Question

Estimate the maximum possible sound level in decibels of sound waves in air. (Hint: Set the pressure amplitude equal to \(1 \mathrm{~atm} .\) )

Short answer

The maximum possible sound level in air is approximately 194 dB.

Step-by-step solution

Understanding the Given Information

It's given that the pressure amplitude is set as 1 atm. We also know that 1 atmosphere (atm) is equivalent to \(1.013 \times 10^{5}~\mathrm{Pa}\). Moreover, the reference pressure (\(p_0\)) for sound in air is specified as \(2 \times 10^{-5}~\mathrm{Pa}\).

Apply the Formula for Sound Level in Decibels

The formula for calculating the sound level in decibels is given by \(L = 20 \log \left(\frac{p}{p_0}\right)\) where \(p\) is the pressure amplitude and \(p_0\) is the reference pressure. The pressure amplitude in this case is equivalent to \(1.013 \times 10^{5}~\mathrm{Pa}\) and the reference pressure is \(2 \times 10^{-5}~\mathrm{Pa}.\)

Substitute the Values into the Formula

Substitute the values of \(p\) and \(p_0\) into the formula, it gives:\[L = 20 \log \left(\frac{1.013 \times 10^{5}}{2 \times 10^{-5}}\right)\]

Calculate the Sound Level

On calculating the value in the logarithmic part of the expression by dividing \(1.013 \times 10^{5}\) by \(2 \times 10^{-5}\), it results in \(5.065 \times 10^{9}\). Taking the logarithm gives 9.70. Multiplying 9.70 by 20 gives a sound level of 194 dB.

Key concepts

Decibels

Decibels are a measurement unit used to quantify the intensity of sound. The decibel scale is logarithmic, which means every increase of 10 decibels represents a tenfold increase in sound intensity. This characteristic makes it easier to represent the vast range of sound pressures that the human ear can detect—from the faintest whisper to the roar of a jet engine.

Understanding how decibels work is crucial for assessing sound levels. It is not an absolute unit like a meter or a kilogram but a relative one that compares different sound pressures. This relative approach simplifies representing sound intensities that otherwise would involve large numbers.

• The formula used to calculate sound level in decibels is: \[ L = 20 \log \left(\frac{p}{p_0}\right) \] where: - \( p \) is the pressure amplitude of the sound wave, - \( p_0 \) is the reference pressure, typically taken to be the threshold of hearing.

In the specific exercise discussed, the formula is aptly utilized to estimate the maximum possible sound level in decibels by substituting appropriate values for the pressure amplitude and reference pressure.

Pressure Amplitude

Pressure amplitude is a term that refers to the maximum pressure variation above and below the ambient atmospheric pressure occurring due to a sound wave. It is a key factor in determining the loudness of a sound. Greater pressure amplitude typically means a louder sound.

In the context of the exercise, the pressure amplitude was set to 1 atmosphere (atm), an immense value when dealing with sound in air. Understanding pressure amplitude is vital because it directly affects the calculation of sound levels in decibels.

• An important conversion: \(1 \, \text{atm} = 1.013 \times 10^{5} \, \text{Pa}\).

This conversion allows us to easily plug into the decibel formula and aids in determining the sound level for a given pressure amplitude. This simplifies calculating sound level estimates, as depicted in the step-by-step exercise solution.

Reference Pressure

Reference pressure is a foundational concept when working with sound measurements. It serves as the baseline for calculating sound levels in decibels. Usually, the reference pressure value is set at \(2 \times 10^{-5} \, \text{Pa}\), which corresponds to the quietest sound that can be perceived by the average human ear.

Having a standard reference pressure is essential because sound levels are often compared to this baseline to determine how loud or soft they are. Without this standardization, comparing sound intensities would be challenging because of the varying ranges involved.

• This reference value allows: - Easy calculation of sound levels using the decibel formula, - Consistency across calculations and different contexts.

In the exercise, the reference pressure is critical to solving the problem, as it is used in the logarithmic calculation to derive the sound level in decibels, thereby determining the maximum possible sound level in air.