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Chapter 19: Problem 15

A certain sound level is increased by an additional \(30 \mathrm{~dB}\). Show that ( \(a\) ) its intensity increases by a factor of 1000 and (b) its pressure amplitude increases by a factor of 32 .

Short Answer

Expert verified
By using the logarithmic property of decibels, when the sound level is increased by 30 dB: (a) the intensity does increase by a factor of 1000, and (b) the pressure amplitude also increases by an approximately a factor of 32.

Step by step solution

01

Establishing the Decibel Formula

Begin by applying the formula of decibel scale, which is: \(L_{dB} = 10 \log_{10} \left(\frac{I}{I_0}\right)\)
02

Deriving the Intensity Factor

Next, in order to prove the claim (a), the decibel increase is given as 30 dB. By substituting in the formula, the equation can be written as \(30 = 10 \log_{10} \left(\frac{I}{I_0}\right)\). Rearranging the equation for \(I\) and \(I_0\), you get \(\frac{I}{I_0} = 10^{\frac{30}{10}}\), which simplifies to \(\frac{I}{I_0} = 10^3\) or \(\frac{I}{I_0} = 1000\). Therefore, the intensity increases by a factor of 1000.
03

Deriving the Pressure Amplitude Factor

To prove claim (b), knowing that the intensity is directly proportional to the square of the pressure amplitude \(I\propto P_{amplitude}^2\), the pressure amplitude increase may be written as \(\sqrt{\frac{I}{I_0}} = \frac{P_{amplitude}}{P_{initial}}\). Substituting \(\frac{I}{I_0} = 1000\) into the equation you get \(\sqrt{1000} =\frac{P_{amplitude}}{P_{initial}}\), which simplifies to \(\frac{P_{amplitude}}{P_{initial}} = 31.62\). Therefore, the pressure amplitude increases by an approximately a factor of 32.

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Most popular questions from this chapter

You are given four tuning forks. The fork with the lowest frequency vibrates at \(500 \mathrm{~Hz}\). By using two tuning forks at a time, the following beat frequencies are heard: \(1,2,3,5,7\), and \(8 \mathrm{~Hz}\). What are the possible frequencies of the other three tuning forks?

If a violin string is tuned to a certain note, by what factor must the tension in the string be increased if it is to emit a note of double the original frequency (that is, a note one octave higher in pitch)?

A sound wave of intensity \(1.60 \mu \mathrm{W} / \mathrm{cm}^{2}\) passes through a surface of area \(4.70 \mathrm{~cm}^{2} .\) How much energy passes through the surface in \(1 \mathrm{~h}\) ?

Show that the sound wave intensity \(I\) can be written in terms of the frequency \(f\) and displacement amplitude \(s_{\mathrm{m}}\) in the form $$ I=2 \pi^{2} \rho v f^{2} s_{\mathrm{m}}^{2} $$

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