We'll need the trigonometric identity below for part c:

$\overline{){\mathbf{S}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{(}}{\mathbf{A}}{\mathbf{-}}{\mathbf{B}}{\mathbf{)}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{S}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{A}}{\mathbf{\xb7}}{\mathbf{S}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{B}}{\mathbf{}}{\mathbf{-}}{\mathbf{}}{\mathbf{C}}{\mathbf{o}}{\mathbf{s}}{\mathbf{}}{\mathbf{A}}{\mathbf{\xb7}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{B}}}$

**(a)**

The equation describes a traveling wave. The negative sign in the sine function (kx - ωt) shows that the wave described by this general solution is traveling in the positive x-direction.

Consider a traveling wave described by the formula

y_{1}(x,t) = Asin (kx - ωt)

This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves.

(A) Which one of the following statements about the wave described in the problem introduction is correct?

- The wave is traveling in the +x direction.
- The wave is traveling in the -x direction.
- The wave is oscillating but not traveling.
- The wave is traveling but not oscillating.

(B) Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0 this new wave should have the same displacement as y_{1}(x,t), the wave described in the problem introduction.

- A cos (k x - ωt)
- A cos (k x + ωt)
- A sin (k x - ωt)
- A sin (k x + ωt)

(C) The principle of superposition states that if two functions each separately satisfy the wave equation, then the sum (or difference) also satisfies the wave equation. This principle follows from the fact that every term in the wave equation is linear in the amplitude of the wave.

Consider the sum of two waves y_{1}(x,t) + y_{2}(x,t), where y_{1}(x,t) is the wave described in Part A and y_{2}(x,t) is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

y_{s} = y_{e}(x) y_{t}(t)

This form is significant because y_{e}(x), called the envelope, depends only on position, and y_{t}(t) depends only on time. Traditionally, the time function is taken to be a trigonometric function with unit amplitude; that is, the overall amplitude of the wave is written as part of y_{e}(x).

Find y_{e}(x) and y_{t}(t). Keep in mind that y_{t}(t) should be a trigonometric function of unit amplitude.

Express your answers in terms of A, k, ω, x, and t.

(D) At the position x=0, what is the displacement of the string (assuming that the standing wave y_{s}(x,t) is present)?

Express your answer in terms of parameters given in the problem introduction.

(E) At certain times, the string will be perfectly straight. Find the first time t_{1}>0 when this is true.

Express t_{1} in terms of ω, k, and necessary constants.

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