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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

The tensile stress in a thick copper bar is \(99.5 \%\) of its elastic breaking point of \(13.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .\) If a \(500-\mathrm{Hz}\) sound wave is transmitted through the material, (a) what displacement amplitude will cause the bar to break? (b) What is the maximum speed of the elements of copper at this moment? (c) What is the sound intensity in the bar?

A tuning fork vibrating at \(512 \mathrm{Hz}\) falls from rest and accelerates at \(9.80 \mathrm{m} / \mathrm{s}^{2} .\) How far below the point of \(\mathrm{re}-\) lease is the tuning fork when waves of frequency \(485 \mathrm{Hz}\) reach the release point? Take the speed of sound in air to be \(340 \mathrm{m} / \mathrm{s}\)

With particular experimental methods, it is possible to produce and observe in a long thin rod both a longitudinal wave and a transverse wave whose speed depends primarily on tension in the rod. The speed of the longitudinal wave is determined by the Young's modulus and the density of the material as \(\sqrt{Y / \rho} .\) The transverse wave can be modeled as a wave in a stretched string. A particular metal rod is \(150 \mathrm{cm}\) long and has a radius of \(0.200 \mathrm{cm}\) and a mass of \(50.9 \mathrm{g} .\) Young's modulus for the material is \(6.80 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .\) What must the tension in the rod be if the ratio of the speed of longitudinal waves to the speed of transverse waves is \(8.00 ?\)

A train is moving parallel to a highway with a constant speed of \(20.0 \mathrm{m} / \mathrm{s} .\) A car is traveling in the same direction as the train with a speed of \(40.0 \mathrm{m} / \mathrm{s} .\) The car horn sounds at a frequency of \(510 \mathrm{Hz},\) and the train whistle sounds at a frequency of \(320 \mathrm{Hz}\). (a) When the car is behind the train, what frequency does an occupant of the car observe for the train whistle? (b) After the car passes and is in front of the train, what frequency does a train passenger observe for the car horn?

A siren mounted on the roof of a firehouse emits sound at a frequency of \(900 \mathrm{Hz}\). A steady wind is blowing with a speed of \(15.0 \mathrm{m} / \mathrm{s} .\) Taking the speed of sound in calm air to be \(343 \mathrm{m} / \mathrm{s}\), find the wavelength of the sound (a) upwind of the siren and (b) downwind of the siren. Firefighters are approaching the siren from various directions at \(15.0 \mathrm{m} / \mathrm{s} .\) What frequency does a firefighter hear \((\mathrm{c})\) if he or she is approaching from an upwind position, so that he or she is moving in the direction in which the wind is blowing? (d) if he or she is approaching from a downwind position and moving against the wind?
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