Ch. 15 - Wave Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 15 - Wave Motion

Problem 29c

Chapter 15, Problem 29c

A 572-Hz longitudinal wave in air has a speed of 345 m/s. At a particular instant, what is the phase difference (in degrees) between two points 4.4 cm apart?

Verified step by step guidance

Determine the wavelength of the wave using the formula:

λ = \frac{v}{f}

, where

v

is the speed of the wave (345 m/s) and

f

is the frequency (572 Hz).

Convert the distance between the two points (4.4 cm) into meters by dividing by 100, since 1 cm = 0.01 m.

Calculate the fraction of the wavelength that corresponds to the distance between the two points using the formula:

fraction = \frac{d}{λ}

, where

d

is the distance (in meters) and

λ

is the wavelength.

Convert the fraction of the wavelength into a phase difference in degrees using the formula:

phase = \frac{fraction \times 360}{1}

, since one full wavelength corresponds to 360 degrees.

Express the phase difference as the final result, ensuring it is within the range of 0 to 360 degrees (if necessary, subtract multiples of 360 to bring it into this range).

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Key Concepts

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

Wave Speed

Wave speed is the distance a wave travels per unit of time. In this case, the speed of the longitudinal wave in air is given as 345 m/s. This speed is crucial for determining how quickly the wave propagates through the medium and is essential for calculating the phase difference between two points.

Wavelength

Wavelength is the distance between successive points of similar phase in a wave, such as crest to crest or trough to trough. It can be calculated using the formula λ = v/f, where v is the wave speed and f is the frequency. For the given wave, the wavelength can be determined, which is necessary for finding the phase difference over a specified distance.

Phase Difference

Phase difference refers to the difference in the phase of two points in a wave, typically measured in degrees or radians. It indicates how far one point is ahead or behind another in the wave cycle. The phase difference can be calculated using the formula Δφ = (2π/λ) * Δx, where Δx is the distance between the two points, allowing us to find the phase difference for the specified distance of 4.4 cm.

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Related Practice

Textbook Question

A 572-Hz longitudinal wave in air has a speed of 345 m/s. How much time is required for the phase to change by 90° at a given point in space?

Textbook Question

A transverse wave with a frequency of 220 Hz and a wavelength of 10.0 cm is traveling along a cord. The maximum speed of particles on the cord is 0.10 the wave speed. What is the amplitude of the wave?

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Textbook Question

A 572-Hz longitudinal wave in air has a speed of 345 m/s. What is the wavelength?

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Textbook Question

Write the equation for the wave in Problem 29 traveling to the right, if its amplitude is 0.020 cm, and D = - 0.020 cm, at t = 0 and x = 0.

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Textbook Question

Suppose two linear waves of equal amplitude and frequency have a phase difference ϕ as they travel in the same medium. They can be represented by: D₁ = A sin (kx - ωt); D₂ = A sin ( kx - ωt + ϕ). What is the amplitude of this resultant wave? Is the wave purely sinusoidal, or not?

Textbook Question

A transverse wave on a cord is given by D(x,t) = 0.12 sin (3.0x - 15.0t), where D and x are in meters and t is in seconds. At t = 0.20s, what are the displacement, velocity, and acceleration of a point on the cord where x = 0.60 m?