**Part A**

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

y(x,T/4) = Asin(kx)sin(ωT/4)

But, ω = 2π/T

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 *x*_{1}, *x*_{2}, and *x*_{3} 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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