Question

A source of sound is located at the center of two concentric spheres, parts of which are shown in the drawing. The source emits sound uniformly in all directions. On the spheres are drawn three small patches that may or may not have equal areas. However, the same sound power passes through each patch. The source produces 2.1 W of sound power, and the radii of the concentric spheres are rA = 0.50 m and rB = 1.2 m.(a) Determine the sound intensity at each of the three patches. (b) The sound power that passes through each of the patches is 1.8 × 10-3 W. Find the area of each patch.

Answer #1

(a) Determine the sound intensity at each of the three patches.

For patch 1, we have

I_{1} = P / A = P / (4r_{A}^{2})

where, P = sound power = 2.1 W

r_{A} = radius of small concentric sphere = 0.50 m

then, we get

I_{1} = (2.1 W) / [4 (3.14) (0.50 m)^{2}]

I_{1} = [(2.1 W) / (3.14 m^{2})]

**I _{1} = 0.668 W/m^{2}**

For patch 2, we have

I_{2} = I_{1}

**I _{2} = 0.668 W/m^{2}**

For patch 3, we have

I_{3} = P / A = P / (4r_{B}^{2})

where, P = sound power = 2.1 W

r_{B} = radius of large concentric sphere = 1.2 m

then, we get

I_{3} = (2.1 W) / [4 (3.14) (1.2 m)^{2}]

I_{3} = [(2.1 W) / (18.0864 m^{2})]

**I _{3} = 0.116 W/m^{2}**

(b) Find the area of each patch.

For patch 1, we have

A_{1} = P / I_{1}
[(1.8 x 10^{-3} W) / (0.668 W/m^{2})]

**A _{1} = 0.0027 m^{2}**

For patch 2, we have

A_{2} = P / I_{2}
[(1.8 x 10^{-3} W) / (0.668 W/m^{2})]

**A _{2} = 0.0027 m^{2}**

For patch 3, we have

A_{3} = P / I_{3}
[(1.8 x 10^{-3} W) / (0.116 W/m^{2})]

**A _{3} = 0.0155 m^{2}**

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