Question

Ignoring attenuation, how does the intensity of a sound change as the distance from the source doubles? (A) \(\quad\) It is four times as intense. (B) \(\quad\) It is twice as intense. (C) It is half as intense. (D) \(\quad\) It is one-quarter as intense.

Short answer

When the distance is doubled, the intensity of the sound is one-quarter as intense.

Step-by-step solution

Understand the Concept of Sound Intensity

Sound intensity is a measure of the power per unit area. The intensity of a sound wave is inversely proportional to the square of the distance from the source. This is known as the inverse square law.

Set Up the Inverse Square Law Equation

The inverse square law can be mathematically expressed as: \[ I \times d^2 = \text{constant} \] where \( I \) is the intensity and \( d \) is the distance from the source.

Apply the Inverse Square Law

When the distance from the source is doubled (i.e., \( d \rightarrow 2d \)), the equation becomes: \[ I_2 \times (2d)^2 = \text{constant} \] Simplifying: \[ I_2 \times 4d^2 = \text{constant} \]

Relate Initial and Final Intensity

Since the initial intensity equation was \( I_1 \times d^2 = \text{constant} \), and the new intensity equation is \( I_2 \times 4d^2 = \text{constant} \), we can set them equal to each other: \[ I_1 \times d^2 = I_2 \times 4d^2 \] This simplifies to: \[ I_1 = 4I_2 \]

Find the Final Intensity

Rearranging the equation \( I_1 = 4I_2 \), we find: \[ I_2 = \frac{I_1}{4} \] This indicates that the intensity \( I_2 \) is one-quarter of the initial intensity \( I_1 \) when the distance is doubled.

Choose the Correct Option

Based on our calculations, the correct answer is option (D): It is one-quarter as intense.

Key concepts

inverse square law

The inverse square law is a fundamental principle in physics that helps us understand how different quantities diminish with increasing distance from their source.
This law states that the intensity of a physical effect such as sound, light, or radiation decreases proportionally to the square of the distance from the source.

For example, if you double the distance from a sound source, the intensity of the sound becomes one-quarter of what it was originally. Mathematically, this is expressed as:

\( I \times d^2 = \text{constant} \)
where \( I \) is the intensity and \( d \) is the distance.

The inverse square law helps us predict how intense a sound will be at different distances. It's like spreading out a fixed amount of energy over a larger area as you move farther away.

sound wave

Sound waves are vibrations that travel through a medium such as air, water, or solids. These waves carry energy from one place to another.
Sound waves are characterized by their frequency (how many times the particles of the medium vibrate in a second) and amplitude (how much the particles are displaced from their resting position).

When you listen to music or someone speaking, you're experiencing sound waves traveling through the air and hitting your eardrums. The intensity of these waves determines how loud they appear.
This intensity of a sound wave diminishes over distance, as described by the inverse square law. This means sound becomes quieter the further you are from the source.

distance attenuation

Distance attenuation refers to the reduction in sound intensity as it travels away from the source.
Because of the inverse square law, as sound moves farther away, it spreads out over a larger area, weakening its intensity.

Here's an example: If you stand right next to a speaker at a concert, the music is extremely loud. But move to the back of the venue, and the volume decreases significantly.
This happens because the sound energy distributes over greater distances, causing the intensity to drop.

Understanding distance attenuation is crucial in acoustics, audio engineering, and designing spaces like theaters or concert halls to ensure sound is heard clearly despite varying distances.