Reflection of waves

A part of the wave is reflected and a part is transmitted into the second medium. If a wave is an incident obliquely on the boundary between two different media the transmitted wave is called the refracted wave. The incident and refracted waves obey Snell’s law of refraction, and the incident and reflected waves obey the usual laws of reflection.

We can thus, summaries the reflection of waves at a boundary or interface between two media as follows:
A traveling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change.
To express the above statement mathematically, let the incident wave be represented by

then, for reflection at a rigid boundary, the reflected wave is represented by,

For reflection at an open boundary, the reflected wave is represented by

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