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A firework charge is detonated many meters above the ground. At a distance of 400m from the explosion, the acoustic pressure reaches a maximum of 10.0 N/m^{2}. Assume that the speed of sound is constant at 343 m/s throughout the atmosphere over the region considered, that the ground absorbs all the sound falling on it, and that the air absorbs sound energy as described by the rate 7.0 dB/km.

What is the sound level (dB) at 4.00km from the explosion?

I know that

[itex]\beta = 10 log \left \frac{I}{I_0} \right [/itex]

and that

[itex]I=\frac{P}{A}=\frac{1}{2} p v w^2 {s^2_{max}}[/itex]

where p is the density of air, v is the speed of sound, w is the angular frequency and s_{max}is the amplitude of the position function s(x,t)=s_{max}cos(kx-wt).

but I am having trouble correctly solving for I, and so I can't get the book answer of B=65.6 dB. Any help is appreciated.

Note: the equation I obtained for this problem taking into account the damping of the sound in air is:

[itex]\beta = 10 log \frac{I}{I_0}+br[/itex]

where b=-7 dB/km and r=4.0km is the distance from the explosion.

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