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To see how two traveling waves of the same frequency create a standing wave. Consider a traveling wave described by the formula
y1(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.
1. Find ye(x) and yt(t). Keep in mind that yt(t) should be a trigonometric function of unit amplitude.
2. At the position x=0, what is the displacement of the string (assuming that the standing wave ys(x,t) is present)?
3. At certain times, the string will be perfectly straight. Find the first time t1>0 when this is true.
4. Which one of the following statements about the wave described in the problem introduction is correct?
A. The wave is traveling in the +x direction.
B. The wave is traveling in the −x direction.
C. The wave is oscillating but not traveling.
D. The wave is traveling but not oscillating.
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=0this new wave should have the same displacement as y1(x,t), the wave described in the problem introduction.
A. Acos(kx−ωt)
B. Acos(kx+ωt)
C. Asin(kx−ωt)
D. Asin(kx+ωt)

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temmydbrain

Answer:

Check the explanation

Explanation:

so basically the standing wave will be created by two waves... y1 and the wave reflected, ie, y2=Asin(wt+Kx)..

so the resulting wave will be Y= y1 + y2 = 2A Sin(Kx) Cos(wt) = y(x) y(t)

so y(x)= 2ASin(Kx) and y(t)=Cos(wt)...

The definition of standing wave and trigonometry allows to find the results for the questions about the waves are:

      1. For the standing wave its parts are: spatial [tex]y_e = A' \ sin \ kx[/tex]  and

         temporal part [tex]y_t = A' \ cos \ wt[/tex]

      2. The string moves with an oscillating motion  y = A’ cos wt.

      3. Thefirst displacement is zero for  [tex]t = \frac{\pi }{2w}[/tex]  

      4. the correct result is:

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

      5. The correct result is:

          D. Asin(kx+ωt)

Traveling waves are periodic movements of the media that transport energy, but not matter, the expression to describe it is:

       y₁ = A sin (kx -wt)

Where A is the amplitude of the wave k the wave vector, w the angular velocity and x the position and t the time.

1. Ask us to find the spatial and temporal part of the standing wave.

To form the standing wave, two waves must be added, the reflected wave is:

       y₂ = A sin (kx + wt)

The sum of a waves

       y = y₁ + y₂

       y = A (sin kx-wt + sin kx + wt)

We develop the sine function and add.

       Sin (a ± b) = sin a cos b ± sin b cos a

The result is:

       y = 2A sin kx cos wt

They ask that the function be unitary therefore

The amplitude  of each string

        A_ {chord} = A_ {standing wave} / 2

The spatial part is

          [tex]y_e[/tex]= A 'sin kx

The temporary part is:

          [tex]y_t[/tex] = A ’cos wt

2. At position x = 0, what is the displacement of the string?

          y = A ’cos wt

The string moves in an oscillating motion.

3. At what point the string is straight.

When the string is straight its displacement is zero x = 0, the position remains.

           y = A ’cos wt

For the amplitude of the chord to be zero, the cosine function must be zero.

           wt = (2n + 1) [tex]\frac{\pi}{2}[/tex]  

the first zero occurs for n = 0

          wt = [tex]\frac{\pi }{2}[/tex]  

           t = [tex]\frac{\pi }{2w}[/tex]

4) The traveling wave described in the statement is traveling in the positive direction of the x axis, therefore the correct statement is:

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

5) The wave traveling in the opposite direction is

            y₂ = A sin (kx + wt)

The correct answer is:

            D.     Asin(kx+ωt)

In conclusion using the definition of standing wave and trigonometry we can find the results for the questions about the waves are:

     1. For the standing wave its parts are: spatial [tex]y_e = A' \ sin \ kx[/tex]  and

         temporal part [tex]y_t = A' \ cos \ wt[/tex]

      2. The string moves with an oscillating motion  y = A’ cos wt.

      3. Thefirst displacement is zero for  [tex]t = \frac{\pi }{2w}[/tex]  

      4. the correct result is:

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

      5. The correct result is:

          D. Asin(kx+ωt)

Learn more about standing waves here:  brainly.com/question/1121886