The engineer who sets about designing an efficient sonar or radar device soon comes up against a problem resulting from the need to make the pulses extremely loud. They have to be loud because when a sound is broadcast its wavefront advances as an ever-expanding sphere. The intensity of the sound is distributed and, in a sense, 'diluted' over the whole surface of the sphere. The surface area of any sphere is proportional to the radius squared. The intensity of the sound at any particular point on the sphere therefore decreases, not in proportion to the distance (the radius) but in proportion to the square of the distance from the sound source, as the wavefront advances and the sphere swells. This means that the sound gets quieter pretty fast, as it travels away from its source, in this case the bat.
When this diluted sound hits an object, say a fly, it bounces off the fly. This reflected sound now, in its turn, radiates away from the fly in an expanding spherical wavefront. For the same reason as in the case of the original sound, it decays as the square of the distance from the fly. By the time the echo reaches the bat again, the decay in its intensity is proportional, not to the distance of the fly from the bat, not even to the square of that distance, but to something more like the square of the square - the fourth power, of the distance. This means that it is very very quiet indeed. The problem can be partially overcome if the bat beams the sound by means of the equivalent of a megaphone, but only if it already knows the direction of the target. In any case, if the bat is to receive any reasonable echo at all from a distant target, the outgoing squeak as it leaves the bat must be very loud indeed, and the instrument that detects the echo, the ear, must be highly sensitive to very quiet sounds - the echoes. Bat cries, as we have seen, are indeed often very loud, and their ears are very sensitive.
