Diffraction occurs for all types of waves, including sound waves. High-frequency sound from a distant source with wavelength 9.00 cm passes through a slit 12.0 cm wide. A microphone is placed 8.00 m directly in front of the center of the slit, corresponding to point O in Fig. 36.5a . The microphone is then moved in a direction perpendicular to the line from the center of the slit to point O. At what distances from O will the intensity detected by the microphone be zero?

Verified step by step guidance

1. Identify the given values: wavelength (λ) of the sound wave is 9.00 cm, width of the slit (a) is 12.0 cm, and the distance from the slit to the microphone (L) is 8.00 m.
2. Convert all measurements to the same unit, preferably meters, for consistency in calculations.
3. Use the formula for the minima in diffraction patterns, which occurs at angles (θ) satisfying the equation \(a \sin(\theta) = m\lambda\), where \(m\) is the order of the minimum (m = ±1, ±2, ±3, ...).
4. Calculate the angle (θ) for each order (m) using the equation from step 3. Remember that \(\sin(\theta) = \frac{m\lambda}{a}\).
5. Determine the positions along the perpendicular direction from point O where the intensity is zero by using the relationship \(x = L \tan(\theta)\), where \(x\) is the distance from point O at which the intensity is zero.

Key Concepts

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through narrow openings. This phenomenon occurs for all types of waves, including sound, light, and water waves. The extent of diffraction depends on the wavelength of the wave relative to the size of the opening or obstacle. In this scenario, the sound waves will diffract as they pass through the slit, affecting the intensity detected by the microphone.

Interference

Interference is the phenomenon that occurs when two or more waves overlap and combine to form a new wave pattern. This can result in constructive interference, where wave amplitudes add together, or destructive interference, where they cancel each other out. In the context of the microphone's position, points of zero intensity correspond to locations where destructive interference occurs due to the path difference between waves arriving from different parts of the slit.

Path Difference

Path difference refers to the difference in distance traveled by two waves arriving at a point from different sources. For destructive interference to occur, the path difference must equal an odd multiple of half the wavelength. In this problem, as the microphone is moved, the varying path differences from the sound waves passing through the slit will determine the points where the intensity is zero, based on the wavelength of the sound and the geometry of the setup.