Sound of a certain wavelength is coming from within a room and passing through a rectangular door (a "slit"). If the width of the doorway is twice the wavelength of the sound, what is the diffraction angle è of the sound leaving the doorway?
Please show me how to do this.
To find the diffraction angle of the sound leaving the doorway, you can use the concept of diffraction. Diffraction occurs when a wave encounters an obstacle or a slit and spreads out. The diffraction angle, denoted as θ, is the angle between the direction of the incident wave and the direction of the diffracted wave.
In this case, the width of the doorway is twice the wavelength of the sound. Let's denote the width of the doorway as D and the wavelength as λ.
The first step is to determine the size of the diffraction pattern. For a single slit, the angular size of the diffraction pattern is given by the formula:
sin(θ) = λ / D
In this case, we know that D = 2λ, so we can substitute this value into the formula:
sin(θ) = λ / (2λ)
sin(θ) = 1/2
Now, to find the diffraction angle θ, we need to take the inverse sine (also known as arcsine) of both sides of the equation:
θ = arcsin(1/2)
Using a calculator or mathematical software, we can find the arcsine of 1/2:
θ ≈ 30 degrees
Therefore, the diffraction angle of the sound leaving the doorway is approximately 30 degrees.