## Wave Equation after Reflection

Hello all,
1. The problem statement, all variables and given/known data

We can represent a mechanical transverse wave by Y=Asin(kx-wt+∅).

Now imagine this wave travelling (towards right as velocity is positive) and meeting up with
two cases

Case 1) Rigid wall.
Case 2) Free end.

The way gets reflected completely( ignoring transmission or any other losses).

Now my question is what is the new equation of wave.?

3. The attempt at a solution

In all text books I have studied with, (Resnick Halliday Krane being one of them)

for equation of wave as Y=Asin(kx-wt+∅), reflected wave (from free end) is written as Y=Asin(wt+kx+∅) and (for rigid we add a phase difference of pi) to make it y=Asin(wt+kx+∅+pi).

From what I realise this is simply done to make the velocity negative.

Now my question is we could have made the velocity negative even by writing the equation as y=Asin(-kx-wt+∅) (for free end) and y=Asin(-kx-wt+∅+pi) for rigid end.

What prompts us to use the earlier mentioned equations more?
whose coefficient has to be negated to form the reflected wave and why is it so?

Also we could represent the original wave by Y=Asin(wt-kx+∅2) {∅2 is a different phase constant}

In this case what will be the equation of reflected wave(from free end)?

Will it be y=Asin(wt+kx+∅2)
or y =Asin(-wt-kx-∅2)

Thanks a ton

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 To shorten the post above my question is if a wave y=Asin(kx-wt) gets reflected from a free end.(Ignore transmission or any other loses.Ideal condition) what will be the equation of reflected wave.? From what I know we make the velocity of original wave negative and since its reflection from free end we add a phase difference of 0. My question is what will the new equation be Will it be y=Asin(kx+wt) or y=Asin(-kx-wt) as velocity can be negated by making negative either the coefficient of x or of t. and how do we decide whose coefficient has to be negated. (Note:Bumping after 24 hours as suggested by admin)
 Hello, I havent got an answer yet so am bumping again. My question simply is how do we find the new equation of a wave after reflection from a free end.(No losses due to transmission or friction) Initial equation is y = Asin(kx-wt + phi) what is reflected wave equation Y= ?

 Tags equation, reflect, reflection, transverse, wave