When a wave travels from one medium to another, it gets reflected at the boundary of the second medium. A common example of this is when a person yells at a distance from a cliff and hears the echo of their voice. The sound waves (longitudinal waves) traveling in the air are reflected from the bounding cliff. Similarly, flipping one end of a string whose other end is tied to a wall causes a pulse (transverse wave) to travel through the string, which gets reflected upon reaching the wall. In this case, the tied end acts as the boundary of the wave, which is not free to move with the oscillations.
Recall Newton's third law of motion, which states that every action has an equal and opposite reaction. As the incident wave encounters the wall, the string exerts an upward force on the wall, and the wall reacts by exerting an equal and opposite force on the string. Thus, in the case of reflection at a fixed boundary, a crest becomes a trough after reflection and vice versa.
If, however, the boundary is not fixed, and is free to move with the wave's oscillations, the phase of the reflected wave does not get inverted. For example, if the string is tied to a solid ring capable of sliding along a frictionless pole, the end of the string is free to move up and down. The wave encounters the free boundary applying an upward force on the ring, moving the ring up. The ring travels up to the maximum height equal to the amplitude of the wave and then accelerates down toward the equilibrium position due to the tension in the string. Thus, if the incident wave were a trough, the reflected wave would also be a trough in the case of a free boundary.
This text is adapted from Openstax, University Physics Volume 1, Section 16.5: Interference of Waves.
