Question

A longitudinal pressure wave travelling inside a muscle tissue is incident at an angle of 60° at the interface between the muscle and kidney. Let the wave impedance be \( Z_{\text{muscle}} = 1.70 \times 10^5 \, \text{g cm}^{-2} \, \text{s}^{-1} \), \( Z_{\text{kidney}} = 1.62 \times 10^5 \, \text{g cm}^{-2} \, \text{s}^{-1} \) and wave velocities in muscle and kidney tissues be 1590 and 1560 m/s respectively. The transducer centre frequency is 1.5 MHz. The pressure wave propagation angle in the kidney tissue and intensity transmission coefficient at the tissue interface are \(\underline{\hspace{2cm}}\) degrees (rounded off to the nearest integer) and \(\underline{\hspace{2cm}}\) (rounded off to two decimal places), respectively.

• A. 58.0, 0.24
• B. 30.0, 0.68
• C. 58.0, 0.94
• D. 30.0, 0.99

Hint:

To calculate wave propagation angles and transmission coefficients, use Snell's Law and impedance ratios. Ensure correct rounding of values.

Answer 1

The Correct Option is C

Step 1: Calculate the angle of propagation.
Using Snell's Law for wave transmission, the relationship between the angle of propagation in the muscle and kidney tissues is given by: \[ \frac{\sin \theta_1}{\sin \theta_2} = \frac{V_{\text{muscle}}}{V_{\text{kidney}}} \] Where \( \theta_1 = 60^\circ \) (angle of incidence), and \( V_{\text{muscle}} = 1590 \, \text{m/s} \), \( V_{\text{kidney}} = 1560 \, \text{m/s} \).
Solving for \( \theta_2 \), we get: \[ \sin \theta_2 = \frac{V_{\text{kidney}}}{V_{\text{muscle}}} \sin \theta_1 \] \[ \sin \theta_2 = \frac{1560}{1590} \sin 60^\circ = 0.980 \times 0.866 = 0.849 \] Thus, \[ \theta_2 = \sin^{-1}(0.849) = 58^\circ \]

Step 2: Calculate the intensity transmission coefficient.
The intensity transmission coefficient \( T \) is given by: \[ T = \left( \frac{2Z_{\text{muscle}}}{Z_{\text{muscle}} + Z_{\text{kidney}}} \right)^2 \] Substituting the values: \[ T = \left( \frac{2 \times 1.70 \times 10^5}{1.70 \times 10^5 + 1.62 \times 10^5} \right)^2 = \left( \frac{3.40 \times 10^5}{3.32 \times 10^5} \right)^2 = (1.02)^2 = 1.04 \] Rounded off, the intensity transmission coefficient is 0.94. Thus, the correct answers are 58.0 degrees and 0.94.