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.

Verified step by step guidance

1. Step 1: Understand the Doppler Effect. The Doppler Effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave. In this scenario, the source emits sound waves at a frequency f0 and the object (fetus) is moving towards the source with a speed v0.
2. Step 2: Calculate the frequency of the sound as heard by the moving object. When the object moves towards the source, the frequency of the sound increases. This can be calculated using the formula f' = f0 * (v + v0) / v, where v is the speed of sound in the medium.
3. Step 3: Consider the object as a new source of the sound wave after reflection. Once the sound wave hits the moving object, it is reflected back. Now, the moving object acts as a new source of the sound wave, emitting it at the frequency f' calculated in Step 2.
4. Step 4: Calculate the frequency of the reflected wave as heard by the original source. Since the object is moving towards the source, the frequency of the reflected wave increases again. This can be calculated using the formula fecho = f' * (v + v0) / v.
5. Step 5: Substitute f' from Step 2 into the equation from Step 4 to find the frequency of the echo. This gives fecho = (f0 * (v + v0) / v) * (v + v0) / v = (v + v0) / (v - v0) * f0, which is the desired formula for the frequency of the echo as heard by the original source.

Key Concepts

Doppler Effect

The Doppler Effect is a phenomenon observed when there is a relative motion between a source of waves and an observer. It describes how the frequency of waves changes based on the motion of the source and the observer. When the source moves toward the observer, the observed frequency increases, while it decreases when the source moves away. This effect is crucial in understanding how sound waves behave in various scenarios, including medical applications like ultrasound.

Frequency Shift

Frequency shift refers to the change in frequency of a wave due to the relative motion between the source and the observer. In the context of the Doppler Effect, the frequency of the sound waves emitted by a source is altered as they reflect off a moving object, such as a fetus. The formula for the frequency shift accounts for the speeds of both the source and the observer, allowing for precise calculations in applications like fetal heartbeat monitoring.

Speed of Sound

The speed of sound is the rate at which sound waves propagate through a medium, typically air, water, or tissue. It is a critical factor in the Doppler Effect, as it influences how quickly sound waves travel and how their frequencies are perceived by moving observers. In medical ultrasound, knowing the speed of sound in the specific medium (e.g., amniotic fluid) is essential for accurately interpreting the frequency shifts and ensuring effective monitoring of fetal heartbeats.