See: Radiolab’s Walls of Jericho podcast from October 2010.
Act 1: [mp3 | 8M] The hosts lay out the story of Jericho, where an Israelite army brought the walls down, supposedly by shofar (a ram’s horn) blasts. Along the way, we learn about the logarithmic decibel scale. In the final seconds of this clip, we get to the question that all my students were already asking: what would it really take?*
Act 2: [mp3 | 8M] Wherein David Lubman, the acoustical scientist consulted by the hosts, reveals how many shofar blowers it would take to bring down the walls.
Act 3: Continue playing the Act 2 file to explore issues of how to focus the sound and the physics of sound cannons.
*At this point, my kids set to the calculations. They wanted to know if there was a faster way. It was a beautiful experience where the kids asked me to take them from brute-force-arithmetic to honest-to-goddess upper level math. Then I hit play on Act 2. In the words of the experts in this podcast, “there’s a problem.” Just as my class noticed (and demanded we contact the Radiolab folks), another teacher noted a problem in Act 2’s big reveal:
Steven from Palo Alto
There appears to be an inconsistency with the explanation of the mathematics that leads to the total number of shofar players needed to me the 177dB target. If every time the number of shofar players is doubled, the dB level increases by three, then the number of shofar players would have to be doubled 29 times between 95 dB (the sound level of one shofar player) and 177dB. 2^29 shofar players is more than 1000x larger than the 407,380 figure that David Lubman gives. I am not trying to be critical, but I was hoping to use this story as an example of exponential growth for a class that I teach, and this is where my demonstration derails in relation to the podcast. Did anyone look as closely at this part of the story as I did?
Your confusion is perfectly understandable.
The example you cited (how many shofar doublings to reach 177 dB @ 1 m assuming 95 dB for 1 shofar) is not pertinent to the the number of shofars required to bring down the wall. The example was intended only to help a dedicated (math) teacher demonstrate exponential growth. You can see that by reading the comments. I believe the number of shofar doublings needed to reach 177 dB from a start of 95 dB is correct.
My original calculation of the number of shofars needed to bring down the wall was made for the Discovery Channel special “Joshua and the Walls of Jericho”, and is much more complicated.
It did not assume a source strength of 95 dB @ 1 m as in our example. It assumed a source strength 20 dB higher: 115 dB @ 1 m. This is almost certainly unrealistically high. At the time, I bent over backwards to give the biblical story the benefit of any doubts.
I’ve outlined below some of the assumptions I made in the original lower-bound estimate of the 400,000 shofars needed.
One vexing reason the example was simplified is the problem of spatially distributed sources. The Radiolab example silently assumes a virtual source of fixed size with an equivalent source strength of 10*log (N), where N is the number of shofars. That gives the maximum sound level at a small spot on the wall . But for more than one shofar, the source is spatially distributed. Even if the total sound energy is the same, but the maximum sound level is lower, and the insonified area is correspondingly larger. Much larger than for a single virtual source we assumed.
Imagine the impossible problem of crowding hundreds of thousands of shofars and their players around a single spot on the wall.
First, try to cover a hemisphere with equi-spaced players on its surface, each equi-distant (say, 2 m) from an arbitrary spot on the wall. (Ignore the problem of those near the top needing ladders!) Then place shofars at equi-spaces spots on concentric hemispheres at increasing radial distances from the source. The good news is that the second hemisphere can accomodate more players. The bad news is that the contribution per shofar is smaller because of larger spreading losses..
Beyond a modest distance the bottom of the hemisphere is interrupted by the ground. At larger distances the hemisphere becomes a quarter-sphere.
It assumed a single source with a source strength equivalent to N shofars.
Initially, it assumed an equivalent source strength of seven shofars, in keeping with the biblical text. That gave an initial source strength of 115 dB + 10 log (7) = 115 dB + 8.5 dB = 123.5 dB @ 1 m.
I arbitrarily assumed a spot on the wall at a distance of 2 m from the equivalent source, so the level at the wall is 6 dB lower [-20log (2)] or 117.5 dB.
Then I assumed the wall interaction would increase sound level at the wall by 3 dB (incoherent addition).
Then I added the estimated sound level of shouting men. I found an estimate for trained shouting male voice as 90 dB @ 1 m from EPA document EPA-600/1-77-025 May, 1977.. To that I arbitrarily added a generous 10 dB, again to give the biblical story the benefit of a doubt.
What about the 10,000 men? This was tough because the shouting men comprise a distributed source as well. I assumed 300 men, the estimated number of men in Gideon’s successful raiding party against the Midianites described in the Book of Judges. I assumed the shouters stood 0.75 m apart 20 rows of 15 men. Ranks farther from the wall added less to the sound level at the wall because of spreading loss. For that reason, increasing the number of shouting males to 10,000 (a factor of 33.3) would increase the level at the wall by far less than the 15 dB one would otherwise assume.
But even adding 30 dB more for the shouting men would not change my conclusion.
Thanks so much for the detailed response, David! I can’t wait to share it with my students tomorrow. They were the ones who demanded — on the spot — that I comment on the podcast when our numbers didn’t mesh. Thank you for helping my students have an incredibly empowering experience.
This is exactly why I blog. And teach. And love my job/kids/life.
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