# Fixed End Reflections Simulation

#### QUIZ

Q) When a wave pulse on a string reflects from a hard boundary, how is the reflected pulse related to the incident pulse?

A. Shape unchanged, amplitude unchanged
B. Shape inverted, amplitude unchanged
C. Shape unchanged, amplitude reduced
D. Shape inverted, amplitude reduced
E. Amplitude unchanged, speed reduced

#### Fixed End Reflections Simulation

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Reflection
In our discussion of waves so far, we have assumed that the waves being analyzed could travel indefinitely without striking anything that would stop them or otherwise change their motion. But what happens to the motion of a wave when it reaches a boundary?

Fixed End Reflections
If a medium is fixed at one end, then when a wave reaches the media boundary a fixed-end reflection occurs. A fixed-end reflection also occurs when a medium is fixed at both ends, as in a harp (Figure 01).
Consider a pulse in a string moving toward a rigid, denser medium such as a wall (Figure 01). When the pulse reaches the fixed end, it is reflected. As you see in Figure 01, the reflected pulse has the same shape as the incoming pulse, but its orientation is inverted. We may explain this inversion as follows. When the pulse reaches the fixed end of the string, it exerts an upward force on the wall. In response, the wall exerts a downward force on the string in accordance with Newton’s third law of motion.
Therefore, the incoming upright pulse is inverted upon reflection.

Figure 01. When a pulse in one medium meets a boundary with a denser medium, the reflected pulse is inverted.

The difference in the media as a wave reaches a media boundary may not be as dramatic as either the free-end or the fixed-end case. In nature there are media boundaries that are neither free-end nor fixed-end. For example, the boundary between water and air and the boundary between air and a tree are neither fixed-end boundaries nor free-end boundaries. If a wave travels from a medium in which its speed is faster to a medium in which its speed is slower, the wave particles can move more freely than they did in the faster medium. Th is may seem opposite to what you might think, but the motion of particles in media where the wave speed is higher is tightly controlled by molecular forces. So the transfer of the wave energy to the next particle is very efficient. Th is is similar to the free-end case, so the reflected wave has the same orientation as the original. Conversely, if a wave travels from a medium in which its speed is slower to a medium in which its speed is faster, the wave cannot move as freely. Th is is similar to the fixed-end case, so the reflected wave is inverted(see Figure 02 on the next).

Media Boundaries: Amplitudes
The amplitude of a wave before it encounters a media boundary is closely related to the wave’s energy. The amplitude does not change if the wave’s energy remains constant.
When a wave encounters a media boundary that is not strictly an ideal free-end or fixed-end boundary, the wave splits into two. One wave is reflected, and the other is transmitted. The term transmission describes the process of a wave moving through a medium or moving from one medium into another medium. The amplitude of the original wave may not be shared equally by the reflected wave and the transmitted wave. However, the sum of the two amplitudes must equal the amplitude of the original wave (Figure 02).

Figure 02. At a media boundary that is neither free-end nor fixed-end, the original wave splits into two waves. (a) If the wave moving along the rope encounters a medium that has a slower wave speed, then the wave splits into two, and one wave is reflected and the other is transmitted. The reflected wave is upright. (b) When a wave moves into a faster medium, then the wave splits into two, and one wave is reflected and the other is transmitted. The reflected wave is inverted.

If the difference between the wave speeds in the two media is small, transmission is preferred—the amplitude of the transmitted wave is closer to the amplitude of the original wave. As a result, the amplitude of the reflected wave is much smaller because of the conservation of energy. For cases in which the wave speed is significantly different between the two media, reflection is preferred—the amplitude of the reflected wave is closer to the amplitude of the original wave.