Rossing 6.3 With one violin playing, the sound level at a certain place is measured as 50 dB. If four violins play equally loudly, what will the sound level most likely be at this place?
Let's call the sound intensity produced by the one violin playing, I1. Then the sound intensity level from this violin that we hear is
LI(1) = 10 log( I1 / I0 ).
When four violins play, assuming they each produce sound intensity I1 at our ear, the total sound intensity is just the sum of the intensities produced separately from each violin, or
Itotal = 4 I1
We can then find the sound intensity level when four violins play
LI (total) = 10 log( Itotal / I0 ) = 10 log( 4I1 / I0 )
We now make use of a property of logarithms, log(A x B) = log A + log B, to write the total sound intensity level as
LI (total) = 10 log( 4 ) + 10 log( I1 / I0 )
The second term is just the sound intensity level from a single violin, LI(1) = 10 log( I1 / I0 ). This means that the sound intensity level from four violins playing equally loudly is
LI (total) = 6 dB + 50 dB = 56 dB
The first term is 10 log( 4 ) = 6 dB. The second term is the sound intensity level from one violin.
S22 The sound level from a single violin playing is 60 dB. We can find its intensity from the definition of level in terms of intensity
L = 10 log ( I / I0)
or, 60 dB = 10 log ( I / I0)
or, 6 = log ( I / I0)
meaning, I / I0 = 106
or, I = 106 x I0 = 106 x 10-12 W/m2
finally, this gives I = 10-6 W/m2
Now, when two violins play together, the total intensity is 2 x 10-6 W/m2. NOTE: WE DO NOT ADD THE SOUND LEVELS FROM THE TWO VIOLINS PLAYING SEPARATELY!!! This would make no sense if we did add the sound levels, since we would conclude that two violins playing would produce a level of 120 dB, equal to the loudness of a nearby jet engine!
The proper thing to do is to add the INTENSITIES for the two sounds:
I2 violins = 2 x I = 2 x 10-6 W/m2
This means that the sound level is, L2 violins = 10 log (2I / I0)
or, L2 violins = 10 log(2) + 10 log(I / I0)
or, L2 violins = 3 dB + 60 dB = 63 dB
S23 Each of four people talking, when speaking individually produce an unknown sound level L1, corresponding to an unknown intensity I1. When all four talk together, the sound level is 70 dB and the total intensity is 4 x I1. So, from the definition of sound level
L4 people = 70 dB = 10 log(4 I1 / I0)
or, using a rule for logarithms:
L4 people = 70 dB = 10 log(4) + 10 log(I1 / I0)
or, L4 people = 70 dB = 10 log(4) + L1
or, 70 dB - 6 dB = L1
finally, L1 = 64 dB
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Last updated: 27 Oct 1999
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