A wave is first reflected by a boundary P and then by a boundary Q without dissipating energy. If the boundary P is rigid and the boundary Q is non-rigid, then find the fraction of the amplitude of the reflected wave by boundary P to the reflected wave by boundary Q.

Reflection of waves:

• When a wave is incident on a boundary between two media and it returns back into the initial medium, then it is called reflection of waves.
  • The phenomenon of echo is an example of reflection by a rigid boundary.
• If the boundary is not completely rigid or is an interface between two different elastic media, the situation is somewhat complicated.
• A part of the incident wave is reflected and a part is transmitted into the second medium.
  • 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.

Reflection by a rigid boundary:

• If a pulse travelling along a stretched string and is reflected by the rigid boundary.
  • Assuming there is no absorption of energy by the boundary, the reflected wave has the same frequency and amplitude as the incident pulse but it suffers a phase change of π or 180° on reflection.
  • This is because the boundary is rigid and the disturbance must have zero displacements at all times at the boundary.
• By the principle of superposition, this is possible only if the reflected and incident waves differ by a phase of π so that the resultant displacement is zero.

Reflection by a non-rigid boundary:

• If the boundary point is not rigid but completely free to move (such as in the case of a string tied to a freely moving ring on a rod), the reflected pulse has the same phase, amplitude, and frequency (assuming no energy dissipation) as the incident pulse.
• The net maximum displacement at the boundary is then twice the amplitude of each pulse. An example of a non-rigid boundary is the open end of an organ pipe.

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