perfect fifth such as the pair of notes G/C. On this piano, GA = Hz and G4/C4 = which is very close to the perfect fifth ratio of 3:2. P6. Pressing the first valve of a trumpet or tuba increases the acoustic length by 12.2% and lowers the pitch a whole tone. The "effective" length of an instrument is the length of an idealized cylinder (or cone) that gives the appropriate frequency for its fundamental mode. If the "effective" length of the trumpet is 1.40 m, cm of tubing should be added to produce this amount of pitch change. P7. The effective length of a trombone is 2.75 m in "first" position with the slide fully retracted. To lower the pitch by one whole tone (by 12% in moving to position 3), the length must increase by cm which means that the slide must move by cm. P8. The active string length for the violin is 33 cm and for the double bass it is 104 cm. Since the maximum spacing of semitones is 0.0595 times the active length, this corresponds to cm for the violin and cm for the double bass