Physics: Two sources of sound are located on the x axis, and each emits power uniformly in all directions. there are no reflections. one source is positioned at the origin and the other at x = +78.0 m. the source at the origin emits four times as much power as the other source. where on the x axis are the two sounds equal in intensity? note that there are two answers.

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Two sources of sound are located on the x axis,

saeedalr366

31.12.2021

The intensity is same at $x_{1}=156.0m$ and at $x_{2}=52.0m$

Explanation:

Let the 2 sources have the same intensity at a distance 'x' from the origin

Since the energy of the wave front emitted will be distributed over a sphere we have

For source 1, located at origin intensity at a distance of 'x' from origin is given by

$e_{1}=\frac{P_{1}}{4\pi x^{2}}$

For source 2, located at x = +78.0 m intensity at a distance of 'x' from origin is given by

$e_{2}=\frac{P_{2}}{4\pi (x-78)^{2}}$

Now it is also given that $P_{1}=4P_{2}$

Thus equating the intensities we get

$\frac{4E_{2}}{4\pi x^{2}}=\frac{E_{2}}{4\pi (x-78)^{2}}$

Solving the equation for x we get

$\frac{4}{x^{2}}=\frac{1}{(x-78)^{2}}\\\\(\frac{x}{x-78})^{2}=4\\\\\therefore \frac{x}{x-78}=\pm 2$

Solving using both the values we get

$\therefore \frac{x_{1}}{x_{1}-78}= 2\\\\2x_{1}-156=x_{1}\\\\\therefore x_{1}=156.0m$

Similarly the other value is obtained as

$\therefore \frac{x_{2}}{x_{2}-78}= -2\\\\-2x_{2}+156=x_{2}\\\\\therefore -3x_{2}=-156.0m\\\\\Rightarrow x_{2}=52.0m$

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