Consider a traveling wave described by the formula
.
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 direction.
The wave is traveling in the 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 this new wave should have the same displacement as , the wave described in the problem introduction.
ANSWER:
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 , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:
.
This form is significant because , called the envelope, depends only on position, and 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 .
Part C
Find and . Keep in mind that should be a trigonometric function of unit amplitude.
Consider a traveling wave described by the formula
.
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?

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 this new wave should have the same displacement as , the wave described in the problem introduction.

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 , where is the wave described in Part A and is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:
.
This form is significant because , called the envelope, depends only on position, and 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 .
Part C  

Find and . Keep in mind that should be a trigonometric function of unit amplitude. 