/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 60 The sound from a trumpet radiate... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 60

The sound from a trumpet radiates uniformly in all directions in 20\(^\circ\)C air. At a distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB. The frequency is 587 Hz. (a) What is the pressure amplitude at this distance? (b) What is the displacement amplitude? (c) At what distance is the sound intensity level 30.0 dB?

Short Answer

Expert verified
(a) Pressure amplitude is about 0.063 Pa, (b) Displacement amplitude is about 3.2 碌m, and (c) new distance is around 28.3 m.

Step by step solution

01

Understand the Intensity Level

The intensity level in decibels (dB) is given by the formula: \[\beta = 10 \log_{10} \left(\frac{I}{I_0}\right)\]where \(I_0\) is the reference intensity, \(10^{-12} \text{ W/m}^2\). For 52.0 dB, calculate the intensity \(I\).
02

Calculate the Intensity (I)

Rearrange the formula to solve for \(I\):\[\frac{I}{I_0} = 10^{\frac{\beta}{10}}\]\(I = I_0 \times 10^{\frac{52.0}{10}}\)Compute \(I\).
03

Calculate the Pressure Amplitude

The pressure amplitude \(P_m\) can be calculated using:\[P_m = \sqrt{2I \rho v}\]where \(\rho\) is the air density \(1.21 \text{ kg/m}^3\) and \(v\) is the speed of sound (approximately \(343 \text{ m/s}\) at \(20^\circ \text{C}\)). Compute \(P_m\).
04

Calculate the Displacement Amplitude

Use the relationship between pressure amplitude and displacement amplitude:\[P_m = B k A_s\]where \(B\) is the bulk modulus of air (approximately \(1.42 \times 10^5 \text{ N/m}^2\)), \(k\) is the wave number \((\frac{2\pi f}{v})\). Rearrange to find displacement amplitude \(A_s\).
05

Intensity Level Change with Distance

For part (c), relate the change in intensity level to distance using the formula:\[\beta_1 - \beta_2 = 10 \log_{10} \left(\frac{r_2^2}{r_1^2}\right)\]Set \(\beta_1 = 52.0\, \text{dB}\),\(\beta_2 = 30.0\, \text{dB}\),\(r_1 = 5.00\, \text{m}\) and solve for \(r_2\).
06

Calculate the New Distance

Substitute the values from Step 5 into the formula to solve for \(r_2\):\[\frac{r_2^2}{r_1^2} = 10^{\frac{\beta_1 - \beta_2}{10}}\]Calculate \(r_2\) for the 30.0 dB intensity level.

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Key Concepts

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

Pressure Amplitude
Pressure amplitude is a crucial concept in understanding sound waves, especially in terms of how energy is distributed through a medium like air. Pressure amplitude, denoted by \( P_m \), gives us an idea of how much the pressure varies from the ambient pressure as the sound wave passes. It involves the root mean square (RMS) of the pressure variation. This value is significant because it helps determine how `loud` a sound is.
  • The pressure amplitude can be related to the intensity \( I \) of the sound wave using the formula \( P_m = \sqrt{2I \rho v} \).
  • Here, \( \rho \) symbolizes the density of air, typically around 1.21 kg/m鲁 at room temperature, and \( v \) is the speed of sound, approximately 343 m/s in 20掳C air.
This relationship indicates that pressure amplitude is not only a factor of sound intensity but also of the medium through which the sound is traveling. This context becomes key when analyzing how sound propagates through different environments.
Displacement Amplitude
The displacement amplitude, linked to a sound wave's other characteristics, describes the maximum distance that particles in the medium are displaced from their mean position as the wave passes.
It is denoted by \( A_s \) and is directly influenced by the pressure amplitude, which is again concerned with the energy carried in the wave.
  • The relationship between the pressure amplitude and displacement amplitude can be described by the equation \( P_m = B k A_s \).
  • In this context, \( B \) stands for the bulk modulus of the medium, approximately 1.42 脳 10\(^5\) N/m虏 for air, and \( k \) is the wave number (\( \frac{2\pi f}{v} \)).
  • \( f \) represents frequency and \( v \) is the velocity of the wave through the medium, which for sound in air is about 343 m/s at 20掳C.
Displacement amplitude plays a crucial role in understanding the mechanical waves as it gives insight into the actual movement and energy transfer in the medium.
Decibel Levels
Decibel (dB) levels are a useful way of describing sound intensity, particularly its loudness. The decibel scale is logarithmic, meaning each 10 dB increase represents a tenfold increase in intensity.
This makes it easier to handle the wide range of human hearing, from the quietest whisper to the loudest concert.
  • The formula \( \beta = 10 \log_{10} \left(\frac{I}{I_0}\right) \) helps calculate the intensity level, where \( I_0 \) is the reference intensity, set at \( 10^{-12} \text{ W/m}^2 \), which is the threshold of hearing.
  • For instance, if the sound has a level of 52.0 dB, the intensity \( I \) can be calculated and understood in relative terms based on \( I_0 \).
Understanding decibel levels is essential in acoustics because they give a quantitative basis for sound measurement, predicting how environments and distances affect perceived loudness.
Frequency
Frequency, a fundamental property of sound waves, indicates how many complete oscillations of a wave occur per second. It is measured in Hertz (Hz) and when dealing with sound, it is closely related to the pitch that is heard.
For example, the standard frequency of a pitch produced by a trumpet can be around 587 Hz, which is a common note in musical contexts.
  • The frequency \( f \) is a vital part of calculating the wave number \( k \), which appears in many equations connecting different sound wave properties, like \( k = \frac{2\pi f}{v} \).
  • Here, \( v \) is the speed of sound through the medium, typically 343 m/s at 20掳C.
Understanding frequency aids in exploring how sounds differ in pitch and how their related phenomena, such as resonance and harmony, occur. It is an essential parameter in sound analysis and acoustics engineering.

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Most popular questions from this chapter

A turntable 1.50 m in diameter rotates at 75 rpm. Two speakers, each giving off sound of wavelength 31.3 cm, are attached to the rim of the table at opposite ends of a diameter. A listener stands in front of the turntable. (a) What is the greatest beat frequency the listener will receive from this system? (b) Will the listener be able to distinguish individual beats?

A car alarm is emitting sound waves of frequency 520 Hz. You are on a motorcycle, traveling directly away from the parked car. How fast must you be traveling if you detect a frequency of 490 Hz?

You have a stopped pipe of adjustable length close to a taut 62.0-cm, 7.25-g wire under a tension of 4110 N. You want to adjust the length of the pipe so that, when it produces sound at its fundamental frequency, this sound causes the wire to vibrate in its second \(overtone\) with very large amplitude. How long should the pipe be?

Example 16.1 (Section 16.1) showed that for sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 \(\times\) 10\(^{-8}\) m produces a pressure amplitude of 3.0 \(\times\) 10\(^{-2}\) Pa. (a) What is the wavelength of these waves? (b) For 1000-Hz waves in air, what displacement amplitude would be needed for the pressure amplitude to be at the pain threshold, which is 30 Pa? (c) For what wavelength and frequency will waves with a displacement amplitude of 1.2 \(\times\) 10\(^{-8}\) m produce a pressure amplitude of 1.5 \(\times\) 10\(^{-3}\) Pa?

Two organ pipes, open at one end but closed at the other, are each 1.14 m long. One is now lengthened by 2.00 cm. Find the beat frequency that they produce when playing together in their fundamentals.
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