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Chapter 16: Problem 66

A \(2.00 \mathrm{MHz}\) sound wave travels through a pregnant woman's abdomen and is reflected from the fetal heart wall of her unborn baby. The heart wall is moving toward the sound receiver as the heart beats. The reflected sound is then mixed with the transmitted sound, and 72 beats per second are detected. The speed of sound in body tissue is \(1500 \mathrm{~m} / \mathrm{s}\). Calculate the speed of the fetal heart wall at the instant this measurement is made.

Short Answer

Expert verified
The speed of the fetal heart wall at the instant this measurement is made is approximately \(0.036 m/s\).

Step by step solution

01

Analyze the given information

From the exercise, the following data can be identified: \nInitial frequency, \(f_i = 2.00 MHz = 2.00 \times 10^6 Hz\),\nDetected beats, \(f_b = 72 Hz\),\nSpeed of sound in the body, \(v = 1500 m/s\).
02

Calculate the frequency of the reflected wave

The detected beat frequency is the difference between the frequencies of the transmitted wave and the reflected wave. So, let's calculate the frequency of the reflected wave, denoted as \(f_r\), using the following equation: \(f_b = f_i - f_r\). Rearranging the formula for \(f_r\) we get: \(f_r = f_i - f_b = 2.00 \times 10^6 Hz - 72 Hz = 1.999928 \times 10^6 Hz\).
03

Apply the Doppler shift formula

We can now apply the Doppler shift formula for the frequency of the wave reflected from the moving heart wall: \(f_r = f_i \times \frac{v + v_0}{v}\), where \(v_0\) is the speed of the heart wall. By rearranging this equation, the speed of the heart wall can be calculated as follows: \(v_0 = \frac{f_r}{f_i} \times v - v.\)
04

Substitute and Solve

Substitute the calculated and given values into the formula: \(v_0 = \frac{1.999928 \times 10^6 Hz}{2.00 \times 10^6 Hz} \times 1500 m/s - 1500 m/s = 0.036 m/s\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acoustics
Acoustics is the science of sound and is crucial in understanding how sound waves behave under various conditions. In the context of ultrasonography, acoustics plays a fundamental role in medical diagnostics. Sound waves, like those used in ultrasonography, have distinct properties such as frequency, wavelength, and speed. These properties help in forming clear images within the body.
The sound waves used in medical ultrasonography are typically high-frequency, ranging from 1 to 20 MHz. Such frequencies are beyond human hearing and provide the resolution required to view minute structures such as a fetal heart. In our exercise, a 2.00 MHz frequency was used, which is sufficient for detailed imaging inside the mother's body.
Key acoustic principles such as the speed of sound, which in bodily tissues is approximately 1500 m/s, aid in calculating crucial details like the Doppler effect involved in our scenario. Understanding these acoustic fundamentals allows for the precise application of ultrasonography in medical settings.
Sound Wave Reflection
Sound wave reflection is a process that occurs when a sound wave hits a surface and bounces back. This principle is used meticulously in ultrasonography. When a sound wave encounters a boundary of a different material or density, such as a fetal heart within a mother's body, part of the wave is reflected back towards the source.
The exercise we examined involves sound waves reflecting off a fetal heart, which is used to gather information about its movement. The frequency difference between the transmitted and reflected wave creates a detectable pattern called the beat frequency, which was 72 beats per second in this scenario.
Reflecting waves are instrumental in diagnostics because they provide data based on how fast or slow tissues are moving. By analyzing these reflected waves, doctors can infer the speed of internal body parts, such as the fetal heart wall's motion, using the Doppler effect as a guide.
Fetal Heart Monitoring
Fetal heart monitoring is a crucial aspect of prenatal care. It involves using acoustic principles and sound wave reflection, as discussed earlier. Through techniques such as Doppler ultrasonography, healthcare providers can monitor and evaluate the fetal heart's condition.
In the exercise, the fetal heart wall's movement was gauged by measuring beat frequency, indicating how the heart is functioning. With the Doppler effect, it's possible to determine the speed at which the heart wall is moving, which we calculated to be approximately 0.036 m/s. This measurement can indicate normal or abnormal conditions, informing necessary medical decisions.
Fetal heart monitoring is vital because it provides a window into the baby's health and growth before birth. The non-invasive nature and reliability of this monitoring make it indispensable in modern obstetric care, offering peace of mind and vital information for both healthcare providers and expectant parents.

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Most popular questions from this chapter

A bat flies toward a wall, emitting a steady sound of frequency \(1.70 \mathrm{kHz}\). This bat hears its own sound plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of \(8.00 \mathrm{~Hz}\) ?

A pipe closed at both ends can have standing waves inside of it, but you normally don't hear them because little of the sound can get out. But you can hear them if you are inside the pipe, such as someone singing in the shower. (a) Show that the wavelengths of standing waves in a pipe of length \(L\) that is closed at both ends are \(\lambda_{n}=2 L / n\) and the frequencies are given by \(f_{n}=m v / 2 L=n f_{1},\) where \(n=1,2,3, \ldots\) (b) Modeling it as a pipe, find the frequency of the fundamental and the first two overtones for a shower \(2.50 \mathrm{~m}\) tall. Are these frequencies audible?

Two powerful speakers, separated by \(15.00 \mathrm{~m}\), stand on the floor in front of the stage in a large amphitheater. An aisle perpendicular to the stage is directly in front of one of the speakers and extends \(50.00 \mathrm{~m}\) to an exit door at the back of the amphitheater. (a) If the speakers produce in-phase, coherent \(440 \mathrm{~Hz}\) tones, at how many points along the aisle is the sound minimal? (b) What is the distance between the farthest such point and the door at the back of the aisle? (c) Suppose the coherent sound emitted from both speakers is a linear superposition of a \(440 \mathrm{~Hz}\) tone and another tone with frequency \(f\). What is the smallest value of \(f\) so that minimal sound is heard at any point where the \(440 \mathrm{~Hz}\) sound is minimal? (d) At how many additional points in the aisle is the \(440 \mathrm{~Hz}\) tone present but the second tone is minimal? (e) What is the distance from the closest of these points to the speaker at the front of the aisle?

The shock-wave cone created by a space shuttle at one instant during its reentry into the atmosphere makes an angle of \(58.0^{\circ}\) with its direction of motion. The speed of sound at this altitude is \(331 \mathrm{~m} / \mathrm{s}\). (a) What is the Mach number of the shuttle at this instant, and (b) how fast (in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{mi} / \mathrm{h})\) is it traveling relative to the atmosphere? (c) What would be its Mach number and the angle of its shock-wave cone if it flew at the same speed but at low altitude where the speed of sound is \(344 \mathrm{~m} / \mathrm{s} ?\)

You live on a busy street, but as a music lover, you want to reduce the traffic noise. (a) If you install special sound-reflecting windows that reduce the sound intensity level (in \(\mathrm{dB}\) ) by \(30 \mathrm{~dB}\), by what fraction have you lowered the sound intensity (in \(\left.\mathrm{W} / \mathrm{m}^{2}\right) ?\) (b) If, instead, you reduce the intensity by half, what change (in \(\mathrm{dB}\) ) do you make in the sound intensity level?
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