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Chapter 20: Problem 33

What are the sound intensity levels for sound waves of intensity (a) \(5.0 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2}\) and (b) \(5.0 \times 10^{-2} \mathrm{W} / \mathrm{m}^{2} ?\)

Short Answer

Expert verified
The sound intensity levels for the given intensities are: (a) ~80 dB and (b) ~130 dB. Please use a log table or calculator to get the detailed value.

Step by step solution

01

Identify the given intensities

The exercise gives us two different sound intensities which are: a) \(5.0 \times 10^{-8} W/m^2\) and b) \(5.0 \times 10^{-2} W/m^2\).
02

Sound Intensity Level Formula

The formula needed to calculate the sound intensity level (L) is \( L = 10 \cdot log10(I / I_0) \). In this formula \( I_0 \) is the reference intensity which is \( 10^{-12} W/m^2 \). Understand that sound intensity level is expressed in decibels (dB).
03

Substitute the values and solve for sound intensity level (a)

For the first intensity, substitute the given value into the formula: \( L = 10 \cdot log10((5.0 \times 10^{-8}) / 10^{-12}) \). Simplify the formula using properties of logarithms to find L.
04

Substitute the values and solve for sound intensity level (b)

For the second intensity, substitute the given value into the formula: \( L = 10 \cdot log10((5.0 \times 10^{-2}) / 10^{-12}) \). Simplify the formula using properties of logarithms to calculate the sound level L.

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Key Concepts

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

Decibels
Decibels are a unit of measurement used to express the intensity of sound. This scale is logarithmic, not linear. This means that each increase of 10 decibels represents a tenfold increase in sound intensity. For example, a sound at 20 decibels is not twice as loud as a sound at 10 decibels; it is actually ten times louder.
  • The decibel scale is relative, based on a reference intensity level which is typically set at the threshold of human hearing, or \(10^{-12} \mathrm{W/m}^2\).
  • This scale allows for a wide range of sound intensities to be represented in a manageable way.
  • Understanding decibels is essential for measuring sound intensity levels effectively.
By using decibels, we can describe very soft sounds like a whisper (around 20 dB) or extremely loud sounds like a jet engine at 100 feet (around 130 dB). This scale helps not only in measuring sound in physics but also in understanding the potential effects of sound on human hearing.
Logarithms
Logarithms are mathematical operations that help us work with exponential functions. They are particularly useful in sound calculations due to the logarithmic nature of human hearing. A logarithm answers the question: What power do we need to raise a certain number (called the base) to get another number?
  • For example, in the case of sound intensity, our base is typically 10, because we use \( \log_{10} \).
  • This is why the sound intensity level formula includes \(10 \cdot \log_{10}(I/I_0)\).
  • By applying the properties of logarithms, we transform a wide range of sound intensities into a comprehensible scale.
    • This transformation converts multiplication into addition, simplifying the understanding of how sound intensities combine.
Logarithms help simplify calculations where numbers can get extremely large or small, which is often the case in physics-related fields like acoustics.
Sound Waves
Sound waves are vibrations that travel through a medium, such as air, water, or solids. These waves can be described in terms of their frequency, wavelength, and amplitude, among other properties. Here are some key elements of sound waves:
  • Frequency: Determines the pitch of the sound; higher frequencies correspond to higher pitches.
  • Amplitude: Related to the loudness of the sound; larger amplitudes mean louder sounds.
  • Speed: Depends on the medium through which the sound is traveling. Sound travels fastest through solids and slowest through gases.
  • Reflection and Refraction: Sound waves can bounce off surfaces or bend when passing through different media, affecting how we perceive sound.
When measuring sound intensity, we are interested in how much sound energy is passing through a unit area. This is why terms like intensity and decibels are vital to understanding sound waves.
Physics Concepts
Physics is the study of the natural world and involves various principles that explain phenomena like sound. Sound is a form of energy that is dependent on varying parameters, including frequency, amplitude, and intensity. Several physics concepts help us understand sound better:
  • The nature of sound as a mechanical wave means it requires a medium to travel.
  • The inverse square law describes how sound intensity decreases with increasing distance from the source. This means the further you are from the sound source, the less intense it becomes.
  • Constructive and destructive interference occurs when multiple sound waves overlap, affecting the resulting sound intensity and quality.
Understanding these basic physics concepts allows students to grasp how sound behaves, interacts, and varies under different conditions. This is crucial for applications not only in acoustics and music but also in engineering and technology, where controlling sound is often necessary.

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Most popular questions from this chapter

A string that is under \(50.0 \mathrm{N}\) of tension has linear density \(5.0 \mathrm{g} / \mathrm{m} .\) A sinusoidal wave with amplitude \(3.0 \mathrm{cm}\) and wavelength \(2.0 \mathrm{m}\) travels along the string. What is the maximum speed of a particle on the string?

A wave travels with speed \(200 \mathrm{m} / \mathrm{s}\). Its wave number is \(1.5 \mathrm{rad} / \mathrm{m} .\) What are its (a) wavelength and (b) frequency?

You are cruising to Jupiter at the posted speed limit of \(0.1 c\) when suddenly a daredevil passes you, going in the same direction, at \(0.3 c .\) At what wavelength does your rocket cruiser's light detector "see" his red tail lights? Is this wavelength ultraviolet, visible, or infrared? Use 650 \(nm\) for the wavelength of red light.

The sound intensity from a jack hammer breaking concrete is \(2.0 \mathrm{W} / \mathrm{m}^{2}\) at a distance of \(2.0 \mathrm{m}\) from the point of impact. This is sufficiently loud to cause permanent hearing damage if the operator doesn't wear car protection. What is the sound intensity for a person watching from 50 m away?

Lasers can be used to drill or cut material. One such laser generates a series of high-intensity pulses rather than a continuous beam of light. Each pulse contains \(500 \mathrm{mJ}\) of energy and lasts 10 \(ns\). The laser fires 10 such pulses per second. a. What is the peak power of the laser light? The peak power is the power output during one of the 10 ns pulses. b. What is the average power output of the laser? The average power is the total energy delivered per second. c. Alens focuses the laser beam to a \(10-\mu\) m-diameter circle on the target. During a pulse, what is the light intensity on the target? d. The intensity of sunlight at midday is about \(1100 \mathrm{W} / \mathrm{m}^{2} .\) What is the ratio of the laser intensity on the target to the intensity of the midday sun?
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