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Chapter 17: Problem 22

The intensity of a sound wave at a fixed distance from a speaker vibrating at a frequency \(f\) is \(I\) (a) Determine the intensity if the frequency is increased to \(f^{\prime}\) while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to \(f / 2\) and the displacement amplitude is doubled. GRAPH CANT COPY

Short Answer

Expert verified
The intensity of the sound wave when the frequency is increased to \(f^{\prime}\) with constant displacement amplitude is given by \(I' = I \times \left( \frac{f^{\prime}}{f} \right)^2\). When the frequency is halved and the displacement amplitude is doubled, the intensity is twice the original intensity, i.e., \(I'' = 2I\).

Step by step solution

01

Determine intensity with increased frequency

When the frequency is increased to \(f'\), the intensity \(I'\) of the sound wave can be determined using the relationship \( I' 鈭� (f^{\prime})^2 \). Here we are maintaining a constant displacement, so the amplitude stays the same. To get the new intensity, we need to square the ratio of the new frequency to the old frequency, and then multiply that by the original intensity. Therefore, \( I' = I \times \left( \frac{f^{\prime}}{f} \right)^2 \).
02

Determine intensity with reduced frequency and increased amplitude

In the second situation, the frequency is reduced to \(\frac{f}{2}\) and the displacement amplitude is doubled. The intensity is proportional to the square of the frequency and the square of the amplitude A (I 鈭� f^2A^2). Thus the new intensity I'' can be calculated as follows: \( I'' = I \times \left( \frac{f/2}{f} \right) ^2 \times \left( 2 A \right)^2 = 2I \). The squaring of the doubled amplitude cancels the squaring of the halved frequency.

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Key Concepts

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

Understanding Frequency in Sound Waves
The concept of frequency in sound waves is foundational to understanding how sounds are perceived. In physics, frequency is the number of times a wave repeats itself within a second, measured in hertz (Hz). It directly impacts the pitch of the sound: higher frequencies correspond to higher pitches, while lower frequencies are associated with lower pitches.

For example, middle C on a piano resonates at approximately 261.6 Hz, meaning the sound wave repeats 261.6 times every second. When the frequency is increased, the waves are more tightly spaced, resulting in a higher pitched sound.
The Role of Amplitude in Sound Waves
Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In the context of sound waves, amplitude is essentially how far the medium (e.g., air) is displaced by the wave. This displacement is what our ears perceive as loudness; greater amplitudes are interpreted as louder sounds.

In daily life, amplitude is often experienced when adjusting the volume on audio devices. Increasing the volume increases the amplitude of the sound waves, causing a more significant displacement of air particles, and hence, a louder sound. Conversely, decreasing the volume reduces the amplitude and perceived loudness.
Intensity Proportional Relationships Between Frequency and Amplitude
Intensity in acoustics is a measure of the energy transmitted by the sound wave per unit of area, per unit of time. It is proportional to both the square of the frequency and the square of the amplitude of the wave, represented by the equation I 鈭� f^2A^2. This means that if you double the frequency while keeping the amplitude constant, the intensity of the sound wave will quadruple because the relationship is quadratic.

Similarly, if the frequency is halved and the amplitude is doubled, the changes counterbalance each other since halving the frequency would quarter the intensity, but doubling the amplitude would quadruple it, resulting in an unchanged overall intensity. Understanding these relationships is crucial in fields like audio engineering and acoustics to control and manipulate sound.

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Most popular questions from this chapter

On a Saturday morning, pickup trucks and sport utility vehicles carrying garbage to the town dump form a nearly steady procession on a country road, all traveling at \(19.7 \mathrm{m} / \mathrm{s} .\) From one direction, two trucks arrive at the dump every 3 min. A bicyclist is also traveling toward the dump, at \(4.47 \mathrm{m} / \mathrm{s}\). (a) With what frequency do the trucks pass him? (b) What If? A hill does not slow down the trucks, but makes the out-of-shape cyclist's speed drop to \(1.56 \mathrm{m} / \mathrm{s} .\) How often do noisy, smelly, inefficient, garbage-dripping, roadhogging trucks whiz past him now?

As a certain sound wave travels through the air, it produces pressure variations (above and below atmospheric pressure) given by \(\Delta P=1.27 \sin (\pi x-340 \pi t)\) in SI units. Find (a) the amplitude of the pressure variations, (b) the frequency, (c) the wavelength in air, and (d) the speed of the sound wave.

The speed of sound in air (in \(\mathrm{m} / \mathrm{s}\) ) depends on temperature according to the approximate expression $$v=331.5+0.607 T_{\mathrm{C}}$$ where \(T_{\mathrm{C}}\) is the Celsius temperature. In dry air the temperature decreases about \(1^{\circ} \mathrm{C}\) for every \(150 \mathrm{m}\) rise in altitude. (a) Assuming this change is constant up to an altitude of \(9000 \mathrm{m},\) how long will it take the sound from an airplane flying at 9000 m to reach the ground on a day when the ground temperature is \(30^{\circ} \mathrm{C} ?\) (b) What If? Compare this to the time interval required if the air were a constant \(30^{\circ} \mathrm{C}\). Which time interval is longer?

A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat's crew blows a loud horn. By the time the plane's sound detector perceives the horn's sound, the plane has traveled a distance equal to half its altitude above the ocean. If it takes the sound 2.00 s to reach the plane, determine (a) the speed of the plane and (b) its altitude. Take the speed of sound to be \(343 \mathrm{m} / \mathrm{s}\)

The intensity of a sound wave at a fixed distance from a speaker vibrating at \(1.00 \mathrm{kHz}\) is \(0.600 \mathrm{W} / \mathrm{m}^{2} .\) (a) Determine the intensity if the frequency is increased to \(2.50 \mathrm{kHz}\) while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to \(0.500 \mathrm{kHz}\) and the displacement amplitude is doubled.
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