Problem: 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) = Asin(kx)sin(ωt), 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, ω is the angular frequency of the wave, and t is time.Part AWhat 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; that is, evaluate ω•T/4 and substitute it in the expression for y(x,t).Part B: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.Express the first three nonzero nodal points in terms of the wavelength λ. List them in increasing order.

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Part A

At t = T/4, the displacement is expressed as:

y(x,T/4) = Asin(kx)sin(ωT/4)

But, ω = 2π/T

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Problem Details

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) = Asin(kx)sin(ωt), 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, ω is the angular frequency of the wave, and t is time.

Part A

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; that is, evaluate ω•T/4 and substitute it in the expression for y(x,t).

Part B:

At which three points x1, x2, and x3 closest to = 0 but with > 0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points.

Express the first three nonzero nodal points in terms of the wavelength λ. List them in increasing order.

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