Problem 94 - Pearson Physics

Question

Explain why the intensity of a point source of sound decreases with the square of the distance from the source.

Step-by-Step Solution

Answer

The intensity decreases with the square of the distance because it spreads over a sphere with area \(4\pi r^2\).

Step 1: Concept of Intensity

Intensity of sound is defined as the power per unit area. It is given by the formula \(I = \frac{P}{A}\), where \(I\) is the intensity, \(P\) is the power, and \(A\) is the area through which the power is spread.

Step 2: Understanding Point Source

A point source emits sound equally in all directions, creating a spherical wavefront around the source. As the sound travels outward, the energy is distributed over the surface of a sphere.

Step 3: Surface Area of a Sphere

The surface area \(A\) of a sphere with radius \(r\) is given by \(A = 4\pi r^2\). The radius \(r\) here represents the distance from the point source to the point at which intensity is being measured.

Step 4: Intensity Formula Substitution

Substitute the expression for the surface area of a sphere into the intensity formula: \(I = \frac{P}{4\pi r^2}\). Here, \(I\) represents the intensity at a distance \(r\) from the point source.

Step 5: Inverse Square Law

From the formula \(I = \frac{P}{4\pi r^2}\), it is clear that intensity \(I\) is inversely proportional to the square of the distance \(r\). Thus, as \(r\) increases, \(I\) decreases with the square of the distance.

Key Concepts

Intensity of Sound

Intensity of sound refers to the amount of energy that sound waves carry per unit area. This can be thought of as how much sound power travels through a given space. When discussing sound intensity, we describe it using the formula: \[ I = \frac{P}{A} \]where:

• \(I\) is the intensity, measured in watts per square meter (W/m²).
• \(P\) is the power of the sound, which is the total energy transferred per second, measured in watts (W).
• \(A\) is the area over which the sound power is distributed.

The importance of intensity lies in how it relates to the loudness perceived by our ears. Higher intensity means more energy and thus, a louder sound. As sound spreads, its intensity diminishes with increasing distance.

Point Source

A point source of sound is a simplification used to understand how sound spreads in three-dimensional space. When we say a sound source is a 'point source', it means that the source of the sound can be considered as a single point emitting sound waves uniformly in all directions.

• This model is useful for understanding basic sound propagation.
• In reality, most sound sources are not perfect point sources but can be approximated as such for simplicity.

A point source creates a series of continuously expanding wavefronts that form spheres around the source. Each expanding sphere represents a further distance from the point source, affecting the intensity of the sound.

Spherical Wavefront

A spherical wavefront comes into play when a point source emits sound. Sound waves move outward in all directions, creating a series of expanding spheres, or spherical wavefronts. As these spheres grow larger, they encompass more area, causing the sound energy to spread out over a larger surface.

• Each spherical wavefront represents all points at the same distance from the sound source at a specific moment in time.
• This spread of energy over a larger area is why the intensity of the sound decreases with distance.

As the wavefront propagates further from the source, it illustrates how the same amount of sound power is distributed over an increasingly larger area, showing the concept behind the inverse square law.

Surface Area of a Sphere

The surface area of a sphere is crucial to understanding how sound intensity diminishes with distance. Given a sphere with radius \(r\), the surface area \(A\) is calculated using the formula: \[ A = 4\pi r^2 \]This formula helps us understand why the intensity of sound from a point source decreases with the square of the distance:

• As \(r\) (the distance from the point source) increases, the surface area over which the sound spreads also increases drastically, specifically by a factor of \(r^2\).
• This relationship is embedded in the formula for sound intensity: \[ I = \frac{P}{4\pi r^2} \]which shows that intensity \(I\) decreases as the square of the radius \(r\) increases.

Understanding the surface area of a sphere clarifies why the sound intensity is inversely related to the square of the distance, which reflects the essence of the inverse square law.