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Chapter 16: Problem 3

Consider a sound wave in air that has displacement amplitude 0.0200 mm. Calculate the pressure amplitude for frequencies of (a) 150 Hz; (b) 1500 Hz; (c) 15,000 Hz. In each case compare the result to the pain threshold, which is 30 Pa.

Short Answer

Expert verified
Pressure amplitudes: 7.77 Pa (150 Hz), 77.7 Pa (1500 Hz), 777 Pa (15,000 Hz). The 1500 Hz and 15,000 Hz frequencies exceed the pain threshold of 30 Pa.

Step by step solution

01

Understand the Relationship between Displacement Amplitude and Pressure Amplitude

The pressure amplitude \( P_0 \) in a sound wave is related to the displacement amplitude \( s \), the angular frequency \( \omega \), the medium density \( \rho \), and the speed of sound \( v \). The formula is given by \( P_0 = \rho v \omega s \).
02

Identify Constants and Quantities

We need the density of air \( \rho = 1.21\, \text{kg/m}^3 \) and the speed of sound in air \( v = 343\, \text{m/s} \). The displacement amplitude is given as \( s = 0.0200\, \text{mm} = 0.0200 \times 10^{-3}\, \text{m} \).
03

Calculate Angular Frequency for Each Frequency

Angular frequency is calculated using \( \omega = 2\pi f \). Compute \( \omega \) for each frequency:(a) For \( f = 150\, \text{Hz} \): \( \omega = 2\pi \times 150 \approx 942 \text{ rad/s} \).(b) For \( f = 1500 \text{ Hz} \): \( \omega = 2\pi \times 1500 \approx 9420 \text{ rad/s} \).(c) For \( f = 15000 \text{ Hz} \): \( \omega = 2\pi \times 15000 \approx 94200 \text{ rad/s} \).
04

Calculate Pressure Amplitude for Each Frequency

Use the formula \( P_0 = \rho v \omega s \) to compute pressure amplitude:(a) For \( 150 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 942 \times 0.0200 \times 10^{-3} \approx 7.77 \text{ Pa} \].(b) For \( 1500 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 9420 \times 0.0200 \times 10^{-3} \approx 77.7 \text{ Pa} \].(c) For \( 15000 \text{ Hz} \): \[ P_0 = 1.21 \times 343 \times 94200 \times 0.0200 \times 10^{-3} \approx 777 \text{ Pa} \].
05

Compare with Pain Threshold

The pain threshold is 30 Pa. Compare calculated pressure amplitudes:(a) For \( 150 \text{ Hz} \), \( 7.77 \text{ Pa} \) is below the pain threshold.(b) For \( 1500 \text{ Hz} \), \( 77.7 \text{ Pa} \) is above the pain threshold.(c) For \( 15000 \text{ Hz} \), \( 777 \text{ Pa} \) is well above the pain threshold.

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Key Concepts

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

Displacement Amplitude
When discussing sound waves, displacement amplitude refers to the maximum amount by which particles in the medium move from their rest position due to the wave. Imagine a tiny object floating on a wave. The distance it moves up and down measures the displacement amplitude.
This is important because it helps quantify how "strong" or "intense" the sound wave is in terms of physical movement. In the context of the original exercise, the displacement amplitude was given as 0.0200 mm or 0.0200 x 10^{-3} m.
  • Displacement amplitude is directly related to how much energy is delivered by the wave.
  • A larger displacement amplitude means the particles are further displaced, often resulting in a louder sound.
Understanding displacement amplitude is crucial for calculating the pressure amplitude, which involves additional characteristics of the medium and the wave.
Frequency and Angular Frequency
Frequency, represented by the letter \( f \), is the number of vibrations or cycles per second of a wave, measured in hertz (Hz). For example, a frequency of 150 Hz means that the wave cycles 150 times per second. Frequency is a fundamental property of waves affecting both the pitch of sound and the wave's energy.
Angular frequency \( \omega \) offers another way to describe how fast the wave oscillates, expressed in radians per second. The angular frequency is calculated using the formula \( \omega = 2\pi f \).
  • Angular frequency is a convenient format in physics, especially when working with trigonometric functions.
  • Both frequency and angular frequency help characterize the timing and periodicity of wave phenomena.
In the exercise, calculating \( \omega \) forms part of determining the wave's pressure amplitude, a direct measure of the force each cycle of the wave imposes.
Density of Air and Speed of Sound
The density of air \( \rho \) and speed of sound \( v \) are essential parameters affecting how sound waves travel. For most calculations involving air, the density is typically around 1.21 kg/m鲁, while the speed of sound is about 343 m/s.
These values are crucial: the combination of air density and speed influences the wave's pressure amplitude and energy transfer.
  • Density reflects how much matter is present in a given volume of air.
  • The speed of sound depends on various factors, such as air temperature, and represents how fast sound travels through the medium.
In the sound wave calculations, these constants determine how efficiently the wave energy is propagated through the air, impacting both the calculated displacement and pressure amplitudes.
Pain Threshold in Acoustics
In acoustics, the pain threshold is the level at which sound becomes physically uncomfortable to hear, typically around 30 Pa (pascals) for humans. This concept helps gauge the intensity of sound pressure that can be tolerated by the ear without causing damage or considerable discomfort.
Sound pressure levels above the pain threshold can cause harm and are considered dangerous. Comparisons against this value are critical, especially in environments where noise exposure risks are evaluated.
  • Understanding this threshold helps establish safe listening environments and set regulations for exposure to loud sounds.
  • It acts as a benchmark to assess the relative safety of sound pressure levels in surrounding environments.
In the exercise, calculated pressure amplitudes are compared against this standard to elucidate whether the sound waves at various frequencies pose a risk to hearing.

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Most popular questions from this chapter

The motors that drive airplane propellers are, in some cases, tuned by using beats. The whirring motor produces a sound wave having the same frequency as the propeller. (a) If one single-bladed propeller is turning at 575 rpm and you hear 2.0-Hz beats when you run the second propeller, what are the two possible frequencies (in rpm) of the second propeller? (b) Suppose you increase the speed of the second propeller slightly and find that the beat frequency changes to 2.1 Hz. In part (a), which of the two answers was the correct one for the frequency of the second single-bladed propeller? How do you know?

Horseshoe bats (genus \(Rhinolophus\)) emit sounds from their nostrils and then listen to the frequency of the sound reflected from their prey to determine the prey's speed. (The "horseshoe" that gives the bat its name is a depression around the nostrils that acts like a focusing mirror, so that the bat emits sound in a narrow beam like a flashlight.) A \(Rhinolophus\) flying at speed \(v_{bat}\) emits sound of frequency \(f_{bat}\); the sound it hears reflected from an insect flying toward it has a higher frequency \(f_{refl}\). (a) Show that the speed of the insect is $$vinsect = v\Bigg[\frac{f_{refl}(v - v_{bat}) - f_{bat}(v + v_{bat})}{f_{refl}(v - v_{bat}) + f_{bat}(v + v_{bat})}\Bigg] $$ where \(v\) is the speed of sound. (b) If \(f_{bat} =\) 80.7 kHz, \(f_{refl} =\) 83.5 kHz, and \(v_{bat} =\) 3.9 m/s, calculate the speed of the insect.

(a) \(\textbf{Whale communication.}\) Blue whales apparently communicate with each other using sound of frequency 17 Hz, which can be heard nearly 1000 km away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 m/s? (b) \(\textbf{Dolphin clicks.}\) One type of sound that dolphins emit is a sharp click of wavelength 1.5 cm in the ocean. What is the frequency of such clicks? (c) \(\textbf{Dog whistles.}\) One brand of dog whistles claims a frequency of 25 kHz for its product. What is the wavelength of this sound? (d) \(\textbf{Bats.}\) While bats emit a wide variety of sounds, one type emits pulses of sound having a frequency between 39 kHz and 78 kHz. What is the range of wavelengths of this sound? (e) \(\textbf{Sonograms.}\) Ultrasound is used to view the interior of the body, much as x rays are utilized. For sharp imagery, the wavelength of the sound should be around one-fourth (or less) the size of the objects to be viewed. Approximately what frequency of sound is needed to produce a clear image of a tumor that is 1.0 mm across if the speed of sound in the tissue is 1550 m/s?

A railroad train is traveling at 30.0 m/s in still air. The frequency of the note emitted by the train whistle is 352 Hz. What frequency is heard by a passenger on a train moving in the opposite direction to the first at 18.0 m/s and (a) approaching the first and (b) receding from the first?

The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?
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