OpenStax College Physics, Chapter 17, Problem 84 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. Blood is moving towards this ultrasound transducer and it's gonna reflect the sound back to the transducer, which is also a detector, and the reflected frequency is going to be different as a result of the Doppler shift because of the motion of this blood towards the transducer. And the two frequencies together the frequency originally emitted and the frequency detected here are going to produce a beat frequency when they are combined and our job in this question is to figure out what is that beat frequency? So it's going to be the absolute value of the difference between the frequency observed by the transducer and the frequency it originally emitted. Now I put a prime on here because there are two Doppler shifts to consider: the first is this blood moving towards this transducer... it's as though this observer is moving towards the source and so there's going to be a Doppler shift in the frequency that this blood, you know, observes so to speak and then the blood will in turn reflect that shifted frequency and this transducer will will perceive a Doppler shift in that reflected frequency because this source now is moving towards the transducer. So we have this frequency originally emitted by the source; we have the Doppler shifted frequency observed by this observer moving towards the transducer— that's what we are gonna calculate here— and then we have this f obs prime which is the frequency observed at the transducer resulting from a source moving towards the transducer. So with the observer moving, we have this first formula which is the original frequency emitted by the source multiplied by the speed of sound in tissue plus the speed of the blood and divide that by the speed of sound and then we have this f obs prime at the transducer which equals the original frequency emitted by the source which now is this blood which had a Doppler shifted frequency compared to the original frequency emitted. So that's f obs here— all of this is gonna get plugged in here in a second— times the speed of sound divided by the speed of sound minus the speed of the blood which is now the source. Okay! So then plug all of this in blue in for f obs and I do that substitution in red here, this v w cancels and we are left with the frequency observed at the transducer is the frequency originally emitted times the sum of these speeds divided by the difference in these speeds. So that's 2.50 times 10 to the 6 hertz times 1540 meters per second— speed of sound in tissue— plus 30.0 times 10 to the minus 2 meters per second— speed of the blood— divided by the speed of sound minus the speed of the blood. So that's 2500974 hertz and we are told that this frequency has precision to seven significant figures. So the beat frequency then will be that frequency observed at the transducer minus the frequency originally emitted and the difference is 974 hertz.