Problem 74 - University Physics with Modern Physics

Question

Ultrasound in Medicine. \(\quad\) A \(2.00-\mathrm{MHz}\) sound wave travels through a pregnant woman's abdomen and is retiected from the fotal 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 85 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.

Step-by-Step Solution

Answer

The fetal heart wall speed is approximately 0.064 m/s.

Step 1: Understand the Doppler Effect

In this problem, we need to apply the Doppler effect for sound waves. The Doppler effect describes how the frequency of sound waves changes when the source or observer is moving. In this case, the fetal heart wall is moving toward the receiver, causing a change in frequency.

Step 2: Identify Known Values

We are given:- Frequency of the ultrasound: \(f_s = 2.00 \text{ MHz} = 2.00 \times 10^6 \text{ Hz}\)- Speed of sound in tissue: \(v = 1500 \text{ m/s}\)- Beat frequency observed: \(f_{beats} = 85 \text{ Hz}\)Remember, the beat frequency \(f_{beats}\) is the difference between the observed frequency \(f'\) and the source frequency \(f_s\).

Step 3: Calculate Observed Frequency

Using the relationship for beat frequency, we have:\[ f' = f_s + f_{beats} \]Substitute the known values:\[ f' = 2.00 \times 10^6 + 85 \]\[ f' = 2.000085 \times 10^6 \text{ Hz} \]

Step 4: Apply the Doppler Effect Equation

The Doppler effect for sound is given by the equation:\[ f' = \left( \frac{v + v_r}{v - v_s} \right) f_s \]In this case, the receiver is stationary \((v_r = 0)\), and the source is the moving fetal heart wall \((v_s = \text{speed of the wall})\). Thus, the equation simplifies to:\[ f' = \left( \frac{v}{v - v_s} \right) f_s \]

Step 5: Solve for Speed of the Heart Wall

Rearrange the simplified Doppler effect equation to solve for \(v_s\):\[ v_s = v - \frac{v \cdot f_s}{f'} \]Substitute the known values:\[ v_s = 1500 - \frac{1500 \times 2.00 \times 10^6}{2.000085 \times 10^6} \]Calculate:\[ v_s \approx 1500 - 1499.936 \approx 0.064 \text{ m/s} \]

Step 6: Final Interpretation

The calculated speed of the fetal heart wall moving toward the receiver is approximately \(0.064 \text{ m/s}\). This value reflects the velocity at which the heart wall approaches the ultrasound transducer at the moment of measurement.

Key Concepts

Ultrasound in Medicine

Ultrasound technology plays a critical role in modern medicine. It uses high-frequency sound waves to create images of organs and tissues inside the body. These non-invasive ultrasound scans are common in prenatal care, where they help visualize and monitor the development of a fetus.
Ultrasound employs frequencies ranging from 2 to 18 MHz, much higher than the range of human hearing. These sound waves are transmitted into the body by a transducer, which also receives the echoes that bounce back after reflecting off internal structures. The time it takes for the echoes to return and their strength provides information to create detailed images of the body's internal components. This method is safe and does not involve radiation unlike other imaging techniques.

Sound Waves

Sound waves are a type of mechanical wave that travels through a medium, such as air, water, or tissue. In the context of ultrasound, these waves propagate through body tissue. They move by vibrating the particles within the medium in which they travel.
Sound waves are characterized by their frequency, which is the number of cycles that occur per second, measured in Hertz (Hz). The frequency determines the pitch of the sound we hear, although in medical ultrasound, frequencies are too high for human hearing. High-frequency sound waves, like those used in ultrasound imaging, provide clearer images of smaller and more precise structures, making them indispensable for medical diagnostics.

Beat Frequency

Beat frequency is an interesting phenomenon that occurs when two sound waves of slightly different frequencies interfere with each other. Instead of hearing two distinct tones, you perceive a beating or pulsating sound, known as the beat frequency.
The beat frequency is calculated as the difference between the two frequencies. In medical ultrasound, this property is utilized in Doppler ultrasound scans to measure the movement of structures like the fetal heart. By sending ultrasound waves into the body and measuring the frequency shifts that occur when these waves reflect off moving objects, the beat frequency becomes a valuable tool for assessing the speed and movement of internal organs and tissues. This is particularly useful in fetal heart monitoring.

Speed of Sound in Tissue

The speed of sound varies depending on the medium it travels through. In human tissue, sound waves typically travel at approximately 1500 m/s. This rate is an average representation of how sound waves propagate through different types of soft tissues in the body.
Understanding the speed of sound in tissue is crucial for accurate ultrasound readings. The ultrasound machine uses this constant speed to compute distances to create precise images. It’s important to note that variations in body composition, such as fat and muscle, can slightly alter sound propagation speed. Nevertheless, 1500 m/s serves as a reliable baseline for calculating the interactions of sound waves within the body during an ultrasound scan.