The Doppler effect using ultrasonic waves of frequency $2.25 \times 10^6 \textrm{ Hz}$ is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 240 Hz is observed. Assuming that the speed of sound in tissue is 1540 m/s, calculate the maximum velocity of the surface of the beating heart.

VIDEO TRANSCRIPT

This is Giancoli Answers with Mr. Dychko. So, the heart tissue of this fetus is moving and then here's the ultra sonic sound source, the wand, and it's emitting the sound through the tissue of the womb there. And the Doppler effect frequency is going to be a speed of sound plus or minus speed of the observer divided by speed of sound minus or plus speed of the source times the original frequency. Now, there's going to be a couple frequencies to think about here, one is the frequency that's, you know, you could say perceived by the heart, and it's not going to be the frequency emitted by the wand because let's suppose the heart tissue is moving away from the wand, it could be moving towards the wand, we don't actually know. But as we'll see at the end, it doesn't really matter that much, it makes a slight slight difference but not enough to be significant. So, we'll just take one assumption. And so with the heart tissue moving away from the wand, the frequency perceived by the heart then is going to be the speed of sound minus the speed of the heart because the heart is the observer here, moving away from the sound source. And so since it's moving away we're going to choose a minus there. And the sound source is stationary. And so on the bottom we have just speed of sound. And this is the frequency emitted by the wand. And then the heart will reflect the same f prime frequency. And but that's not what the wand gonna notice though, the wand will perceive f double prime because now the wand sees that a sound source is moving away from it. And so but. So, we have a stationary observer being the wand now. And a moving sound source, the heart moving away from the wand. So, that's f double prime and so we have observer being stationary in the second case, and the observer is the wand now. And we're dividing by speed of sound. And we're going to choose a plus or a minus here such that we reduce the perceived frequency because since the tissue's moving away, that would tend to reduce the perceived frequency, it's f double prime. And so we have to choose a plus here in order to increase the denominator so that we decrease the quotient. And that's going to get multiplied by f prime because that's the frequency reflected by the heart. And f prime is all this. So, we plug that in down here. And the speed of sound cancels there and we're left with f double prime. Frequency measured by the wand is going to be speed of sound minus beat of the heart divided by speed of sound plus the speed of the heart times the frequency emitted by the wand. Now the wand does not measure this f double prime, instead it measures the beat frequency, this interference frequency between the emitted frequency and f double prime. And so that's going to be, the beat frequency is the difference in frequencies. So, we have... Now, since, and the b frequency is always some positive number. And now since we've got the tissue moving away from the wand, that means f double prime is going to be lower than f. And so I've written it this way around, where you have f minus f double prime. And we substitute in f double prime here. And factor out the f. And then we'll write this as a single fraction by multiplying this 1 by the sound of plus v heart. And that makes me sound plus v heart, all over v sound plus v heart and then minus v sound and then plus v heart because you assume that there's a bracket there. And this minus sign gets distributed into the brackets so the first v sound becomes minus and then the second one becomes, v heart becomes plus. So, the v sounds have opposite signs there and they disappear, we;re left with 2 times v heart. And multiply both sides by this denominator and you have beat frequency times and v sound and then beat frequency times v heart equals 2 f v heart. And then factor out v heart after you move this to the right hand side which makes it a minus. So, that's 2 f v heart minus fb v heart. And factor out the v heart and you have it multiplied by 2 f minus fb, and that equals this on its own fb times v sound which, you know, it ends up in the right side cause I flipped the sides around as well. And then divide both sides by this bracket, and we get an expression finally for the speed of the heart. And is the beat frequency times the speed of sound divided by 2 times the frequency emitted by the wand minus the beat frequency. So, as 240 hertz times 1540 meters per second speed of sound within the tissue divided by 2 times 2.25 mega hertz written as times 10 to the 6 hertz minus 240 hertz, which is 0.082 meters per second. Now, if you thought that the heart was moving towards the wand, it might be. Then the difference would be that you'd have a minus here and an a plus there and you'd also switch the order of these things around so that you have f double prime because it would now be greater than f. And if you follow through with all that, you would end up with a plus sign down here instead of a minus. So, it would give a different answer but the difference will not be significant because this number is, it's 6, I guess... It's 4 orders of magnitude greater than this number. And so since we're only going to take, you know, 2 significant digits here. If you have a plus there or minus, that's only gonna affect the... You know, digits, that's over here somewhere, the 4 significant digit which is not significant, so it's not going to make a difference.