A row of seats is parallel to a stage at a distance of 8.7 m from it. At the center and front o

step 1 In this problem drawing the diagram first, so just look at it carefully. After that I will simpl

A row of seats is parallel to a stage at a distance of 8.7...

A row of seats is parallel to a stage at a distance of $8.7 \mathrm{~m}$ from it. At the center and front of the stage is a diffraction horn loudspeaker. This speaker sends out its sound through an opening that is like a small doorway with a width $D$ of $7.5 \mathrm{~cm} .$ The speaker is playing a tone that has a frequency of $1.0 \times 10^{4} \mathrm{~Hz}$. The speed of sound is $343 \mathrm{~m} / \mathrm{s}$. What is the distance between two seats, located near the center of the row, at which the tone cannot be heard?

Key Concepts

Small-Angle Approximation

The small-angle approximation is used when the diffraction angles are small enough that sin? and tan? can be approximated by the angle itself (in radians). This simplification facilitates the calculation of the spatial locations of minima on a distant screen or seating row, by relating the angular position to a linear displacement using the distance from the aperture.

Wave Diffraction

Diffraction is the phenomenon where waves bend around obstacles or spread after passing through a narrow aperture. This behavior leads to the formation of characteristic patterns of intensity, with regions of constructive and destructive interference. Understanding diffraction is essential when analyzing how waves, such as sound, navigate openings and cause variations in signal strength at different positions.

Wavelength and Frequency Relationship

The relationship between wavelength, frequency, and wave speed is fundamental in wave mechanics. The wavelength is determined by dividing the speed of the wave by its frequency. This concept allows one to calculate the distance over which a wave's phase repeats, which is critical when predicting interference and diffraction patterns.

Single-Slit Diffraction Pattern

In single-slit diffraction, waves emerging from different parts of the slit interfere with each other, creating a central bright fringe accompanied by a series of minima and secondary maxima. The positions of the minima are given by the condition sin? = m?/D, where m is a nonzero integer, ? is the wavelength, and D is the width of the slit. This principle is used to determine the locations where the intensity of the wave (or sound) is reduced or cancelled.

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