Superposition of Waves - Standing Waves

In summary, the conversation discusses a situation where a wave traveling in the negative x-direction encounters a barrier and is reflected. This results in a standing wave where the amplitude of the reflected wave is the same as the incident wave. The equations for the resultant wave are given, and it is shown that when a standing wave occurs, the amplitude is positive. The conversation also addresses a discrepancy in the book's solution, which is resolved by realizing that the substitution made was incorrect.
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Homework Statement



Consider a situation in which a wave is traveling in the negative x-direction encounters a barrier and is reflected. Assume an ideal situation in which none of the energy is lost on reflection nor absorbed by the transmitting medium. This permits us to write both waves with the same amplitude. I will represent these equations as

[itex]E_{1} = E_{0}sin(ωt + kx)[/itex]
[itex]E_{2} = E_{0}sin(ωt - kx - θ_{R})[/itex]

Here [itex]θ_{R}[/itex] is included to account for possible phase shifts upon reflection. The resultant wave of the two waves can be represented as

[itex]E_{R} = E_{1} + E_{2} = E_{0}[sin(ωt + kx) + sin(ωt - kx - θ_{R})][/itex]

Next I make the substitution

[itex]β_{+} = ωt + kx[/itex] and [itex]β_{+} = ωt - kx - θ_{R}[/itex]

and employ the identity

[itex]sin(β_{+}) + sin(β_{-}) = 2sin(\frac{1}{2}(β_{+} + β_{-}))cos(\frac{1}{2}(β_{+} + β_{-}))[/itex]

This yields

[itex]E_{R} = 2E_{0}cos(kx + \frac{θ_{R}}{2})sin(ωt - \frac{θ_{R}}{2})[/itex]

Consider the situation in which a standing wave results when [itex]\frac{θ_{R}}{2} = \frac{∏}{2}[/itex] and you get

[itex]E_{R} = 2E_{0}sin(kx)cos(ωt)[/itex]

Homework Equations


The Attempt at a Solution



This is what my book claims. The only problem I have is that it looks like it made the substitution

[itex]sin(x - \frac{∏}{2}) = -cos(x)[/itex] and [itex]cos(x - \frac{∏}{2}) = sin(x)[/itex]

The problem is that when I make these substitutions I get

[itex]-2E_{0}sin(kx)cos(ωt)[/itex]

I'm not exactly sure how it's supposed to be positive. Thanks for any help.
 
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  • #2
if cos(x-pi/2) = sin(x), what does cos(x+pi/2) equal?
 
  • #3
Oh wow. I can't believe I didn't see that thanks.
 

FAQ: Superposition of Waves - Standing Waves

1. What is the principle of superposition?

The principle of superposition states that when two or more waves pass through the same medium, the resulting wave is the sum of the individual waves. This means that the waves will add together to form a new wave with a different amplitude and phase.

2. What is a standing wave?

A standing wave is a type of wave that forms when two waves with the same amplitude and frequency travel in opposite directions through the same medium. The resulting wave appears to be standing still, with points of maximum and minimum amplitude that do not move.

3. How are standing waves formed?

Standing waves are formed when two waves with the same frequency and amplitude travel in opposite directions and interfere with each other. When the peaks and troughs of the two waves align, they reinforce each other and create points of maximum amplitude. When they are out of phase, they cancel each other out and create points of minimum amplitude.

4. What is the difference between a node and an antinode in a standing wave?

A node is a point in a standing wave where the amplitude is always zero. This is where the two waves interfere destructively and cancel each other out. An antinode, on the other hand, is a point in a standing wave where the amplitude is always at a maximum. This is where the two waves interfere constructively and reinforce each other.

5. How is the wavelength of a standing wave related to the length of the medium?

The wavelength of a standing wave is related to the length of the medium in which it is traveling. Specifically, the wavelength is equal to twice the length of the medium. This means that the length of the medium must be a multiple of half the wavelength for a standing wave to form.

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