Subjective Type

A point isotropic sound source is located in the perpendicular to the plane of a ring drawn through the centre $$O$$ of the ring. The distance between the point $$O$$ and the source is $$l=1.00\ m$$ the radius of the ring is $$R=0.50\ m.$$ Find the mean energy flow across the area enclosed by the ring if at the point $$O$$ the intensity of sound is equal to $$I_{0}=30\mu\ W/m^{2}.$$ The damping of the waves is negligible.

Solution

The power of output of the source much be
$$4\pi l^2 I_0=Q\ Watt$$.
The required flux of acoustic power is then $$: Q=\dfrac {\Omega }{4\pi}$$
Where $$\Omega$$ is the solid angle subtended by the disc enclosed by the ring at $$S$$. This solid angle is
$$\Omega =2\pi (1-\cos \alpha)$$
So flux $$\Phi =I_0 I_0 \left( 1-\dfrac {l}{\sqrt{r^2+R^2}}\right) 2\pi l^2$$
Substitution gives $$\Phi =2\pi \times 30\left( 1-\dfrac {1}{\sqrt{1+\dfrac 14}}\right) \mu W=1.99\ \mu W$$.
Eqn (1) is a well known result stich is derived as follows; Let $$SO$$ be the polar axis. Then the required solid angle is the area of that part of the surface of a sphere of much radius whose colatitude is $$\le \alpha$$.
Thus $$ \Omega =\int_0^{\alpha}{2\pi \sin \theta d\theta }=2\pi (1-\cos \alpha )$$.

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