A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed vv, frequency ff, amplitude AA, and wavelength λ\lambda. What is the amplitude of the motion at the points located at (i) x=λ/2x = λ/2, (ii) x=λ/4x = λ/4, and (iii) x=λ/8x = λ/8, from the left-hand end of the string?

Ch 15: Mechanical Waves

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 15: Mechanical Waves

Problem 46b

Chapter 15, Problem 46b

A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v$ , frequency $f$ , amplitude $A$ , and wavelength $λ$ . What is the amplitude of the motion at the points located at (i) $x = λ / 2$ , (ii) $x = λ / 4$ , and (iii) $x = λ / 8$ , from the left-hand end of the string?

Verified step by step guidance

Understand that the fundamental mode of a vibrating string is a standing wave pattern, which is formed by the superposition of two traveling waves moving in opposite directions.

In a standing wave, nodes are points of zero amplitude, and antinodes are points of maximum amplitude. The distance between two consecutive nodes or antinodes is λ/2.

For a point located at x = λ/2, it coincides with a node, where the amplitude of motion is zero.

For a point located at x = λ/4, it coincides with an antinode, where the amplitude of motion is maximum, equal to the amplitude A of the traveling waves.

For a point located at x = λ/8, it is between a node and an antinode. The amplitude of motion at this point can be determined using the equation for a standing wave: y(x, t) = 2A sin(kx) cos(ωt), where k = 2π/λ. The amplitude at x = λ/8 is 2A sin(π/4).

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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 traveling waves moving in opposite directions with the same frequency and amplitude. In a string fixed at both ends, nodes (points of zero amplitude) and antinodes (points of maximum amplitude) are created. Understanding the distribution of nodes and antinodes is crucial for determining the amplitude at specific points on the string.

Fundamental Mode

The fundamental mode of vibration is the simplest form of standing wave on a string, characterized by having one antinode in the center and nodes at the ends. The wavelength of the fundamental mode is twice the length of the string, and it determines the positions of nodes and antinodes, which are essential for calculating amplitudes at specific points.

Wave Amplitude

Wave amplitude refers to the maximum displacement of points on the wave from their equilibrium position. In standing waves, amplitude varies along the string, reaching maximum at antinodes and zero at nodes. To find the amplitude at specific points, one must understand the spatial relationship between these points and the nodes/antinodes in the wave pattern.

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Textbook Question

A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v$ , frequency $f$ , amplitude $A$ , and wavelength $\lambda$ . Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) $x = λ/2$ , (ii) $x = λ/4$ , and (iii) $x = λ/8$ , from the left-hand end of the string.

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Textbook Question

One string of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.765 m? (Assume that the break-ing stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

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Textbook Question

A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v$ , frequency $f$ , amplitude $A$ , and wavelength $\lambda$ . How much time does it take the string to go from its largest upward displacement to its largest downward displacement at the points located at (i) $x = λ/2$ , (ii) $x = λ/4$ , and (iii) $x = λ/8$ , from the left-hand end of the string.

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Textbook Question

The wave function of a standing wave is $y(x,t)=4.44$\text{ mm}\]\sin$[(32.5$\text{ rad/m}$)x]$\sin$[(754$\text{rad/s}$)t]$ . For the two traveling waves that make up this standing wave, find the wave speed.

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