Sound Intensity: Videos & Practice Problems

Sound intensity level, measured in decibels (dB), uses a logarithmic scale to simplify the wide range of sound intensities humans can hear. The formula for sound intensity level is β = 10 log₁₀(I/I₀), where I₀ is the reference intensity of 1 x 10^-12 W/m². For example, a siren with a power of 9 mW at 3 meters produces a sound level of approximately 79 dB, illustrating how sound intensity relates to perceived loudness.

Sound Intensity Level and the Decibel Scale

Understanding sound intensity and its measurement is crucial in acoustics. The intensity of a wave, applicable to all types of waves, is quantified using the equation:

I = \frac{P}{4 \pi r^2}

where I represents intensity in watts per meter squared, P is the power of the sound source, and r is the distance from the source. However, when discussing sound, we often refer to the sound intensity level, which is measured in decibels (dB). This is because the human ear can detect a vast range of sound intensities, from the faint rustling of leaves (approximately \(1 \times 10^{-11}\) W/m²) to the loud roar of a jet engine (around \(1 \times 10^{1}\) W/m²).

To simplify these values, we use a logarithmic scale defined by the equation for sound intensity level:

\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)

In this equation, \(\beta\) is the sound intensity level in decibels, I is the intensity of the sound source, and I₀ is a reference intensity, typically set at \(1 \times 10^{-12}\) W/m², which represents the threshold of hearing for humans.

To illustrate this concept, consider a siren producing a power of 9 milliwatts (0.009 W). To find the sound level at a distance of 3 meters, we first calculate the intensity:

I = \frac{0.009}{4 \pi (3)^2} = 7.96 \times 10^{-5} \text{ W/m}^2

Next, we substitute this intensity into the sound intensity level equation:

\beta = 10 \log_{10} \left(\frac{7.96 \times 10^{-5}}{1 \times 10^{-12}}\right)

Calculating this gives:

\beta \approx 79 \text{ dB}

This result indicates that the sound level of the siren at 3 meters is approximately 79 decibels, a significant sound level that can be easily perceived by the human ear. Understanding these principles allows for better comprehension of sound dynamics in various environments.

A sound wave from a police siren has an intensity of 0.01 W/m² at a certain distance. A second sound wave from an ambulance has a sound intensity level 8 dB greater than the police siren, when measured at the same distance. What is the sound intensity level (in dB) of the sound wave due to the ambulance?

• A. 8.01 dB
• B. 108 dB
• C. 0.063 dB

Answer: (B)

Example 2

When dealing with sound intensity and its relationship to distance, it's essential to understand how sound levels change as you move away from a source. In this scenario, we have a sound source emitting sound uniformly in all directions, and we want to determine how the sound intensity level, measured in decibels, changes when the distance from the source is doubled.

To analyze this, we start by defining two distances: r₁ (the initial distance) and r₂ (the distance after moving away, which is twice r₁). The corresponding sound intensities at these distances are I₁ and I₂. The sound intensity level in decibels is calculated using the formula:

β = 10 \log_{10} \left( \frac{I}{I_0} \right)

where I₀ is the reference intensity. To find the change in sound intensity level, we need to calculate the difference between the two levels:

Δβ = β₁ - β₂

Substituting the formula for β, we have:

Δβ = 10 \log_{10} \left( \frac{I₁}{I_0} \right) - 10 \log_{10} \left( \frac{I₂}{I_0} \right)

Factoring out the common term gives us:

Δβ = 10 \left( \log_{10} \left( \frac{I₁}{I₂} \right) \right)

Using the inverse square law, we know that the intensity ratio can be expressed in terms of the distances:

\frac{I₁}{I₂} = \frac{r₂^2}{r₁^2}

Since r₂ is twice r₁, we can substitute:

\frac{I₁}{I₂} = \frac{(2r₁)^2}{r₁^2} = \frac{4r₁^2}{r₁^2} = 4

Substituting this back into our equation for Δβ gives:

Δβ = 10 \log_{10}(4)

Calculating this yields:

Δβ = 10 \times 2 = 20 \text{ dB}

However, since we are interested in the decrease in sound intensity level when moving away, we note that the sound intensity level decreases by 6 dB for each doubling of distance. Therefore, moving from r₁ to r₂ results in a decrease of 6 dB.

This relationship illustrates that sound intensity diminishes non-linearly with distance, specifically following the inverse square law. Each time the distance is doubled, the sound intensity level decreases by 6 dB, highlighting the significant impact of distance on sound perception.

Here’s what students ask on this topic:

What is the formula for calculating sound intensity level in decibels?

The formula for calculating sound intensity level in decibels (dB) is given by:

$\beta =10log{10}_{}(\frac{I}{I}$

Here, $\beta$ is the sound intensity level in decibels, $I$ is the intensity of the sound, and $I_0$ is the reference intensity, which is $1\times {10}^{-} W/m^2$.

How do you convert sound intensity to decibels?

To convert sound intensity to decibels, use the formula:

$\beta =10log{10}_{}(\frac{I}{I}$

Here, $\beta$ is the sound intensity level in decibels, $I$ is the intensity of the sound, and $I_0$ is the reference intensity, which is $1\times {10}^{-} W/m^2$. By plugging in the values of $I$ and $I_0$, you can calculate the sound intensity level in decibels.

What is the reference intensity used in the sound intensity level formula?

The reference intensity used in the sound intensity level formula is $1\times {10}^{-} W/m^2$. This value, denoted as $I_0$, represents the lower threshold of human hearing. It serves as a reference point for comparing other sound intensities and is crucial for calculating the sound intensity level in decibels.

How do you calculate the sound intensity level of a siren producing 9 mW of power at a distance of 3 meters?

To calculate the sound intensity level of a siren producing 9 mW of power at a distance of 3 meters, follow these steps:

1. Convert power to watts: $9 \mathrm{mW}=0.009 W$.

2. Calculate intensity using $I=\frac{P}{}$: $\frac{0.009}{}$ = $7.96\times {10}^{-} W/m^2$.

3. Use the formula $\beta =10log{10}_{}(\frac{I}{I}$:

$10log{10}_{}(\frac{7.96}{\times }$ = $79 \mathrm{dB}$.

Why is a logarithmic scale used for measuring sound intensity levels?

A logarithmic scale is used for measuring sound intensity levels because the range of sound intensities that humans can hear is extremely wide. The logarithmic scale compresses this vast range into a more manageable and comprehensible set of numbers. For example, instead of dealing with intensities ranging from $1\times {10}^{-} W/m^2$ to $1\times {10}^1 W/m^2$, the logarithmic scale converts these to a more practical range of 0 to 130 decibels. This makes it easier to compare and understand different sound levels.