What Is the Equation of a Wave Reflected from a Rigid Boundary?

[Amith2006]

Sir,
Consider a progressive wave represented by the equation,
y = A[sin(wt – kx)]
If it is reflected from a wall, what will probably be the equation of the reflected wave?
I think it is y = A1[sin(wt + kx)]
Is it right? Should a negative sign be given to the expression? Will there be a phase change of pi assuming the wall to be rigid boundary?

 

[maverick280857]

Sir,
Consider a progressive wave represented by the equation,
y = A[sin(wt – kx)]
If it is reflected from a wall, what will probably be the equation of the reflected wave?
I think it is y = A1[sin(wt + kx)]
Is it right? Should a negative sign be given to the expression? Will there be a phase change of pi assuming the wall to be rigid boundary?

There will be a phase change of $\pi$ if the wall is of a denser medium than the incident medium. Since the wall is rigid, this is correct. $A_{1}$ is determined from considerations of boundary conditions. That will tell you whether there should be a minus sign or not. You don't have to "give" it a negative sign.

 

[kyle8921]

Yes, your equation is correct. The reflected wave will have the same amplitude as the incident wave, but the sign of the wave vector (k) will be opposite, resulting in a reflected wave traveling in the opposite direction. Therefore, the equation for the reflected wave will have a positive sign for kx, as shown in your equation.

As for the phase change, it depends on the type of boundary the wall represents. If the wall is a rigid boundary, there will be a phase change of pi (180 degrees) as the wave reflects off of it. This means that the reflected wave will be shifted by half a wavelength compared to the incident wave. However, if the wall is a soft boundary, there may not be a phase change or it may be different depending on the properties of the boundary.