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Chapter 15: Problem 12

The \(1 / R^{2}\) dependency for intensity can be thought of to be due to the fact that the same power is being spread out over the surface of a larger and larger sphere. What happens to the intensity of a sound wave inside an enclosed space, say a long hallway?

Short Answer

Expert verified
Answer: In an enclosed space, the intensity of a sound wave does not follow the \(1 / R^2\) dependency as it does in an open space. Instead, it depends on the geometry of the space, the reflective properties of the surfaces, and the absorption level of the materials in the room. This is because sound waves interact with the walls, floor, and ceiling, causing the sound to reflect and reverberate throughout the space, resulting in a more complex sound field. In a long hallway, the intensity of the sound will generally be higher and more uniform than in an open space due to the sound energy being constrained by the dimensions of the hallway.

Step by step solution

01

Understand the difference between open and enclosed spaces

In an open space environment, the intensity of a sound wave follows a \(1 / R^2\) dependency, meaning the intensity decreases with the square of the distance from the sound source. This is because the sound energy is spreading out over an ever-increasing surface area as it travels away from the source. In an enclosed space, however, the sound waves will interact with the walls, floor, and ceiling, causing the sound to reflect and reverberate throughout the space.
02

Effects of reflection and reverberation

When sound waves encounter a boundary, such as a wall or other solid surface, some of the energy is absorbed by the surface, and some is reflected back into the space. This is known as reverberation. Reverberation causes the sound waves coming directly from the source to mix with the reflected sound waves, resulting in a more complex sound field. This means that the intensity at a particular point in the enclosed space depends not only on the direct sound from the source but also on the reflected and reverberant sound energy.
03

Resulting intensity distribution in an enclosed space

As a result of reflection and reverberation, the intensity of a sound wave inside an enclosed space does not follow the \(1 / R^2\) dependency. Instead, it depends on the geometry of the space, the reflective properties of the surfaces, and the absorption level of the materials in the room. In general, the intensity of sound in a long hallway will be higher and more uniform than in an open space, because the sound energy cannot spread out in 3D; it is constrained by the dimensions of the hallway.
04

Importance of room acoustics

The analysis of sound intensity in enclosed spaces is crucial for determining the acoustical properties of rooms, such as concert halls, classrooms, and lecture halls. To achieve desired acoustics, the room's size, shape, and materials used must be taken into consideration to minimize undesirable echoes, maximize sound clarity, and maintain an even distribution of sound intensity throughout the room.

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