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The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move). Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave: y(x, t) = = A sin(kx) sin(wt), where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wave number, w is the angular frequency of the wave, and t is time.

The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move). Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave: y(x, t) = = A sin(kx) sin(wt), where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wave number, w is the angular frequency of the wave, and t is time.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Part C What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string? Express the displacement in terms of A, x, k, and other constants; evaluate w. and substitute it in the expression for y (x, t). ► View Available Hint(s) y (x, 7) = Submit Part D VE ΑΣΦ X1, X2, X3 = At which three points x₁, x2, and x3 closest to x = 0 but with x > 0 will the displacement of the string y (x, t) be zero for all times? These are the first three nodal points. Submit ? Express the first three nonzero nodal points in terms of the wavelength X. List them in increasing order, separated by commas. You should enter only the factors that multiply X. Do not enter for each one. ► View Available Hint(s) ΠΑΠ ΑΣΦ that is, ? λ
Transcribed Image Text:Part C What is the displacement of the string as a function of x at time T/4, where T is the period of oscillation of the string? Express the displacement in terms of A, x, k, and other constants; evaluate w. and substitute it in the expression for y (x, t). ► View Available Hint(s) y (x, 7) = Submit Part D VE ΑΣΦ X1, X2, X3 = At which three points x₁, x2, and x3 closest to x = 0 but with x > 0 will the displacement of the string y (x, t) be zero for all times? These are the first three nodal points. Submit ? Express the first three nonzero nodal points in terms of the wavelength X. List them in increasing order, separated by commas. You should enter only the factors that multiply X. Do not enter for each one. ► View Available Hint(s) ΠΑΠ ΑΣΦ that is, ? λ
The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move). Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave: y(x, t) = A sin(kx) sin(wt), where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wave number, w is the angular frequency of the wave, and t is time. Part A Which one of the following statements about such a wave as described in the problem introduction is correct? This wave is traveling in the + direction. This wave is traveling in the direction. This wave is oscillating but not traveling. This wave is traveling but not oscillating. Submit Part B At time t = Request Answer 0, what is the displacement of the string y (x, 0) ? Express your answer in terms of A, k, and other previously introduced quantities. y (x,0) = Submit ΠΟΠ ΑΣΦ VO Request Answer ?
Transcribed Image Text:The nodes of a standing wave are points at which the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example, that the point at which a string is tied to a support has zero displacement at all times (i.e., the point of attachment does not move). Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave: y(x, t) = A sin(kx) sin(wt), where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, k is the wave number, w is the angular frequency of the wave, and t is time. Part A Which one of the following statements about such a wave as described in the problem introduction is correct? This wave is traveling in the + direction. This wave is traveling in the direction. This wave is oscillating but not traveling. This wave is traveling but not oscillating. Submit Part B At time t = Request Answer 0, what is the displacement of the string y (x, 0) ? Express your answer in terms of A, k, and other previously introduced quantities. y (x,0) = Submit ΠΟΠ ΑΣΦ VO Request Answer ?
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