Problem 74 - University Physics with Modern Physics

Question

BIO Ultrasound in Medicine. A \(2.00-\) 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 beals. 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.

Step-by-Step Solution

Answer

The speed of the fetal heart wall is approximately 0.054 m/s.

Step 1: Understand the Doppler Effect

When the heart wall moves towards the receiver, the frequency of the returning sound waves increases due to the Doppler effect. This increase in frequency is what causes the beats detected, which are the difference in frequency between the transmitted and received sound waves.

Step 2: Determine Beat Frequency

The beat frequency given is 72 Hz. Beat frequency is the absolute difference between the received (reflected) frequency and the transmitted frequency: \( f_{beat} = |f_{r} - f_{t}| \). Given that the transmitted frequency \( f_t \) is 2 MHz, or \(2 \times 10^{6}\) Hz, we will proceed to find the speed of the heart wall.

Step 3: Use the Doppler Shift Formula for Moving Source

For a source moving towards the observer, the received frequency is \( f_r = f_t \left(\frac{v + v_o}{v - v_s}\right) \). Here, \(v\) is the speed of sound (1500 m/s), \(v_o\) is the speed of the observer (zero, since the receiver is stationary), and \(v_s\) is the speed of the heart wall. We need to solve this for \(v_s\).

Step 4: Relate Beat Frequency to Doppler Shift

Since \( f_{beat} = f_r - f_t \), we substitute for \( f_r \) using the Doppler formula: \( 72 = 2 \times 10^6 \left(\frac{1500}{1500 - v_s}\right) - 2 \times 10^6 \). Rearrange this to solve for \( v_s \).

Step 5: Solve the Equation for the Speed of Heart Wall

Simplify the equation: \[72 = 2 \times 10^6 \times \left( \frac{1}{1500 - v_s} \times 1500 \right) - 2 \times 10^6\]This equation can be further simplified and solved for \(v_s\):\[72 = 2 \times 10^6 \times \left(\frac{1}{1500 - v_s}\right) - 1 \]Solving this will give us \(v_s = 0.054 \) m/s.

Step 6: Calculate the Result

Rearrange the equation to isolate \( v_s \):\[72 + 2 \times 10^6 = 2 \times 10^6 \cdot \frac{1500}{1500-v_s} \]Then, solve for \(v_s\) using algebraic manipulation. Simplifying this further gives the speed of the fetal heart wall:\[ v_s = \frac{(72/2\times10^6) \times 1500}{2\times10^6 - 72/2\times10^6}\]Finally, calculate to find the value of \( v_s = 0.054 \) m/s.

Key Concepts

Ultrasound in Medicine

Ultrasound is a fascinating tool used in medical diagnostics. It involves sound waves with frequencies higher than the human hearing range. In medicine, ultrasound is often used to create images, as it can safely penetrate body tissues. This allows doctors to observe internal structures like the development of a fetus in a woman's uterus.

The key to ultrasound in medicine is its ability to reflect sound waves off different tissues. These reflections create images based on the strength and timing of the returning waves. This non-invasive method is safe for both mother and baby, making it invaluable in prenatal care.

Beat Frequency

In the context of sound waves, a beat frequency occurs when two similar frequencies interfere. This phenomenon is easily heard when tuning musical instruments, but it's also crucial in medical physics.

When sound waves from a transmitter meet those reflected back from a moving surface, like a fetal heart wall, they overlap. The difference in frequency, known as the beat frequency, is detected when these waves are combined. This frequency provides essential information about the velocity of the heart wall. By analyzing it, medical practitioners can gather critical insights into the health and movement of body tissues.

Sound Wave Propagation

Sound wave propagation is the movement of sound waves through a medium. In the case of ultrasound, the medium is body tissue. Sound travels as a wave of pressure variations, transiting through different layers within the body.

When these waves move through tissues, their speed can be influenced by factors like tissue density and composition. To calculate things like the speed of a moving heart wall, understanding the propagation of sound waves is crucial. The medium's properties determine how fast the waves travel, which affects calculations, such as the speed of the fetal heart wall using the Doppler effect.

Medical Physics

Medical physics is a field that combines physics principles with medical insight to innovate and improve healthcare. It includes the application of physics in diagnosis and treatment, ensuring safety and effectiveness.

One prominent example is the use of the Doppler effect in ultrasound. By leveraging changes in frequency due to movement, it allows for real-time analysis of organs and tissues. This helps in monitoring physiological and pathological processes.

• Ensures precise diagnostic techniques.
• Facilitates non-invasive procedures.
• Promotes patient safety and comfort.

Thus, medical physics is pivotal in evolving healthcare technologies and methodologies.