Problem 37 - University Physics with Modern Physics

Question

Physicians use high-frequency (\(f\) = 1\(-\)5 MHz) sound waves, called ultrasound, to image internal organs. The speed of these ultrasound waves is 1480 m\(/\)s in muscle and 344 m\(/\)s in air. We define the index of refraction of a material for sound waves to be the ratio of the speed of sound in air to the speed of sound in the material. Snell's law then applies to the refraction of sound waves. (a) At what angle from the normal does an ultrasound beam enter the heart if it leaves the lungs at an angle of 9.73\(^\circ\) from the normal to the heart wall? (Assume that the speed of sound in the lungs is 344 m\(/\)s.) (b) What is the critical angle for sound waves in air incident on muscle?

Step-by-Step Solution

Answer

The angle in the heart is approximately 48.27°, and the critical angle is around 13.34°.

Step 1: Understanding the Relationship Using Snell's Law

Snell's law is given by \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \), where \( n_1 \) and \( n_2 \) are the indices of refraction of the initial and final substances, respectively, and \( \theta_1 \) and \( \theta_2 \) are the angles of incidence and refraction. Here, \( \theta_1 = 9.73^\circ \) is the angle of refraction in the lungs, where the index of refraction \( n_1 = \frac{v_{air}}{v_{lungs}} = 1 \). \( n_2 = \frac{v_{air}}{v_{muscle}} = \frac{344}{1480} \).

Step 2: Applying Snell's Law to Find Angle in the Heart

Using Snell's law, \( \sin\theta_1 = n_2 \sin\theta_2 \). Substitute \( \theta_1 = 9.73^\circ \), \( n_2 = \frac{344}{1480} \), and solve for \( \theta_2 \):\[\sin\theta_2 = \frac{\sin 9.73^\circ}{\frac{344}{1480}} = \left(\frac{1480}{344}\right) \sin 9.73^\circ\]

Step 3: Calculating the Angle in the Heart

Compute \( \sin 9.73^\circ \) and \( \frac{1480}{344} \), then find \( \sin\theta_2 \):\[\sin\theta_2 = 4.30233 \times \sin 9.73^\circ \approx 0.742\]To find \( \theta_2 \), use \( \theta_2 = \arcsin(0.742) \approx 48.27^\circ \).

Step 4: Understanding Critical Angle

The critical angle \( \theta_c \) is the angle of incidence at which the angle of refraction is 90 degrees. It is given by \( \sin\theta_c = \frac{v_{muscle}}{v_{air}} = \frac{1480}{344} \).

Step 5: Calculating the Critical Angle

Substitute the values and solve for \( \theta_c \):\[\theta_c = \arcsin\left(\frac{344}{1480}\right) \approx 13.34^\circ\]

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in physics that describes how waves, including sound waves, change direction as they pass through different materials. This is especially useful in understanding ultrasound imaging, a common medical procedure. According to Snell's Law, the relationship between the angles of incidence and refraction is described by the equation: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \). Here, \( n_1 \) and \( n_2 \) represent the indices of refraction for two different media, such as lungs and muscle in an ultrasound setting.

• \( \theta_1 \) is the angle of incidence, the angle at which the ultrasound enters the first medium.
• \( \theta_2 \) is the angle of refraction, the angle at which it enters the second medium.

In ultrasound physics, understanding Snell's Law helps predict how sound waves will travel through the body, assisting in creating clearer images. From the exercise, we use the relationship to find the angle at which sound enters the heart when leaving the lungs.

Index of Refraction

The index of refraction for sound waves provides insight into how sound travels through different substances compared to air. Expressed as \( n = \frac{v_{air}}{v_{material}} \), it compares the speed of sound in air (\( v_{air} \)) to that in another medium (\( v_{material} \)). It helps determine how much bending occurs as sound enters a new medium.

• For an ultrasound moving from air to muscle, the speed in air is 344 m/s, and in muscle, it's 1480 m/s.
• The index of refraction for muscle would be \( \frac{344}{1480} \), indicating slower speed in the denser muscle compared to air.

This ratio is crucial in applying Snell's Law correctly, as it influences how sound waves bend when transitioning from one medium to another. With a lower index of refraction, we expect significant bending, guiding the accurate placement of ultrasound probes for optimal imaging.

Critical Angle

The concept of critical angle is vital in ultrasound physics, especially when determining when total internal reflection occurs. The critical angle, \( \theta_c \), is defined as the angle of incidence in a denser medium where the refracted angle becomes 90 degrees. For this condition to be met:\[\sin \theta_c = \frac{v_{muscle}}{v_{air}} = \frac{1480}{344}\]At the critical angle, sound waves cease to enter the second medium (e.g., from muscle to air), instead reflecting entirely within the first medium. This phenomenon is used strategically in diagnosics, as it affects the direction of sound waves and helps create clearer images. Understanding the critical angle allows technicians to adjust the ultrasound beam accurately, enhancing the quality of diagnostic imaging. The calculated critical angle of 13.34° in the exercise helps prevent signal loss by ensuring optimal wave directionality.