Final answer:
The diffraction angle of sound can be determined using the equation sin(θ) = λ / w, where λ is the wavelength and w is the doorway width. With a frequency of 652 Hz and a speed of sound of 343 m/s, the wavelength is approximately 0.526 m. The diffraction angles are found using the arcsin function for the doorway width of 0.850 m when one door is open, and 1.700 m when both doors are open.
Step-by-step explanation:
To determine the diffraction angle of sound passing through a doorway, we can use the equation for the first minimum of a single-slit diffraction pattern, which is sin(θ) = λ / w, where θ is the diffraction angle, λ is the wavelength of sound, and w is the width of the slit (in this case, the doorway).
First, we need to calculate the wavelength (λ) of the sound wave using the formula λ = c / f, where c is the speed of sound and f is the frequency. Given that the speed of sound c is 343 m/s and the frequency f is 652 Hz for the sound coming through the room, the wavelength is:
λ = 343 m/s / 652 Hz = approximately 0.526 m.
When one door is open (0.850 m wide), the diffraction angle θ for the first minimum can be found as follows:
sin(θ) = λ / w = 0.526 m / 0.850 m
∴ θ = arcsin(0.526 m / 0.850 m).
Similarly, when both doors are open (1.700 m wide), the diffraction angle θ is computed in the same manner:
sin(θ) = λ / w = 0.526 m / 1.700 m
∴ θ = arcsin(0.526 m / 1.700 m).
To get the actual angles in degrees, one would need to use a calculator to compute the arcsin values.