BIO Ultrasound has many medical applications, one of which is to monitor fetal heartbeats by reflecting ultrasound off a fetus in the womb. a. Consider an object moving at speed vo toward an at-rest source that is emitting sound waves of frequency f0 . Show that the reflected wave (i.e., the echo) that returns to the source has a Doppler-shifted frequency fecho = (v+v0 / v-vo) fo where v is the speed of sound in the medium.

Video transcript

Welcome back, everyone. We are making observations about multiple or a siren here on a blazing house and we are told that it emits sound at a frequency. F now, a firefighter is approaching the siren on the blazing house at a speed of V F. And we are tasked with driving an expression for F R which is going to be the frequency of reflected waves off of the truck back to the siren. And we also need to take into account the speed of sound in air denoted by V. All right. Well, what we are going to do here is we are going to treat this uh under the Doppler effect shifted to higher values. So what does this mean here? Well, first, let's go ahead and derive an expression for the frequency of the waves hitting the truck. This is going to be the frequency being emitted from the house times one plus the velocity of the truck divided by the speed of sound. I'm gonna name this equation one here. This will be equation one. All right. And then the waves are going to bounce off of the truck and be reflected back towards the blazing house. So what we will have is that our frequency R is equal to frequency T times one minus V of F over V inverse. And I'm gonna name this equation too. Now, let's go ahead and combine these two equations. And what we get is that F R is equal to our F F T, which is F times one plus V F over V divided by one minus V of F over V. What I can do here is I can combine what's inside the parentheses and the numerator and denominator into one fractions. And you will see some cancellation occur. And what we end up with is we get V plus V F divided by V minus V F all times our frequency F which corresponds exactly to our answer choice of B. Thank you all so much for watching. I hope this video helped. We'll see you all in the next one.