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

[Gear300]

A firework charge is detonated many meters above the ground. At a distance of 400 m from the explosion, the acoustic pressure reaches a maximum of 10.0 N/m². 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 7.00 dB/km. What is the sound level (in dB) at 4.00 km from the explosion?

I've been on this question for a while...can't seem to get the answer. The answer I'm supposed to get is 65.6 dB.

Since the sound wave seems as though it'd be a spherical wave, the area in the equation for power and the area the power is distributed over (in the intensity equation) are different...so they do not cancel out. From what I understand from the question, some of the sound energy is being absorbed by the air to produce the drop of 7.00 dB/km...which will cause the drop in intensity over a distance to be higher than with it simply decreasing due to distance alone. What would I do from here?

 

[Redbelly98]

I would start by first ignoring the 7.00 dB/km drop, and think about how sound intensity or pressure amplitude vary with distance from the source of the sound.

 

[Gear300]

I would start by first ignoring the 7.00 dB/km drop, and think about how sound intensity or pressure amplitude vary with distance from the source of the sound.

Hmmm...since 4000 is a factor of 10 greater than 400...the pressure should also be a maximum at 4000 km at that point in time...but how exactly would I use this...I still keep running into loops.

 

[Redbelly98]

The factor-of-10 greater distance is the key.

Your textbook (assuming it is covering this material adequately) should have a discussion or equation about sound intensity vs. distance from the source -- or perhaps sound pressure vs. distance from the source.

 

[Gear300]

I found a procedure to get the answer, and the process involves this:
they ended up taking the maximum intensity to be at 400 m somewhere in the process...how did they come to assume this?

 

[Redbelly98]

When they say "At a distance of 400 m from the explosion, the acoustic pressure reaches a maximum of 10.0 N/m2", I interpreted that to mean the amplitude of pressure oscillations is 10.0 N/m^2 at 400m from the explosion.

I.e., at any location the pressure oscillates, and has a maximum value (at that location) equal to the amplitude.

The overall maximum occurs right at the sound source, and decreases as you move farther away.