Problem 93 - Physics

Question

A listener doubles his distance from a source that emits sound uniformly in all directions. By how many decibels does the sound intensity level change?

Step-by-Step Solution

Answer

The sound intensity level decreases by 6 decibels.

Step 1: Understand the Initial Intensity Formula

Sound intensity ($$I$$) at a distance ($$r$$) from a point source is given by the formula $$I = \frac{P}{4\pi r^2}$$, where $$P$$ is the power of the source.

Step 2: Sound Intensity's Relation with Distance

If the distance from the source is doubled ($$r \rightarrow 2r$$), the new intensity $$I'$$ becomes $$I' = \frac{P}{4\pi (2r)^2} = \frac{P}{16\pi r^2}$$. Since $$I = \frac{P}{4\pi r^2}$$, we can express the new intensity as $$I' = \frac{I}{4}$$.

Step 3: Calculate Change in Intensity Level

The sound intensity level $$\beta$$ in decibels is given by $$\beta = 10 \log_{10}\left(\frac{I'}{I_0}\right)$$, with $$I_0$$ as the reference intensity. The change in intensity level when intensity changes from $$I$$ to $$I'$$ is $$\Delta \beta = 10 \log_{10}(I'/I)$$.

Step 4: Solve for Change in Intensity Level

Using $$I' = \frac{I}{4}$$, substitute in to find: $$\Delta \beta = 10 \log_{10}\left(\frac{I' = \frac{I}{4}}{I}\right)$$. This simplifies to: $$\Delta \beta = 10 \log_{10}\left(\frac{1}{4}\right)$$. Further simplifying gives: $$\Delta \beta = 10 \times (-0.6) = -6\, \text{dB}$$.

Key Concepts

Decibel Scale

The decibel scale is a logarithmic way to measure sound intensity levels. It helps us understand how loud a sound is compared to a reference value. This scale uses base 10 logarithms, which means that every 10 dB represents a tenfold change in intensity.
If a sound increases by 10 dB, it is ten times more intense. Conversely, if a sound decreases by 10 dB, it is one-tenth as intense.

• This scale is useful because our ears respond logarithmically to changes in sound intensity.
• It allows us to quantify sound intensity changes in a way that is meaningful to human perception.
• The formula to calculate the decibel level (\(\beta\)) is:\(\beta = 10 \log_{10}\left(\frac{I}{I_0}\right)\), where \(I_0\) is the reference intensity, typically the threshold of hearing.

Sound Intensity Formula

The sound intensity formula is a critical part of understanding how sound behaves at different distances from its source. The formula given is:\(I = \frac{P}{4 \pi r^2}\), where \(I\) is the sound intensity, \(P\) is the power of the sound source, and \(r\) is the distance from the source.

• Sound intensity is the power flowing through a unit area perpendicular to the direction of sound waves.
• As the distance from the sound source increases, the intensity decreases because the power spreads over a larger area.
• The sphere around the sound source increases as you move away, which affects the intensity captured by a listener.

When you double the distance from the source to \(2r\), the formula becomes:\(I' = \frac{P}{4 \pi (2r)^2} = \frac{I}{4}\), showing how intensity decreases with distance.

Inverse Square Law

The inverse square law is a fundamental principle that describes how sound intensity decreases with distance from the source. According to this law, sound intensity is inversely proportional to the square of the distance from the source.
This means that if you double the distance (\(r\rightarrow2r\)), sound intensity becomes one-fourth as strong:\(I' = \frac{I}{4}\).

• It explains why sounds become quieter as you move farther away from the sound source.
• This principle applies not only to sound but also to light and other radiation forms.
• This is because the same amount of energy spreads out over a larger area, reducing the intensity at any given point.

Understanding this law is key to many applications in acoustics and electronics.

Sound Propagation

Sound propagation refers to the process by which sound waves travel through a medium, such as air, water, or solids. The speed and intensity of sound can vary based on the medium's properties.

• Sound travels by vibrating particles in the medium, transferring energy from one particle to another.
• The denser the medium, the faster sound travels. However, the intensity can diminish over greater distances, as dictated by the medium’s properties and the inverse square law.
• Environmental factors like temperature, humidity, and wind can also affect how sound propagates.

For example, sound travels faster on a warm day than on a cold day because particles move more quickly in warm air.
As sound travels, its intensity diminishes due to energy spreading and losses from interactions with the medium.