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Chapter 16: Problem 27

The sound level in decibels is typically expressed as \(\beta=10 \log \left(I / I_{0}\right),\) but since sound is a pressure wave, the sound level can be expressed in terms of a pressure difference. Intensity depends on the amplitude squared, so the expression is \(\beta=20 \log \left(P / P_{0}\right),\) where \(P_{0}\) is the smallest pressure difference noticeable by the ear: \(P_{0}=2.00 \cdot 10^{-5} \mathrm{~Pa}\). A loud rock concert has a sound level of \(110 . \mathrm{dB}\), find the amplitude of the pressure wave generated by this concert.

Short Answer

Expert verified
Answer: The amplitude of the pressure wave generated by the rock concert is approximately \(1.58 \mathrm{Pa}\).

Step by step solution

01

Write down the given information

We have the following information: - Sound level at the rock concert, \(\beta = 110\mathrm{dB}\) - Smallest pressure difference noticeable by the ear, \(P_{0} = 2.00 \cdot 10^{-5} \mathrm{Pa}\)
02

Write down the expression for sound level in decibels

The expression for sound level in decibels (\(\beta\)) is given by: $$ \beta=20 \log \left(\frac{P} {P_{0}}\right) $$
03

Plug in the given information into the expression for \(\beta\)

We will plug in the given values of \(\beta\) and \(P_{0}\) into the expression for \(\beta\): $$ 110 = 20 \log \left(\frac{P} {2.00 \cdot 10^{-5}}\right) $$
04

Solve for \(P\)

Now, we need to solve this equation for \(P\). Follow these steps: 1. Divide both sides by 20: $$ 5.5 = \log \left(\frac{P} {2.00 \cdot 10^{-5}}\right) $$ 2. Apply anti-logarithm to eliminate the logarithm: $$ 10^{5.5} = \frac{P} {2.00 \cdot 10^{-5}} $$ 3. Multiply both sides by \(2.00 \cdot 10^{-5}\): $$ P = 10^{5.5} \times 2.00 \cdot 10^{-5} \mathrm{Pa} $$ 4. Calculate the value of \(P\): $$ P \approx 1.58 \mathrm{Pa} $$ Thus, the amplitude of the pressure wave generated by the rock concert is approximately \(1.58 \mathrm{Pa}\).

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Most popular questions from this chapter

Compare the intensity of sound at the pain level, \(120 \mathrm{~dB}\), with that at the whisper level, \(20 \mathrm{~dB}\).

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