Problem 13 - College Physics

Question

BIO A person wears a hearing aid that uniformly increases the intensity level of all audible frequencies of sound by \(30.0 \mathrm{~dB}\). The hearing aid picks up sound having a frequency of \(250 \mathrm{~Hz}\) at an intensity of \(3.0 \times 10^{-11} \mathrm{~W} / \mathrm{m}^{2}\). What is the intensity delivered to the eardrum?

Step-by-Step Solution

Answer

After performing all the calculations, the final sound intensity delivered by the hearing aid to the eardrum would be the result obtained from Step 4.

Step 1: Understand the decibel scale

Decibels are a logarithmic measure of sound intensity. The formula to convert between sound intensity \(I\) in watts per square metre (W/m²) and sound level \(L\) in decibels (dB) is \(L = 10 \log_{10} \left( \frac{I}{I_0} \right)\), where \(I_0\) is the reference intensity, equal to \(10^{-12}\) W/m², which is the quietest sound that the average human ear can hear.

Step 2: Find the initial intensity level in decibels

First, we need to find out the initial intensity level of the sound before it is increased by the hearing aid. Using the formula from Step 1 and replacing \(I\) with the given initial intensity level (3.0 x \(10^{-11}\) W/m²), we get: \(L = 10 \log_{10} \left( \frac{3.0 \times 10^{-11}}{10^{-12}} \right)\)

Step 3: Add the increase in sound level

The hearing aid increases the sound level by 30.0 dB. This increase is added to the initial sound level we calculated in the previous step. This gives us the final sound level \(L'\) that the hearing aid delivers to the ear.

Step 4: Convert back to intensity

Now we are given a sound level in decibels and we need to find out the corresponding sound intensity. This requires rearranging the formula from Step 1 to solve for \(I\). The rearranged formula is: \(I = I_0 \times 10^{L'/10}\)

Key Concepts

Decibel Scale

The decibel scale is a way to measure sound intensity in a more manageable form. Sound intensity itself is measured in watts per square meter (W/m²),
which can vary over several magnitudes, making it difficult to compare directly. By using a logarithmic scale, decibel (dB) levels can make understanding and calculations simpler. Logarithms reduce these wide ranges to easier, more practical terms.
A sound’s decibel level is calculated using the equation:\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]where:

• \( L \) is the sound level in decibels
• \( I \) is the intensity of the sound
• \( I_0 \) is the reference intensity level, which is typically \( 10^{-12} \)W/m²

This approach allows for expressing the sound levels in smaller, consistent units, facilitating easier communication and comparison.

Hearing Aids

Hearing aids are devices that help individuals with hearing impairments by amplifying sound. When a hearing aid increases sound intensity by a certain decibel value, it affects all frequencies evenly. In the original exercise, the hearing aid amplifies the sound intensity by 30 dB.
This increase ensures that the user can hear important sounds at a louder intensity. Hearing aids typically gather external sounds, process the sound waves, and amplify them before delivering to the eardrum.
The task is to ensure that conversation and environmental sounds are heard more clearly, aiming to improve the user’s hearing experience. By increasing intensity in decibels, hearing aids offer clearer and more distinguishable sound.

Sound Frequency

Sound frequency refers to the number of wave cycles that pass a point per unit time, measured in hertz (Hz). It essentially determines the pitch of the sound we hear.
In the original problem, the sound frequency mentioned is 250 Hz. This corresponds to a low-pitched sound, like a deep voice or a bass note.
When paired with the intensity information, it tells us how loud that particular frequency would be. Different frequencies can respond differently through a hearing aid depending on the technology, but basic amplification covers the audible range uniformly.
Hence, understanding both frequency and intensity is critical for devices like hearing aids, as they need to amplify varied frequencies appropriately for clearer hearing.

Logarithmic Measure

A logarithmic measure is a tool used to compare quantities that differ widely in size. In sound, it is ideal for dealing with large intensity ranges.
The decibel scale is logarithmic, so every increase by 10 dB represents a tenfold increase in intensity. This makes it useful in many fields, from acoustics to earthquake measurement.
In our problem-solving steps, the logarithmic nature allowed us to manipulate the formulas to easily solve for intensity or intensity level. To convert back from decibels to watts per square meter, the equation rearranges to:\[ I = I_0 \times 10^{L'/10} \]where:

• \( I \) is the final intensity in W/m²
• \( I_0 \) is the reference intensity
• \( L' \) is the final sound level in decibels

Understanding this concept helps us navigate sound intensity issues and easily adjust levels when needed.