Plane waves of ultrasound, f = 51.0 kHz, impinge on a flat plane with a 1.00 cm-wide slit. What is the total angular width (2θ) of diffracted sound waves when the speed of sound in air is 340 m/sec.
A) 167°
B) 83.6°
C) 19.5°
D) 41.8°
E) 0.11°

Question

Quizplus · Accepted Answer

The total angular width (2θ) of diffracted sound waves can be calculated using the formula for the first minimum in a single-slit diffraction pattern, which is given by $\sin(\theta) = \frac{\lambda}{w}$, where $\lambda$ is the wavelength of the sound and $w$ is the width of the slit. The wavelength $\lambda$ can be found using the relation $\lambda = \frac{v}{f}$, where $v$ is the speed of sound and $f$ is the frequency. Substituting the given values, $\lambda = \frac{340 \, \text{m/s}}{51000 \, \text{Hz}} = 0.00667 \, \text{m} = 6.67 \, \text{mm}$. Since the slit width $w = 1.00 \, \text{cm} = 10 \, \text{mm}$, we find $\sin(\theta) = \frac{6.67 \, \text{mm}}{10 \, \text{mm}} = 0.667$. Solving for $\theta$ gives $\theta \approx 41.8^\circ$, so the total angular width $2\theta \approx 83.6^\circ$.

Plane Waves of Ultrasound, F = 51