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Ch 16: Traveling Waves
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
Chapter 16, Problem 65

A 1000 Hz sound wave traveling through 20°C air causes the pressure to oscillate around atmospheric pressure by ±0.050%. What is the maximum speed of an oscillating air molecule? Give your answer in mm/s.

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Step 1: Understand the relationship between the maximum speed of an oscillating air molecule and the amplitude of the pressure oscillation. The maximum speed of the molecule is given by the formula: vmax = ωA, where ω is the angular frequency and A is the amplitude of the displacement.
Step 2: Calculate the angular frequency ω using the formula ω = 2πf, where f is the frequency of the sound wave. Substitute f = 1000 Hz into the formula.
Step 3: Relate the pressure amplitude to the displacement amplitude A. The pressure amplitude is given as ±0.050% of atmospheric pressure. Atmospheric pressure at sea level is approximately 1.013 × 105 Pa. Calculate the pressure amplitude Pmax as 0.0005 × 1.013 × 105 Pa.
Step 4: Use the relationship between pressure amplitude and displacement amplitude: Pmax = ρvsoundωA, where ρ is the density of air, vsound is the speed of sound in air, and ω is the angular frequency. At 20°C, the density of air is approximately 1.204 kg/m³, and the speed of sound is approximately 343 m/s. Rearrange the formula to solve for A: A = Pmax / (ρvsoundω).
Step 5: Substitute the values for Pmax, ρ, vsound, and ω into the formula for A. Then use the formula vmax = ωA to calculate the maximum speed of the oscillating air molecule. Convert the result to mm/s by multiplying by 1000.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sound Wave Properties

Sound waves are longitudinal waves that propagate through a medium, such as air, by causing oscillations in the pressure and density of the medium. The frequency of a sound wave, measured in hertz (Hz), indicates how many oscillations occur per second. In this case, a 1000 Hz sound wave means that the pressure oscillates 1000 times each second.
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Pressure Oscillation

The pressure oscillation in a sound wave refers to the periodic increase and decrease in pressure around the atmospheric pressure level. In this scenario, the pressure oscillates by ±0.050%, which indicates the extent of the fluctuation. This oscillation is crucial for understanding how sound energy is transmitted through the air and affects the motion of air molecules.
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Molecular Speed in Sound Waves

The maximum speed of an oscillating air molecule in a sound wave can be derived from the wave's properties, including its frequency and the amplitude of pressure oscillation. This speed is related to how far the molecules move from their equilibrium position during the oscillation. The formula for calculating this speed involves the wave's frequency and the displacement amplitude, allowing us to express the oscillation in terms of linear motion.
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Related Practice
Textbook Question

A battery-powered siren emits 0.50 W of sound power at 1000 Hz. It is dropped from 100 m directly over your head on a 20°C day. 4.0 s after it is released, what are (a) the frequency and (b) the sound intensity level you hear?

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Textbook Question

FIGURE P16.57 shows a snapshot graph of a wave traveling to the right along a string at 45 m/s. At this instant, what is the velocity of points 1, 2, and 3 on the string?

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Textbook Question

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

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Textbook Question

LASIK eye surgery uses pulses of laser light to shave off tissue from the cornea, reshaping it. A typical LASIK laser emits a 1.0-mm-diameter laser beam with a wavelength of 193 nm. Each laser pulse lasts 15 ns and contains 1.0 mJ of light energy. During the very brief time of the pulse, what is the intensity of the light wave?

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Textbook Question

The string in FIGURE P16.59 has linear density μ. Find an expression in terms of M, μ, and θ for the speed of waves on the string.

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Textbook Question

Some modern optical devices are made with glass whose index of refraction changes with distance from the front surface. FIGURE P16.72 shows the index of refraction as a function of the distance into a slab of glass of thickness L. The index of refraction increases linearly from n1 at the front surface to n₂ at the rear surface. Evaluate your expression for a 1.0-cm-thick piece of glass for which n1 = 1.50 and n2 = 1.60.