Ultrasound, which consists ultrasour of sound waves with...

Ultrasound, which consists ultrasour of sound waves with frequencies above the human audible range, can Artery be used to produce an image of the interior of a human body. Moreover, ultrasound can be used to measure Figure 17-47 Problem $83 .$ the speed of the blood in the body; it does so by comparing the frequency of the ultrasound sent into the body with the frequency of the ultrasound reflected back to the body's surface by the blood. As the blood pulses, this detected frequency varies. Suppose that an ultrasound image of the arm of a patient shows an artery that is angled at $\theta=20^{\circ}$ to the ultrasound's line of travel (Fig. 17-47). Suppose also that the frequency of the ultrasound reflected by the blood in the artery is increased by a maximum of $5495 \mathrm{~Hz}$ from the original ultrasound frequency of $5.000000 \mathrm{MHz} .$ (a) In Fig. $17-47$, is the direction of the blood flow rightward of leftward? (b) The speed of sound in the human arm is $1540 \mathrm{~m} / \mathrm{s}$. What is the maximum speed of the blood? (Hint: The Doppler effect is caused by the component of the blood's velocity along the ultrasound's direction of travel.) (c) If angle $\theta$ were greater, would the reflected frequency be greater or less?

Ultrasound, which consists ultrasour of sound waves with frequencies above the human audible range, can Artery be used to produce an image of the interior of a human body. Moreover, ultrasound can be used to measure Figure 17-47 Problem $83 .$ the speed of the blood in the body; it does so by comparing the frequency of the ultrasound sent into the body with the frequency of the ultrasound reflected back to the body's surface by the blood. As the blood pulses, this detected frequency varies.

Suppose that an ultrasound image of the arm of a patient shows an artery that is angled at $\theta=20^{\circ}$ to the ultrasound's line of travel (Fig. 17-47). Suppose also that the frequency of the ultrasound reflected by the blood in the artery is increased by a maximum of $5495 \mathrm{~Hz}$ from the original ultrasound frequency of $5.000000 \mathrm{MHz} .$
(a) In Fig. $17-47$, is the direction of the blood flow rightward of leftward? (b) The speed of sound in the human arm is $1540 \mathrm{~m} / \mathrm{s}$. What is the maximum speed of the blood? (Hint: The Doppler effect is caused by the component of the blood's velocity along the ultrasound's direction of travel.) (c) If angle $\theta$ were greater, would the reflected frequency be greater or less?

Key Concepts

Ultrasound Imaging

Ultrasound imaging is a diagnostic technique that uses high-frequency sound waves to create images of the internal structures of the body. The sound waves are emitted by a transducer and reflected off tissues, with the returning echoes being used to form an image. This method is non-invasive, safe, and particularly useful in visualizing soft tissues and fluids, such as blood flow in vessels.

Doppler Effect

The Doppler effect refers to the change in frequency or wavelength of waves relative to an observer moving relative to the source of the waves. In medical ultrasound, this effect is utilized to measure the frequency shifts that occur when sound waves reflect off moving blood cells. The magnitude of the shift is directly related to the speed and direction of the blood flow, allowing clinicians to assess cardiovascular conditions.

Velocity Component and Angle Dependence

When measuring blood flow using Doppler ultrasound, only the component of the blood’s velocity that is aligned with the ultrasound beam contributes to the observed frequency shift. This means that the angle between the blood flow direction and the ultrasound beam significantly influences the measurement. Accurately accounting for this angle is crucial to correctly determine the true speed of the blood flow.

Speed of Sound in Medium

The speed of sound in a medium, such as human tissue, is a fundamental property that affects how ultrasound waves propagate. In diagnostic ultrasound, knowing the speed of sound is essential for accurate distance and velocity calculations, as it determines how quickly the sound waves travel through the body and return to the sensor after reflection.

Transcript

00:01 For this problem on the topic of waves, we ought to suppose that the ultrasound image of the arm of a person shows an artery that is angled at 20 degrees to the ultrasound's line of travel, as we can see in the figure.

00:14 The frequency of the ultrasound reflected by blood in the artery is increased by 5 ,495 hertz from its original frequency of 5 megahertz.

00:25 We want to know if the direction of blood flow is rightward or leftward.

00:29 We want to know the maximum speed of the blood, given the speed of sound in the human arm to be 1 ,540 meters per second.

00:38 And we want to know if the angle theta were greater, would the reflected frequency be greater or less? now, the direction of blood flow can be determined by the doppler shift in frequency.

00:50 The reception of the ultrasound by the blood and the subsequent remitting of the signal by the blood back toward the detector is a two -step process, which we can write as f plus delta f is equal to the observe the source frequency f over v plus vs divided by v plus vx divided by v minus vx, where vx is equal to the x is equal to the x component of the speed of the blood, which is, the speed of the blood v blood times the cosine of theta.

01:40 Now, if you write the ratio of frequencies r to be f plus delta f divided by f, then the solution to the above equation for the speed of the blood, v blood, is equal to r minus 1 times the speed of sounding blood v blood, divided by r plus 1 times the cosine of theta.

02:18 So for part a, we can see that the blood is moving towards the right, which is towards the detector, because the doppler shift in frequency is an increase...

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