The Nature of Sound
Practice
practice problem 1
- Calculate the length of the 16 cycle wave train.
- Complete the following table. (Entries marked n/a are not applicable.)
duration
(ms)round trip
distance (mm)distance from
ranger (mm)corresponding
specification16 λ n/a n/a n/a rest 2.38 listening full cycle n/a n/a n/a
Source: Physics and technical characteristics of ultrasonic sonar systems.
solution
There are two ways to find the length of the 16 cycle wave train. One is to calculate the length of one wave…
λ = v f λ = 343 m/s 49,400 Hz λ = 0.00694 m = 6.94 mm and then multiply by 16 to get the length of the whole train.
∆s = 6.94 mm × 16
∆s = 111 mmThe other is to determine the time for 16 cycles…
∆t = 16/49,400 Hz
∆t = 3.24 × 10−4 sand then multiply by speed to get distance traveled.
∆s = v∆t
∆s = (343 m/s)(3.24 × 10−4 s)
∆s = 111 mmWe've already computed the duration of the 16 cycle wavetrain. Let me rewrite it. Two decimal places should provide adequate precision.
∆t = 16/49,400 Hz
∆t = 3.24 × 10−4 s
∆t = 0.32 msNext we'll work on the full operational cycle. Period is the inverse of frequency.
T = 1 f T = 1 50 Hz T = 0.020 s = 20.0 ms The listening phase is the total time minus the 16 cycles and the rest phase.
∆t = 20.0 ms − 0.32 ms − 2.38 ms
∆t = 17.3 msTo compute round trip distances (srt), multiply speed (v) by round trip time (trt).
rest ∆srt = v∆trt
∆srt = (343 m/s)(2.38 ms)
∆srt = 816 mmlistening ∆srt = v∆trt
∆srt = (343 m/s)(17.3 ms)
∆srt = 5930 mmDivide the round trip distances by 2 to get the one way distances (sow).
rest ∆sow = ½∆srt
∆sow = ½(816 mm)
∆sow = 408 mmlistening ∆sow = ½∆srt
∆sow = ½(5930 mm)
∆sow = 2970 mmSince the ultrasonic ranger can't listen for an echo until it stops ringing it can't detect objects closer than about ½ meter (408 mm) — the minimum measurable distance. Since the ultrasonic ranger has to stop listening to start a new cycle of measurements it can't detect objects farther away than about 3 meters (2970 mm) — the maximum measurable distance. We may now complete the table.
duration
(ms)round trip
distance (mm)distance from
ranger (mm)corresponding
specification16 λ 0.32 n/a n/a n/a rest 2.38 816 408 minimum measurable distance listening 17.3 5930 2970 maximum measurable distance full cycle 20.0 n/a n/a n/a
practice problem 2
solution
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practice problem 3
solution
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practice problem 4
solution
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