Intensity and Sound Level — Question 3

Question 3

Background sound must be subtracted in linear units

A meter reads $70.0\,\mathrm{dB}$ with a machine operating and $64.0\,\mathrm{dB}$ with it off. Treat machine and background as incoherent at the receiver. Both readings are intensity levels using the same reference.

1. Find the machine-only level.

2. What fraction of the operating intensity comes from the background? Explain why subtracting $64$ from $70$ is not the machine level.

Question 3 - Solution

Background sound must be subtracted in linear units

(a) Method and calculation

Convert before subtraction:$$I_{\mathrm{on}}/I_0=10^7,\qquad I_b/I_0=10^{6.4}.$$$$I_m/I_0=10^7-10^{6.4}=7.4881\times10^6,$$$$\beta_m=10\log_{10}(7.4881\times10^6)=68.744\,\mathrm{dB}.$$

(b) Interpretation and further calculation

The background fraction is$$I_b/I_{\mathrm{on}}=10^{(64-70)/10}=0.25119.$$The $6\,\mathrm{dB}$ difference represents the ratio $I_{\mathrm{on}}/I_b=3.981$, not an intensity remaining after subtraction. If the readings were nearly equal, small measurement errors would strongly affect the inferred machine intensity.

Physical check

The machine-only level is below the operating level but above the background level. The subtraction assumes background stays unchanged.