Propagation constant k:

$\overline{){\mathbf{k}}{\mathbf{=}}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\lambda}}}$

Period, T:

$\overline{){\mathbf{T}}{\mathbf{=}}\frac{\mathbf{2}\mathbf{\pi}}{\mathbf{\omega}}}$

$\overline{)\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{t}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{t}}}$

Part A

v_{p} = λ/T = (2π/k)/(2π/ω) = ω/k

Two Velocities in a Traveling Wave

Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself).

Consider a transverse wave traveling in a string. The mathematical form of the wave is

y(x,t) = Asin(Kx - wt)

Part A

Find the velocity of propagation v_{p} of this wave.

Express the velocity of propagation in terms of some or all of the variables A, k,and w.

Part B

Find the *y* velocity v_{y}(x,t) of a point on the string as a function of x and t.

Express the *y* velocity in terms of w, A,k,x,and t.

Part C

Which of the following statements about, the *x* component of the velocity of the string, is true?

a) v_{x}(x;t) = v_{p}

b) v_{x}(x;t) = v_{y}(x;t)

c) v_{x}(x;t) has the same mathematical form as v_{y}(x;t) but is 180° out of phase.

d) v_{x}(x;t) = 0

Part D

Find the slope of the string as a function of position x and time t.

Express your answer in terms of A,k,w, x,and t.

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