Problem 8 - College Physics

Question

\(\bullet\) (a) If the amplitude in a sound wave is doubled, by what factor does the intensity of the wave increase? (b) By what fac- tor must the amplitude of a sound wave be increased in order to increase the intensity by a factor of 9\(?\)

Step-by-Step Solution

Answer

(a) 4 times. (b) Increase amplitude by a factor of 3.

Step 1: Understanding Intensity and Amplitude

First, note the relation between intensity (I) and amplitude (A) for a sound wave: \[ I \propto A^2 \]This means intensity is proportional to the square of the amplitude.

Step 2: Calculating Intensity Change for Doubling Amplitude

If the amplitude of the sound wave is doubled, we replace \(A\) with \(2A\) in the relationship:\[ I' \propto (2A)^2 = 4A^2 \]Thus, the new intensity \(I'\) is 4 times the original intensity \(I\).

Step 3: Solving for Amplitude Change to Increase Intensity by 9

To find out by what factor amplitude must be increased to change intensity by a factor of 9, set the new intensity \(I'\) proportional to 9 times the original intensity:\[ I' = 9I \propto (kA)^2 \]Solving for \(k\), we equate:\[ 9A^2 = (kA)^2 \]\[ k^2 = 9 \]\[ k = 3 \]This means the amplitude must be increased by a factor of 3 to achieve this change in intensity.

Key Concepts

Understanding Intensity in Sound Waves

Sound waves carry energy as they travel through a medium, and this energy is what we refer to as intensity. Intensity (\( I \)) of a sound wave is the power transmitted per unit area, usually measured in watts per square meter (\( W/m^2 \)). It describes how much energy passes through a point in a certain period. The more intense a sound wave, the louder it is perceived. The concept of intensity is pivotal in fields like acoustics and physics.
Understanding intensity helps in various applications such as estimating the loudness of sound in a concert or minimizing noise pollution in urban settings. When studying intensity, it is vital to remember its relationship with amplitude.

The Role of Amplitude in Sound Waves

Amplitude is a crucial characteristic of sound waves, representing the maximum displacement of particles from their rest position during wave oscillations. It is essentially the 'height' of the wave, indicating how much the particles in the medium are disturbed as the wave passes.
A larger amplitude means more energy is transferred by the wave, leading to a louder sound. This is often why turning up a speaker increases both the sound's amplitude and our perception of loudness.
Amplitude not only affects loudness but is also tied directly to intensity. As per the equation \( I \propto A^2 \), intensity is proportional to the square of the amplitude. This means small changes in amplitude can result in substantial differences in perceived loudness.

Understanding Proportional Relationships

In physics, a proportional relationship indicates that two quantities increase or decrease together at a constant rate. For sound waves, the intensity (\( I \)) is proportional to the square of the amplitude (\( A^2 \)). This specific relationship is powerful because it means doubling the amplitude results in a quadruple increase in intensity.

• If amplitude increases, intensity skyrockets.
• If amplitude halves, intensity plummets to one-quarter.

Such relationships are not just abstract mathematical principles; they have real-world implications. When dealing with sound systems, understanding proportional relationships helps engineers design equipment that can produce the desired sound levels without distortion.

Solving Physics Problems Involving Sound Waves

Physics problems involving sound waves often use relationships like the one between intensity and amplitude to solve practical issues. These problems offer insight into how sound behaves and how we can manipulate it for desired results.
In the given exercise, the task was to determine how changes in amplitude affect sound intensity, and vice versa. This involves recognizing that:

• Doubling the amplitude (\( 2A \)) results in four times the intensity (\( 4I \)).
• Increasing intensity by nine times requires tripling the amplitude (\( 3A \)).

Such problems highlight important physics concepts but also aid in skills like critical thinking and application of mathematical formulas. Whether you're solving textbook problems or working in sound engineering, understanding these relationships is essential.