Disaster Area, the “loudest band in the galaxy” from Douglas Adams’ The Restaurant at the End of the Universe, is remarkably loud.
So loud that their audience prefers to listen to them in a concrete bunker 37 miles away from the stage. So loud that Disaster Area play their instruments remotely from a spaceship that’s orbiting a different planet. So loud that they have trouble booking shows because their PA violates ordnance regulations.
If their audience prefers a reasonable OSHA-approved listening level of 83 dB SPL, and their listening bunker has one-foot thick concrete walls, how loud is it at ground zero of the Disaster Area stage if the concert took place on Earth?
We’ve done some napkin math to figure it out, so try it yourself and then compare numbers after the cut!
How do we measure how loud something is?
Almost everyone is familiar with the decibel as an indicator of volume: more is louder. A large enough number means it hurts and listening to it long enough causes some permanent damage that makes it hard to hear quiet things for a while (or forever!). However, a decibel is much more than just an indicator of loudness: it’s an important measurement of distance between two values.
Decibels are in common usage in many fields to express ratios of difference that are otherwise cumbersome to throw around. Decibels have a few properties that make them really useful to us:
- They use a logarithmic scale, meaning that each doubling of amplitude corresponds to an increase of about 6 dB.
- They represent a ratio, so they’re always specified in terms of some reference value.
Since decibels measure a change in loudness, there are a few different ways of measuring them for different applications.
For example, the amplitude of sound in the real world is measured in dB SPL (sound pressure level). The amplitude of audio data in a computer is measured in dBFS (full scale). In dB SPL, 0 dB means 0.00002 Pascals: a tiny amount of air pressure. In dBFS, 0 dB means the largest sample value that can possibly be expressed by the wordlength of your bit depth, or a sample value of 1 if floating point is used.
For real world audio, SPL is measured with a sound pressure level meter (as one would expect), which is basically a calibrated microphone with some readout and parameters for how often a measurement is taken and whether or not some frequencies are filtered out before measuring. There’s a lot of pretty good SPL meter apps available for phones, which makes it pretty easy to measure your subway ride and e-mail yourself a CSV so you can nerd out on the data.
dB SPL Reference
Because the decibel is all about the reference, we can build some meaningful perceptual references for dB SPL values. Here are some common world dB values (many of which are cribbed from this handy list):
- 0 dB SPL: threshold of hearing, a mosquito 10 feet away, the eardrum moves less than 1/100 the length of an air molecule
- 10 dB SPL: absolute silence, AT&T-Bell Laboratory “Quiet Room”
- 40 dB SPL: whispered conversation
- 60 dB SPL: normal conversation
- 83 dB SPL: average loudness in a THX-certified cinema program; perception of frequencies is equalized around this loudness (this is where you perceive bass and treble about as well as you perceive mids)
- 85 dB SPL: permanent hearing damage after eight hours of exposure
- 110 dB SPL: average loudness at a typical rock concert or construction site
- 120 dB SPL: threshold of pain, permanent hearing damage after less than a minute of exposure
This general range is why we say that human hearing has a dynamic range of 120 dB. We can go even farther and say that the effective dynamic range of our perception is even lower (probably about 60 dB) because we can’t hear stuff buried by everyday noise like HVACs or wind, and we don’t like listening to loud things for very long (or, at least, we probably shouldn’t). This is probably something to consider, given a recent trend toward placing an importance on high bit depth audio delivery formats, where the dynamic range can be 144 dB or above. In our DAWs we have an insane amount of dynamic range available at the mix stage: about 196 dB!
The “threshold of pain” bit is somewhat subjective, but really this is around the ceiling of where it’s safe for humans to experience sound without protection. The point is, sound is pressure, and levels past this threshold make bad things happen to your hearing forever. Of course, that doesn’t stop rock and roll:
- 130 dB SPL: Front row of an AC/DC concert
- 150 dB SPL: The Who’s sound system in 1976: measured at 120 dB 50m away from the speaker, audible 100 miles away
- 154 dB SPL: The loudest sound system on Earth (used to test if spaceflight vehicles can withstand the forces of takeoff)
- 155 dB SPL: Loud enough to blur vision and cause difficulty breathing
- 194 dB SPL: 1 ATM (14.7 PSI) of pressure, rupturing of eardrums and probably some other distressing events as well
194 dB is pretty much the ceiling for audio on Earth. Past this point, pressure essentially clips, which would probably sound awesome except for you’d be dead.
While we’re talking relative loudness, it’s interesting to note that a sound needs to be at least roughly 6 dB louder than whatever it’s competing with to be perceived, and probably 12 dB louder to hear it well. If your subway ride has a noise floor of 80 dB SPL (it probably does), you’re rocking your jams between 86 and 92 dB SPL to hear them. This phenomenon is known as auditory masking, and a more sophisticated model is exploited by audio codecs like MP3 to discard sound that cannot be perceived.
90dB SPL, by the way, is about where it becomes really important to pay attention to your sound level exposure: permanent hearing damage can occur after just two hours , and safe exposure time halves for every additional 3dB.
OK so how loud is Disaster Area?
The cultured audience of Disaster Area fans sipping Pan Galactic Gargle Blasters inside their listening bunker are enjoying the noise at a refined, safe loudness of 83 dB SPL.
The bunker’s walls
The mass of the bunker’s concrete walls absorbs much of the energy present outside the structure. We can quantify this property as transmission loss, which varies by building material and density.
We take a typical density of concrete (2400 kg/m3), make it 1 foot thick (.3 meters) and get a mass of 720 kg/m2. Eyeball that value on this handy chart and we get a transmission loss of about 54 dB. If the level in the room is 83 dB SPL, we can add the transmission loss to get the level outside the Disaster Area bunker: 137 dB SPL.
Between the PA and the bunker
Sound halves in power every doubling of a distance (more or less: there are some caveats here about point sources and line sources and positioning against reflective surfaces but for our purposes we’ll consider the Disaster Area PA an unencumbered point source in space). A 2x change in acoustic power is about 6 dB.
There are 37 miles between the PA and the bunker, which is pretty close to the beautifully round value of 60,000 meters. There’s about 15 doublings between the stage and our Disaster Area concertgoers (log2 60000 ≈ 15.87).
15 doublings × 6 dB is 90 dB. Add that to the level just outside the bunker wall and we get the level at the stage: 227 dB SPL, louder than Earth’s atmosphere can support.
What happens above 194 dB SPL?
Is it meaningful to measure SPL above 194 dB? We’ve already seen that we can use decibels to compare the ranges of acoustic and digital systems. We can abuse decibels a little further to make observations about the power of natural disasters, explosions, and other phenomena we can observe with sound (but would like very much to be standing far away from). This gives us a context for comparing a Disaster Area concert on Earth to some other similarly catastrophic event, like the eruption of Krakatoa or the yield of Ivy Mike.
We’re not really dealing with sound at this point: sound is the pushing and pulling of pressure waves and our only reference for the kind of noise Disaster Area makes is the shockwaves resulting from heavy ordnance. We have to consider Disaster Area’s stage as the source of an explosion and estimate the shockwave force of the event.
This means that, above 194 dB SPL, instead of losing 6 dB per doubling of distance, we look at how explosions behave, where the shockwave loses around 18 dB (at maximum — there’s some transfer here but we’re keeping it simple for napkin math).
So, considering that:
- There are 10 doublings between bunker and stage until we hit around 194 dB (the “threshold of atmosphere”): 10 × 6 dB = 60 dB
- There are 5 remaining doublings between that point and the stage. 5 × 18 dB = 90 dB
- We don’t care about the fancy pants math or reality, really, because we’re talking about an imaginary space band that competes with rocket launches for loudness.
The total level difference between bunker and stage is 150 dB, which brings Disaster Area to a very pretty 287 dB SPL.
- 194 dB SPL: 1 ATM (14.7 PSI) of pressure, rupturing of eardrums
- 200 dB SPL: Instantaneous human death from pressure waves
- 210 dB SPL: Explosion of 1 ton of TNT
- 220 dB SPL: Saturn V rocket launch
- 286 dB SPL: Eruption of Mount St. Helens, which knocked down trees for 16 miles around it and blew out windows ~200 miles away in the Seattle-Tacoma area
- 287 dB SPL: Disaster Area concert (stage)
- 310 dB SPL: Eruption of Krakatoa, which created an anti-node of pressure on the other side of the planet
That’s our best rough calculation. Did you get another number? Got an acoustic insight we’ve missed? Leave a comment and let us know!