Intensity and Sound Level — Question 2

Question 2

Adding identical sources depends on their coherence

At a receiver one source produces intensity level $60.0\,\mathrm{dB}$ relative to $I_0=10^{-12}\,\mathrm{W/m^2}$. Four sources can each produce that same individual level. Compare mutually incoherent operation with coherent, in-phase operation at the receiver. Assume locally copropagating waves and maintained amplitudes.

1. Find the total intensity and level in both cases.

2. Explain why neither total is obtained by adding four decibel readings. What happens in the coherent case if two contributions reverse phase?

Question 2 - Solution

Adding identical sources depends on their coherence

(a) Method and calculation

One source gives $I_1=I_0 10^6=10^{-6}\,\mathrm{W/m^2}$. Incoherent intensities add, while coherent in-phase pressures add:$$I_{\mathrm{inc}}=4I_1,\quad\beta_{\mathrm{inc}}=60+10\log_{10}4=66.021\,\mathrm{dB},$$$$I_{\mathrm{coh}}=16I_1,\quad\beta_{\mathrm{coh}}=60+20\log_{10}4=72.041\,\mathrm{dB}.$$

(b) Interpretation and further calculation

Decibels are logarithms; the physical quantities must be combined before taking the logarithm. Reversing two equal coherent contributions gives$$\widetilde p=P+P-P-P=0.$$The ideal local intensity is then zero, so no finite logarithmic level exists there. Real imperfections yield a finite residual, and a local null does not imply silence everywhere.

Physical check

The coherent result describes the specified receiver and maintained source amplitudes; it is not a universal factor for total source power.