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Creating a Standing Wave
Learning Goal: To see how two traveling waves of the same frequency create a standing wave.
Consider a traveling wave described by the formula

y_1(x,t) = A \sin(k x - \omega t).

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.

Part A  
Which one of the following statements about the wave described in the problem introduction is correct?
ANSWER:
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.
Part 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.
ANSWER:
A \cos (k x - \omega t)
A \cos (k x + \omega t)
A \sin (k x - \omega t)
A \sin (k x + \omega t)

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_{\rm s}(x,t) = y_{\rm e}(x) y_{\rm 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).

Part C  
Find y_e(x) and y_t(t). Keep in mind that y_t(t) should be a trigonometric function of unit amplitude.
Hint C.1 A useful identity
A useful trigonometric identity for this problem is

\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B).

Hint C.2 Applying the identity
Since you really need an identity for \sin(A- B), simply replace B by -B in the identity from Hint C.1, keeping in mind that \sin(-x) = -\sin(x).
Express your answers in terms of A, k, x, omega, and t. Separate the two functions with a comma.
ANSWER:
  y_e(x), y_t(t) =  2*A*sin(k*x)   cos(omega*t) 
 sin(k*x)   2*A*cos(omega*t) 
Part D  
Which one of the following statements about the superposition wave y_s(x,t) is correct?
ANSWER:
This wave is traveling in the +x direction.
This wave is traveling in the -x direction.
This wave is oscillating but not traveling.
This wave is traveling but not oscillating.
A wave that oscillates in place is called a standing wave. Because each part of the string oscillates with the same phase, the wave does not appear to move left or right; rather, it oscillates up and down only.
Part E  
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.
ANSWER:
  y_s(x=0,t)  0 
This could be a useful property of this standing wave, since it could represent a string tied to a post or otherwise constrained at position x=0. Such solutions will be important in treating normal modes that arise when there are two such constraints.
Part F  
At certain times, the string will be perfectly straight. Find the first time t_1>0 when this is true.
Hint F.1 How to approach the problem
The string can be straight only when \cos(\omega t) = 0, for then y_{\rm s}(x,t) = 0 also (for all x). For any other value of \cos(\omega t), y_s(x,t) will be a sinusoidal function of position x.
Express t_1 in terms of omega, k, and necessary constants.
ANSWER:
  t_1 =  pi/(2*omega) 
Part G  
From Part F we know that the string is perfectly straight at time t=\frac{\pi}{2\omega}. Which of the following statements does the string's being straight imply about the energy stored in the string?
  1. There is no energy stored in the string: The string will remain straight for all subsequent times.
  2. Energy will flow into the string, causing the standing wave to form at a later time.
  3. Although the string is straight at time t=\frac{\pi}{2\omega}, parts of the string have nonzero velocity. Therefore, there is energy stored in the string.
  4. The total mechanical energy in the string oscillates but is constant if averaged over a complete cycle.
ANSWER:
abcd
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