## Sound pressure level decibel decay over distance calculation

This information goes hand-in-hand with Cost of Timing and will help you calculate the sound pressure dampening across different distances!

For the below calculations we’re going to use the altitude at sea level, use a scientific pitch of C6, and imagine the playing field is an anechoic chamber, put simply an isolated area without echo. This allows us to simplify the calculation and we can then use that relative information in reference to environments on the marching field. We’re also going to do any calculations with one a single snare drum player in mind (since using an entire snareline or drumline might make things slightly more difficult). We also need to be careful not to use the terms ‘loudness’ or ‘volume’ because what we’re measuring is the sound pressure level. That all being said, the calculations aren’t actually too bad when using decibels, let’s jump in!

First thing we need to understand is the inverse square law. This tells us that every time you double your distance from the sound source (or in our case the sound source from the audience) the effect of the decay will multiplied by a factor of 4. So on our marching field if you were standing 1 foot from the snare drum sound source you are going to hear the exact dynamic they’re playing on the drum. But now imagine your’re standing on the front sideline and the snare drum is being played on the back sideline. The drum is producing the same dynamic at the back, but the dynamic you’re hearing is going to actually be far fewer decibels at the front!

Now, let’s put some musical terminology into the equation! This might be a little tricky because there is no universal definition of how many dB forte is or mezzo-piano, but we can make a rough estimate that will still benefit us and take away some useful data relative to whatever we personally use for dynamic levels. So, for the sake of argument we’re going to use 80dB for forte, and 60dB for piano. So let’s extrapolate a full dynamic range table for this data.

fff | 100dB |

ff | 90dB |

f | 80dB |

mf | 70dB |

mp | 60dB |

p | 50dB |

pp | 40dB |

ppp | 30db |

The only thing here we need to keep in mind is that the calculation is considering the highest dB at 1ft away from the drum, but audience members usually are never this close, so we need to understand that any differential sound pressure levels we see in our calculations is compared to the full dynamic coming off the drum, not from where you’d normally stand comfortably in front of the drumline. Quick example, if your ear is literally 1ft from the drum head when a snare drum hits it at fff it would obviously be louder (painful even) than when you’re standing even just a few feet away. The below calculations are starting at the ear shattering closeness, in this example, and not the comfortable distance of that dynamic that we might be used to as instructors in front of a drumline.

The calculation we’ll be using the rest of the way down this path is as follows:

*L*_{p2} = *L*_{p1} + 20 log_{10} ( *r*_{1} / *r*_{2} ) dB

_{p2}

_{p1}

_{1}

_{2}

Broken down ** L_{p2}** is the decibel of the distance to the audience, the one we’re calculating for.

**is the decibels at distance**

*L*_{p1}*from the drum, in our case it will be at 1ft. And*

**r**_{1}*is the distance from the sound source to the audience*.*

**r**_{2}*Another caveat that is going to help glue this idea to our cost of timing equation is that all of our dynamics on the field will decay at the same rate once they pass the furthest sound source forward. This means that if a snare drum on the back sideline is playing fortississimo a snare drum on the front sideline would have to play mezzo-piano to balance that sound pressure level (again, to reiterate if that sounds soft, that’s mezzo-piano with your ear 1ft from the front drum which still louder than you’d normally think of mezzo-piano from a comfortable listening or instructing distance). But back to the main point here, both of those sounds will continue to decay as they travel to the audience, however, they’ll decay at the same rate from here forward which means that we only need to factor the distance from the furthest back sound source to the furthest forward sound source for the purposes of our needs here.

We can actually make our equation even simpler because we know our first distance is always going to be 1ft from the drum. Based on the equation above we know that every time the distance doubles we decay 6dB.

Our new simplified equation now looks like this:

*L*_{p2} = 20 log_{10 }r_{2}

_{p2}

New example, snare 2 is standing 67 steps behind the front sideline while snare 1 is standing on the front sideline. The step size from front to back is 22.857 inches. So since the distance doubled 7 times we find that we lose 42.1dB by the time that the snare 2 sound reaches snare 1 up front. If we put these numbers into our simplified equation we see the result is the same.

## 20 log_{10} 128ft = 42.1

This now means we can calculate our sound pressure decibel loss based on our dynamic for anywhere on the field, as long as we know our distance. Then we just simply subtract our decibel loss from our original decibel figure to give us our decibel reading at the front of the field. Then just look up the perceived dynamic based on the chart above.

Let’s go back to the original example of both snares standing on the front and back sidelines. The distance is 160ft so the decay is about 44dB. We said the back snare was playing at fortississimo which is 100dB and so by the time that sound arrived at the front snare we lost 44dB. This means that the front snare had to play at 56dB to balance the sound which, according to our chart, is roughly mezzo-piano if we rounded up a touch.

And now, the final major question we’ve all been asking (except for any math teachers): How do I calculate logarithms with my calculator?! Well to do what we need to do it’s actually quite easy, but understanding what’s going on would be an explanation for another time. In our previous example above we needed to simply calculate 20 log 160. (Note: the base 10 log is called Common Log, so unless you’re using a different base number you don’t actually have to write the 10 in the equation.) To do this in your calculator app all you need to punch in is the 160 value first, press the log button, then multiply the answer by 20. This should give you 44.0824, which rounds up to just 44 for our purposes. Again this is the sound pressure level decibel loss that we then subtract from the source decibels (100dB) to give us the 56dB as our arrival sound pressure level at the front of the field.

**Save this web app to your phone’s home screen for fast access to reference on the field!**