t = 2x / v ( time echo), t = 2.9 10⁻² s
Explanation:
In this case we can use the uniform motion relationships, since the sound wave has a constant speed. Let's start by calculating the time it takes for the click to reach the detector.
v = d / t
t = d / v
where d is the distance from the speaker to the detector and v the speed of sound (v = 340 m / s)
Now let's analyze the echo, it is produced by a reflection of the sound from a large obstacle in the direction of the sound, therefore if the distance to the obstacle is x, the echo travels a distance of 2x in this time (to)
2x = v to
2x = v (d / v)
d = 2x
if we substitute in the first equation
t = 2x / v ( time echo)
Let's analyze these results, if the distance relationship is fulfilled, the detector (microphone) is not able to distinguish between a click and the echo of the previous click
For a numerical result suppose that the distance from the loudspeaker to the detector is d = 10 m, we obtain that the obstacle must be at a distance from the loudspeaker of
x = 5 m
t = 2 5/ 340
t = 2.9 10⁻² s
This is the time the echo has to return in this speaker-microphone configuration