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Chapter 12: Problem 8

\(\bullet\) (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\(?\)

Short Answer

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

Step by step solution

01

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.
02

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 \(I\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental frequency and the first two over-tones, determine the locations along the pipe (measured from the left end) of the displacement nodes if (a) the pipe is open atboth ends; (b) the pipe is closed at the left end and open at the right end.

\(\cdot\) The electromagnetic spectrum. Electromagnetic waves, which include light, consist of vibrations of electric and magnetic fields, and they all travel at the speed of light. (a) FM radio. Find the wavelength of an FM radio station signal broadcasting at a frequency of 104.5 \(\mathrm{MHz}\) . (b) \(\mathrm{X}\) rays. X rays have a wavelength of about 0.10 \(\mathrm{nm}\) . What is their frequency? (c) The Big Bang. Microwaves with a wavelength of 1.1 \(\mathrm{mm}\) , left over from soon after the Big Bang, have been detected. What is their frequency? (d) Sunburn. Sunburn (and skin cancer) are caused by ultraviolet light waves having a frequency of around \(10^{16} \mathrm{Hz}\) . What is their wavelength? (e) SETI. It has been suggested that extraterrestrial civilizations (if they exist) might try to communicate by using electromagnetic waves having the same frequency as that given off by the spin flip of the electron in hydrogen, which is 1.43 GHz. To what wave-length should we tune our telescopes in order to search for such signals? (f) Microwave ovens. Microwave ovens cook food with electromagnetic waves of frequency around 2.45 \(\mathrm{GHz}\) . What wavelength do these waves have?

A bat flies toward a wall, emitting a steady sound of frequency 2000 \(\mathrm{Hz}\) . The bat hears its own sound, plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 10.0 \(\mathrm{Hz}\) ? (Hint: Break this problem into two parts, first with the bat as the source and the wall as the listener and then with the wall as the source and the bat as the listener.)

A stecl wire 4.00 \(\mathrm{m}\) long has a mass of 0.0600 \(\mathrm{kg}\) and is stretched with a tension of 1000 \(\mathrm{N}\) . What is the speed of prop- agation of a transverse wave on the wire?

\(\bullet\) The siren of a fire engine that is driving northward at 30.0 \(\mathrm{m} / \mathrm{s}\) emits a sound of frequency 2000 \(\mathrm{Hz}\) . A truck in front of this fire engine is moving northward at 20.0 \(\mathrm{m} / \mathrm{s}\) . (a) What is the frequency of the siren's sound that the fire engine's driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?
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