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

I₁ = P / A = P / (4r_A²)

where, P = sound power = 2.1 W

r_A = radius of small concentric sphere = 0.50 m

then, we get

I₁ = (2.1 W) / [4 (3.14) (0.50 m)²]

I₁ = [(2.1 W) / (3.14 m²)]

I₁ = 0.668 W/m²

For patch 2, we have

I₂ = I₁

I₂ = 0.668 W/m²

For patch 3, we have

I₃ = P / A = P / (4r_B²)

where, P = sound power = 2.1 W

r_B = radius of large concentric sphere = 1.2 m

then, we get

I₃ = (2.1 W) / [4 (3.14) (1.2 m)²]

I₃ = [(2.1 W) / (18.0864 m²)]

I₃ = 0.116 W/m²

(b) Find the area of each patch.

For patch 1, we have

A₁ = P / I₁ [(1.8 x 10^-3 W) / (0.668 W/m²)]

A₁ = 0.0027 m²

For patch 2, we have

A₂ = P / I₂ [(1.8 x 10^-3 W) / (0.668 W/m²)]

A₂ = 0.0027 m²

For patch 3, we have

A₃ = P / I₃ [(1.8 x 10^-3 W) / (0.116 W/m²)]

A₃ = 0.0155 m²