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 63

Chapter 15, Problem 63

When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave?

Verified step by step guidance

Identify the relationship between the frequency, wavelength, and wave speed. The formula to use is:

v = f λ

, where

v

is the wave speed,

f

is the frequency, and

λ

is the wavelength.

Recognize that the standing wave in the tub corresponds to the fundamental mode of oscillation. In this mode, the wavelength

λ

is twice the width of the tub. Thus, calculate the wavelength as:

λ = 2 \times 0.45 = 0.90 m

Substitute the given frequency

f = 0.85 Hz

and the calculated wavelength

λ = 0.90 m

into the wave speed formula:

v = f λ

Perform the multiplication to find the wave speed:

v = 0.85 \times 0.90

. This will give the speed of the water wave in meters per second.

Ensure the units are consistent throughout the calculation (meters for wavelength, Hz for frequency, and m/s for speed) and verify the result for accuracy.

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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. It can be calculated using the formula v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. In the context of standing waves, understanding wave speed is crucial for determining how quickly the wave propagates through the medium, which in this case is water.

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere with each other. This results in specific points called nodes, where there is no movement, and antinodes, where the movement is maximum. In the given scenario, the water in the tub creates a standing wave pattern due to the sloshing motion at a specific frequency.

Frequency and Wavelength Relationship

The relationship between frequency and wavelength is fundamental in wave mechanics. For waves, the frequency (f) is inversely related to the wavelength (λ) when the wave speed (v) is constant, expressed as v = fλ. In this problem, knowing the frequency allows us to find the wave speed, provided we can determine the wavelength of the standing wave in the tub.

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Related Practice

Textbook Question

The displacement of a standing wave on a string is given by D = 2.4sin(0.60x)cos(42t), where x and D are in centimeters and t is in seconds. Give the amplitude, frequency, and speed of each of the component waves.

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Textbook Question

A guitar string is supposed to vibrate at 247 Hz, but is measured to actually vibrate at 262 Hz. By what percentage should the tension in the string be changed to get the frequency to the correct value?

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Textbook Question

(II) For any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle θ_iM corresponds to an angle of refraction equal to 90°. If θᵢ > θ_iM, all the wave is reflected at the boundary and none is refracted, because this would correspond to sin θᵣ > 1 (where is the angle θᵣ of refraction), which is impossible.

(a) Find a formula for θ_iM using the law of refraction, Eq. 15–19.

Textbook Question

A particular violin string plays at a frequency of 294 Hz. If the tension is increased 22%, what will the new frequency be?

Textbook Question

A longitudinal earthquake wave strikes a boundary between two types of rock at a 41° angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus (Section 15–2)is the same for both types of rock, determine the angle of refraction.

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