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Chapter 17: Problem 24

Suppose that the sound level of a conversation is initially at an angry \(70 \mathrm{~dB}\) and then drops to a soothing \(50 \mathrm{~dB}\). Assuming that the frequency of the sound is \(500 \mathrm{~Hz}\), determine the (a) initial and (b) final sound intensities and the (c) initial and (d) final sound wave amplitudes.

Short Answer

Expert verified
The initial intensity is \(1 \times 10^{-5} \mathrm{~W/m^2}\) and the final intensity is \(1 \times 10^{-7} \mathrm{~W/m^2}\). Amplitudes depend on the medium and are \(A_1 = \sqrt{\frac{1 \times 10^{-5}}{k}}\), \(A_2 = \sqrt{\frac{1 \times 10^{-7}}{k}}\).

Step by step solution

01

Understand the Formula for Sound Level

The sound level \( L \) in decibels (dB) is given by the formula: \[L = 10 \log_{10}\left( \frac{I}{I_0} \right) \]where \( I \) is the intensity in \( W/m^2 \) and \( I_0 = 1 \times 10^{-12} \mathrm{~W/m^2} \) is the reference intensity.
02

Calculate Initial Sound Intensity

Substitute \( L = 70 \mathrm{~dB} \) into the formula to find the initial intensity \( I_1 \). Rearrange to get:\[70 = 10 \log_{10}\left( \frac{I_1}{1 \times 10^{-12}} \right)\]\[\log_{10}\left( \frac{I_1}{1 \times 10^{-12}} \right) = 7\]\[I_1 = 10^7 \times 10^{-12} = 1 \times 10^{-5} \mathrm{~W/m^2}\]
03

Calculate Final Sound Intensity

Substitute \( L = 50 \mathrm{~dB} \) into the formula to find the final intensity \( I_2 \). Rearrange to get:\[50 = 10 \log_{10}\left( \frac{I_2}{1 \times 10^{-12}} \right)\]\[\log_{10}\left( \frac{I_2}{1 \times 10^{-12}} \right) = 5\]\[I_2 = 10^5 \times 10^{-12} = 1 \times 10^{-7} \mathrm{~W/m^2}\]
04

Understand Amplitude Relation to Intensity

The intensity \( I \) of a sound wave is related to the amplitude \( A \) of the wave by the formula: \[I = kA^2\]where \( k \) is a constant depending on the medium and frequency.
05

Calculate Initial Sound Amplitude

Assume \( I_1 = kA_1^2 \), rearranging gives:\[A_1 = \sqrt{\frac{I_1}{k}}\]Substitute \( I_1 = 1 \times 10^{-5} \mathrm{~W/m^2} \):\[A_1 = \sqrt{\frac{1 \times 10^{-5}}{k}}\]
06

Calculate Final Sound Amplitude

Assume \( I_2 = kA_2^2 \), rearranging gives:\[A_2 = \sqrt{\frac{I_2}{k}}\]Substitute \( I_2 = 1 \times 10^{-7} \mathrm{~W/m^2} \):\[A_2 = \sqrt{\frac{1 \times 10^{-7}}{k}}\]

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Key Concepts

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

Decibel Level
The decibel (dB) level is a unit of measurement for sound intensity, which is a measure of how powerful or strong a sound is.
When you hear terms like "70 dB" or "50 dB," it refers to the loudness of a sound.

Decibels use a logarithmic scale because the human ear perceives sound intensity differently at varying loudness levels.
  • A 70 dB sound is quite louder than 50 dB, even though numerically the difference seems small.
  • This is because the decibel scale multiplies rather than adds intensity.
  • Every 10-decibel increase doubles perceived loudness.
The intensity of sound is related logarithmically to its intensity in watts per square meter \[ L = 10 \log_{10}\left( \frac{I}{I_0} \right) \]where \( I_0 \) is the reference intensity. This makes it an excellent way to express various sound levels, such as whispering or shouting, in a way that is easily comprehensible.
This understanding is important for appreciating changes in sound levels, such as in the exercise where the sound intensity drops from 70 dB (angry conversation) to 50 dB (soothing conversation).
Sound Amplitude
Sound amplitude is a critical component of sound waves, contributing to the perception of loudness. It refers to the height of the sound wave's peaks and directly impacts the sound wave's energy.
Greater amplitude means a louder sound, but it's essential to note that sound amplitude is not linearly related to loudness perception, rather, it's an essential factor in the sound's intensity.
  • The relationship between amplitude \(A\) and intensity \(I\) is described by the equation: \[ I = k A^2 \]
  • This equation indicates that sound intensity is proportional to the square of the amplitude.
In practical terms, as the amplitude increases, the energy carried by the wave increases, resulting in increased intensity.
For example, if the amplitude of a 70 dB sound is represented as \(A_1\), reducing sound intensity to 50 dB will result in a different amplitude \(A_2\), calculated using the known intensities and assuming the constant \(k\) remains unchanged. This showcases the direct influence of amplitude on how powerful the sound is perceived.
Frequency of Sound Waves
Frequency of a sound wave refers to how often the sound wave cycles per second and it's measured in hertz (Hz). It determines the pitch of the sound and can drastically affect how we perceive it.
For instance, in the exercise, the frequency is given as 500 Hz, which is quite common for middle C or the easiest range for human ears to hear and recognize.

Sound frequency is not directly related to sound intensity or amplitude, but it's essential in understanding the nature of sound. Here鈥檚 why frequency is crucial:
  • Higher frequencies result in higher pitches, whereas lower frequencies produce deeper sounds.
  • Frequency affects how different sounds are perceived even at the same amplitude and intensity.
  • Recognizing frequency is important when analyzing sound for different tonal qualities, such as in music or language.
This is important in various real-world applications, like in settings where high fidelity sound reproduction is essential, ensuring that sound is clear and distinct. Understanding frequency can also be useful for tuning instruments and audio equipment for the desired sound output.

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Most popular questions from this chapter

A girl is sitting near the open window of a train that is moving at a velocity of \(10.00 \mathrm{~m} / \mathrm{s}\) to the east. The girl's uncle stands near the tracks and watches the train move away.The locomotive whistle emits sound at frequency \(500.0 \mathrm{~Hz}\). The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? A wind begins to blow from the east at \(10.00 \mathrm{~m} / \mathrm{s} .\) (c) What frequency does the uncle now hear? (d) What frequency does the girl now hear?

On July \(10,1996,\) a granite block broke away from a wall in Yosemite Valley and, as it began to slide down the wall, was launched into projectile motion. Seismic waves produced by its impact with the ground triggered seismographs as far away as \(200 \mathrm{~km} .\) Later measurements indicated that the block had a mass between \(7.3 \times 10^{7} \mathrm{~kg}\) and \(1.7 \times 10^{8} \mathrm{~kg}\) and that it landed \(500 \mathrm{~m}\) vertically below the launch point and \(30 \mathrm{~m}\) horizontally from it. (The launch angle is not known.) (a) Estimate the block's kinetic energy just before it landed. Consider two types of seismic waves that spread from the impact point - a hemispherical body wave traveled through the ground in an expanding hemisphere and a cylindrical surface wave traveled along the ground in an expanding shallow vertical cylinder (Fig. 17-49). Assume that the impact lasted \(0.50 \mathrm{~s}\), the vertical cylinder had a depth \(d\) of \(5.0 \mathrm{~m},\) and each wave type received \(20 \%\) of the energy the block had just before impact. Neglecting any mechanical energy loss the waves experienced as they traveled, determine the intensities of (b) the body wave and (c) the surface wave when they reached a seismograph \(200 \mathrm{~km}\) away. (d) On the basis of these results, which wave is more easily detected on a distant seismograph?

Approximately a third of people with normal hearing have ears that continuously emit a low-intensity sound outward through the ear canal. A person with such spontaneous otoacoustic emission is rarely aware of the sound, except perhaps in a noisefree environment, but occasionally the emission is loud enough to be heard by someone else nearby. In one observation, the sound wave had a frequency of \(1665 \mathrm{~Hz}\) and a pressure amplitude of \(1.13 \times 10^{-3} \mathrm{~Pa}\). What were (a) the displacement amplitude and (b) the intensity of the wave emitted by the ear?

A person on a railroad car blows a trumpet note at \(440 \mathrm{~Hz}\). The car is moving toward a wall at \(20.0 \mathrm{~m} / \mathrm{s}\). Find the sound frequency (a) at the wall and (b) reflected back to the trumpeter.

In pipe \(A\), the ratio of a particular harmonic frequency to the next lower harmonic frequency is \(1.2 .\) In pipe \(B,\) the ratio of a particular harmonic frequency to the next lower harmonic frequency is 1.4. How many open ends are in (a) pipe \(A\) and (b) pipe \(B ?\)
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