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Reflection of Wave

In previous sections we have discussed wave propagation in unbounded media. What happens when a pulse or a travelling wave encounters a rigid boundary? It is a common experience that under such a situation the pulse or the wave gets reflected. An everyday example of the reflection of sound waves from a rigid boundary is the phenomenon of echo. If the boundary is not completely rigid or is an interface between two different elastic media, the effect of boundary conditions on an incident pulse or a wave is somewhat complicated. A part of the wave is reflected and a part is transmitted into the second medium. If a wave is 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.

To illustrate the reflection of waves at a boundary, we consider two situations. First, a string is fixed to a rigid wall at its left end, as shown in Figure (a). Second, the left end of the string is tied to a ring, which slides up and down without any friction on a rod, as shown in Figure (b). A pulse is allowed to propagate in both these strings, the pulse on reaching left end gets reflected, the state of disturbance in the string at various times is shown in Figure.

Figure - (a) A pulse incident from the right is reflected at the left end of the string, which is tied to a wall. Note that the reflected pulse is inverted from the incident pulse. (b) Here the left end is tied to a ring that can slide up and down without friction on the rod. Now the reflected pulse is not inverted by the reflection

In Figure (a), the string is fixed to the wall at its left end. When the pulse arrives at that end, it exerts an upward force on the wall. By Newton's third law, the wall exerts an opposite force of equal magnitude on the string. This second force generates a pulse at the support (the wall), which travels back along the string in the direction opposite to that of the incident pulse.

In a reflection of this kind, there must be a node at the support as the string is fixed there. The reflected and incident pulses must have opposite signs, so as to cancel each other at that point. Thus in case of a travelling wave, the reflection at a rigid boundary will take place with a phase reversal or with a phase difference of 180o or
π radians.

In Figure (b), the string is fastened to a ring, which slides without friction on a rod. In this case, when the pulse arrives at the left end, the ring moves up the rod. As the ring moves, it pulls on the string, stretching the string and producing a reflected pulse with the same sign and amplitude as the incident pulse. Thus, in such a reflection, the incident and reflected pulses reinforce each other, creating an antinode at the end of the string: the maximum displacement of the ring is twice the amplitude of either of the pulses. Thus the reflection is without any additional phase shift. In case of a travelling wave the reflection at an open boundary, such as the open end of an organ pipe, the reflection takes place without any phase change.

We can thus summarise the reflection of waves at a boundary or interface between two media as follows:
A travelling 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

     yi (x, t) = ym sin (kx - ωt)

then, for reflection at a rigid boundary the reflected wave is represented by,
     yr (x, t) = ym sin (kx + ωt + π)
                         = - ym sin (kx + ωt)


For reflection at an open boundary, the reflected wave is represented by
        yr (x, t) = ym sin (kx + ωt)

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