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 55b

Chapter 15, Problem 55b

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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The given equation for the standing wave is \( D = 2.4 \sin(0.60x) \cos(42t) \). This represents a standing wave formed by the superposition of two traveling waves. To analyze the component waves, we need to rewrite the equation in terms of traveling waves using the trigonometric identity \( \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \).

Using the identity, rewrite the equation as \( D = 1.2[\sin(0.60x + 42t) + \sin(0.60x - 42t)] \). This shows that the standing wave is the result of two traveling waves with equal amplitudes of 1.2 cm, traveling in opposite directions.

The amplitude of each component wave is 1.2 cm, as seen from the rewritten equation. This is half the amplitude of the standing wave (2.4 cm).

The angular frequency \( \omega \) of the component waves is given as 42 rad/s. To find the frequency \( f \), use the relation \( f = \frac{\omega}{2\pi} \). Substitute \( \omega = 42 \) rad/s to calculate \( f \).

The wave number \( k \) is given as 0.60 rad/cm. The speed \( v \) of the component waves can be found using the wave equation \( v = \frac{\omega}{k} \). Substitute \( \omega = 42 \) rad/s and \( k = 0.60 \) rad/cm to calculate \( v \).

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Key Concepts

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

Standing Waves

Standing waves are formed by the interference of two waves traveling in opposite directions with the same frequency and amplitude. They are characterized by nodes, where there is no displacement, and antinodes, where the displacement is maximum. The equation for a standing wave typically combines sine and cosine functions, representing the spatial and temporal components of the wave.

Amplitude

Amplitude refers to the maximum displacement of points on a wave from their rest position. In the context of the given wave equation, the amplitude can be identified as the coefficient of the sine function, which indicates how far the wave oscillates from its equilibrium position. For the standing wave described, the amplitude is 2.4 cm.

Frequency and Wave Speed

Frequency is the number of cycles of a wave that occur in a unit of time, typically measured in hertz (Hz). In the wave equation, the term associated with time (cos(42t)) indicates the angular frequency, which can be converted to frequency using the formula f = ω/(2π). Wave speed is determined by the product of frequency and wavelength, and can be calculated once the frequency and the spatial component of the wave are understood.

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Related Practice

Textbook Question

A guitar string is 91 cm long and has a mass of 3.2 g. The vibrating portion of the string from the bridge to the support post is ℓ = 64cm and the string is under a tension of 520 N. What are the frequencies of the fundamental and first two overtones?

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Textbook Question

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?

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Textbook Question

One end of a horizontal string is attached to a small-amplitude mechanical 60.0-Hz oscillator. The string’s mass per unit length is 3.9 x 10⁻ ⁴ kg/m. The string passes over a pulley, a distance ℓ = 1.50 m away, and weights are hung from this end, Fig. 15–38. What mass m must be hung from this end of the string to produce five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

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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

The displacement of a standing wave on a string is given by D = 2.4 sin ( 0.60x ) cos (42t) , where x and D are in centimeters and t is in seconds. What is the distance (cm) between nodes?

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