Sound, inverse laws and inverse square laws I am a maths teacher designing a project about noise. There is a plan to build a cement factory close to the school. Using maths we are going to try and persuade local residents that the lorries travelling to and from the factory will be too noisy.
Please could you help me understand how noise propagates? 
1) What information do I need to know about the lorry to know how much sound it will produce at source (either as intensity or pressure)
2) How do I then use this to work out the sound at a distance r away (I am confused about whether this is a inverse or inverse square law)
3) At which stage in this process do I convert into decibels? (I think I am right in saying that the final answer should be in decibels to compare to standard regulations).
I have been unable to find a reference online. Ideally you would give me a step by step guide to how to calculate the sound, in decibels, distance r away, of the lorry. We would explore the maths behind this guide, and then use to write a report.
Thank you
Jeremy
 A: This is actually more difficult than you might expect.
The 3rd question is actually the easiest to address: you convert to decibels when it is convenient to do so, and that is often as early as possible.  Decibels are convenient because they keep the numbers in human-understandable regions and are consistent with how the human ear hears, so they are inherently intuitive.  However, being a math teacher, you may wish to keep things in terms of sound pressure levels (Pascals) instead, depending on how comfortable your students are with logarithms.  As an engineer, I'm very comfortable with them, so I would convert it to decibels as soon as I possibly can.  Your students may not wish to do so.
You will need to know how much sound the lorries produce.  The easiest way to do that is to measure their noise level at a known distance.  This would be a pressure at a distance, measured in either Pascals or dB depending on how you want to approach it.  You can use this to determine how much sound energy was being emitted by assuming the lorry is a point source emitting equally in all directions.
That all being said, the next step is the one which will give you trouble.  Calculating the sound level in a real environment is extremely difficult.  In the real world, you have to consider effects like reflections.  For example, if there is a thermal inversion layer in the atmosphere above you, it causes a lot of the energy to reflect back down.  While sound usually obeys an inverse square law, if it's reflecting in a channel like this, it obeys an inverse law instead!  This can be much louder.
This was an issue for Mythbusters back in the day.  Their famous episode with the exploding cement truck was done in the presence of explosives experts who are well aware of this issue and were monitoring atmospheric conditions.  However, despite all that oversight, they still made a mistake.  Something like an inversion layer caused reflections, and the resulting sound pressure knocked out windows in the nearby town!
A: I'm going to disagree slightly with Cort.  Convert to decibels only when you need to (e.g., when you need to compare with something with a regulation which is stated in decibels). The reason for this is people have a misguided understanding of how to combine sounds and how that changes the sound intensity level (SIL) which is measured in bels or decibels (dB), versus the sound intensity amplitude which is measured in watts/meter$^2$ or pressure amplitude measured in pascals (Pa). Note that the intensity is proportional to the square of the pressure amplitude:$$\mathcal{I}\propto P^2$$ and the level, $\beta$, in decibels is
$$\beta = 10\log \frac{\mathcal{I}}{\mathcal{I}_\textrm{ref}},$$ where $\mathcal{I}_\textrm{ref}=1\times10^{-12}$ W/m$^2$.
The level involves logarithms, which compresses the relative pressures in such a way that a level change of 1 dB which correlates to about the smallest perceptible change of pressure amplitude for a human.
Consider a lorry which has an average SIL of 85 dB at 5 meters. Two such lorries  would have an average SIL of 88 dB at 5 meters.  The common mistake is that people add separate SILs to get a total. What they should add is the intensity of each lorry before converting to a level, unless they are familiar with the quick-and-dirty rules for changing levels:


*

*Doubling the source intensity adds 3 dB, and halving the intensity subtracts 3 dB.

*Every multiplication of the source intensity by 10 adds 10 dB. It takes a source intensity multiplication of 100 to add 20 dB.


Most sound measuring devices already measure in dB, so if you need to incorporate distance effects, you need to convert that to intensity:
$$\mathcal{I}=\mathcal{I}_\textrm{ref}10^{\beta/10}.$$ Then apply the effects.
For trucks on a road I would recommend finding the range of values between inverse square and inverse.  Environmental factors such as temperature, temperature inversions, and humidity can alter the values for individual residents by producing refraction effects.  Such effects can make the vehicles inaudible some days and sounding like they are next door on others.
