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DodongoBongo

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## Homework Statement

Learning Goal: To see how two traveling waves of the same frequency create a standing wave.

Consider a traveling wave described by the formula

[tex]y_1(x,t) = A \sin(k x - \omega t)[/tex].

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.

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 [tex]y_1(x,t)[/tex] is the wave described in Part A and [tex]y_2(x,t)[/tex] is the wave described in Part B. These waves have been chosen so that their sum can be written as follows:

[tex]y_{\rm s}(x,t) = y_{\rm e}(x) y_{\rm t}(t)[/tex].

This form is significant because [tex]y_e(x)[/tex], called the envelope, depends only on position, and [tex]y_t(t)[/tex] 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 [tex]y_e(x)[/tex].

Find [tex]y_e(x)[/tex] and [tex]y_t(t)[/tex]. Keep in mind that [tex]y_t(t)[/tex] should be a trigonometric function of unit amplitude.

Express your answers in terms of A, k, x, [tex]\omega[/tex], and t.

## Homework Equations

[tex]y_1(x,t) = A \sin(k x - \omega t)[/tex]

Part A: The wave is traveling in the +x direction.

Part B: [tex]A \sin (k x + \omega t)[/tex]

## The Attempt at a Solution

I tried using the trig identity [tex]sin(A-B) = sin(A)cos(B) - cos(A)sin(B)[/tex] to try to break up [tex]y_1(x,t) = A \sin(k x - \omega t)[/tex]. I also know I need to find y(x) and y(t), so I tried solving for y(x,0) and y(0,t). So now I have:

[tex]y_1(x,t) = A \sin(k x - \omega t) = A(\sin(k x) \cos(\omega t) - \cos(k x) \sin(\omega t))[/tex]

and Part A + Part B = [tex]A \sin(k x - \omega t) + A \sin(k x + \omega t)[/tex]

and [tex]y(x,0) = A \sin(k x - \omega (0) ) = A \sin(k x)[/tex],

[tex]y(0,t) = A \sin(k (0) - \omega t) = A \sin(- \omega t)[/tex]

I tried entering: [tex]2A\sin(k x), \sin(\omega t)[/tex], but it told me to "Check my trigonometry on term 2" . (It's MasteringPhysics)

I honestly don't know if what I've done so far is a step in the right direction or how to proceed to get y(x,t) = y(x)y(t). Can anyone let me know if I'm doing this right or if I've missed something obvious? I'm also not sure what they mean by "[tex]y_t(t)[/tex] should be a trigonometric function of unit amplitude".

I understand the Principle of Superposition; I know that wave functions are added together, I just don't understand how you can break them apart into a product like that.

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