practice problem 1
- Calculate the length of the 16 cycle wave train.
- Complete the following table. (Entries marked n/a are not applicable.)
16 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. Dan MacIsaac and Ari Hämäläinen. The Physics Teacher. Vol. 40 No. 1 (January 2002): 39–46.
- 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 ƒ 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 mm
The other is to determine the time for 16 cycles…
∆t = 16/49,400 Hz ∆t = 3.24 × 10−4 s
and then multiply by speed to get distance traveled.
∆s = v∆t ∆s = (343 m/s)(3.24 × 10−4 s) ∆s = 111 mm
- We'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 ms
Next we'll work on the full operational cycle. Period is the inverse of frequency.
T = 1 ƒ 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 ms
To 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 mm listening ∆srt = v∆trt ∆srt = (343 m/s)(17.3 ms) ∆srt = 5930 mm
Divide the round trip distances by 2 to get the one way distances (sow).
rest ∆sow = ½∆srt ∆sow = ½(816 mm) ∆sow = 408 mm listening ∆sow = ½∆srt ∆sow = ½(5930 mm) ∆sow = 2970 mm
Since 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.
16 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
practice problem 3
practice problem 4