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When heard independently, two sound wave...

When heard independently, two sound waves produce sensation of 60 db and 55 db respectively. How much will the sensation be if those are sounded together, perfectly in phase?

Text Solution

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`L_1 = 60 db = 10 log_(10) ""(l_1)/(I_0) therefore (l_1 )/( I_0) = 10^6 or I_1 = 10^6 I_0`
Similarly, ` I_2 = 10^(5.5 ) I_0`
As the waves combine perfectly in phase, the vector addition of their amplitudes will be given by ` A^2 = (A_1 +A_2)^2 = A_(1)^(2) +A_(2)^(2) + 2A_1 A_2`
As intensity is proportional to square of the amplitude.
`therefore I=I_1 +I_2 +2 sqrt( 2 sqrt(I_1I_2)`
` = 10^5 I_0 (10^1 + 10 ^(0.5 ) + 2 sqrt(10^(1.5 ))`
`=10^5 I_0 (10 +3 .1623 + 2 xx 10^(0.75))`
` = 24.41 xx 10^5 I_0 = 2.441 xx 10^6 I_0`
` therefore L=10 log_(10) ""((I )/(I_0)) = 10 log_(10) ( 2.441 xx 10^6)`
` = 10 [ log_(10) "" ( 2.441 ) + log ""( 10^6)]`
` + 10 ( 0.3876 =6)`
`=63.876 db ~~ 64 db`
It is interesting to note that there is only a marginal increase in the loudness.
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