Please explain how to solve in detail! thank you As Section 17.3 discusses, high

Question

Please explain how to solve in detail! thank you

As Section 17.3 discusses, high-frequency sound waves exhibit less diffraction than low-frequency sound waves do. However, even high frequency sound waves exhibit much more diffraction under normal circumstances than do light waves that pass through the same opening. The highest frequency that a healthy ear can typically hear is 2.0 times 10^4 Hz. Assume that a sound wave with this frequency travels at 342 m/s and passes through a doorway that has a width of 0.93 m. (a) Determine the angle that locates the first minimum to either side of the central maximum in the diffraction pattern for the sound, (b) Suppose that yellow light (wavelength - 551 nm, in vacuum) passes through a doorway and that the first dark fringe in its diffraction pattern is located at the angle determined in part (a). How wide would this hypothetical doorway have to be? (a) Number Units (b) Number Units

Explanation / Answer

a) The angle that locates the first minimum in the diffraction pattern can be obtained by

sin = / w

= sin^-1 ( / w)

angle , = sin^-1(lamda/width)

lamda, = velocity / frequency
           = v/f

Therefore

angle, = sin-1(v / f*w)

angle, = sin-1[ (342m/s) / (2.0E4)(0.93) ]

            = 1.05 degrees

b) sin = / w

w = lamda/sin

= (551E-9m) / sin(1.05)

= 3.0 E-5 m

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