Question

An audio engineer takes decibel readings at distances of r1 = 11 m and r2 = 21 m from a concert stage speaker during a sound check. When he is r1 from the speaker, the engineer registers a decibel level of β1 = 110 dB on his loudness meter.

r1 = 15 m
r2 = 21 m
β1 = 101 dB

a. What is the intensity of the sound, I1, in watts per square meter, that is measured by the loudness meter when the engineer is a distance of r1 from the speaker?
b. How much power P, in watts, is coming from the speaker during the sound check at distance r1?
c. Assuming that the speaker output does not change between the two measurements at r1 and r2, what sound intensity level β2, in decibels, will the loudness meter report when the engineer is at a distance r2 from the speaker?

Answer 1

Answer:

a) $I = 0.01258 W/m^{2}$

b) $P_{1} = 35.6 W$

c) $I_{2} = 0.00641 W/m^{2}$

Explanation:

r1 = 15m

r2 = 21m

β1 = 101 dB

Calculating intensity I1 on the loudness meter using the equation:

 a)             $\beta = 10 log( \frac{I}{I_0} )$

              $I_0 = 10^{-12} W/m^{2}$ (corresponding to β=0)

⇒          $101 = 10 log( \frac{I}{10^{-12} } )$

⇒          $10.1 = log( \frac{I}{10^{-12} } )$

⇒          $\frac{I}{10^{-12} } = 10^{10.1}$

⇒          $I = 10^{10.1}$ × $10^{-12}$

⇒         $I = 0.01258 W/m^{2}$

b)      Power $P_{1} = A_{1} I_{1}$

  where $A_{1}$ is the area of sphere of radius r1

⇒              $P_{1} = (I_{1} )(4\pi r_1^{2} )$

⇒               $P_{1} = 0.01258 * 4\pi * 15^{2}$

⇒                $P_{1} = 35.6 W$

c)      $\frac{I_2}{I_1} =(\frac{r_1}{r_2} )^{2}$

⇒      $\frac{I_2}{0.01258} = (\frac{15}{21} )^{2}$

    Simplifying

⇒      $I_{2} = 0.00641 W/m^{2}$

Answer 2

A) The intensity when the engineer is a distance of r1 from the speaker is; I₁ = 0.01259 W/m²

B) The power coming from the speaker during the sound check at distance r1 is; P₁ = 35.597 W

C) The intensity when the engineer is a distance of r2 from the speaker is; I₂ = 0.00642 W/m²

We are given;

Distance for decibel readings;

r₁ = 15 m

r₂ = 21 m

When the audio engineer is at r₁ decibel level is; β₁ = 101 dB

A) To calculate the Intensity at r₁ , we will use the formula;

β₁ = 10 log(I₁/I₀)

At β = 0, threshold intensity I₀ = 10⁻¹² W/m²

Thus;

101 = 10 log(I₁/10⁻¹²)

log(I₁/10⁻¹²) = 101/10

I₁/10⁻¹² = 10^(10.1)

I₁ = 10⁻¹² × 10^(10.1)

I₁ = 0.01259 W/m²

B) At at r₁, power is calculated from the formula;

P₁ = I₁*A₁

where A is area = 4πr₁²

A₁ = 4 * π * 15²

Thus;

P₁ = 0.01259 * 4 * π * 15²

P₁ = 35.597 W

C) To get the intensity when he is at r₂ from the speaker, we will use the formula;

I₂ = I₁(r₁/r₂)²

Thus;

I₂ = 0.01259 × (15/21)²

I₂ = 0.00642 W/m²

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