Reflection of Waves:- Reflection from Fixed End, and End

Reflection of Waves: To understand the movement of a wave, let us consider an example of skipping ropes. Have you seen how the rope moves during the skipping of ropes? Do you know what the reason behind such movement of the rope is? It is because of the standing wave. We also encounter different situations involving the reflection of waves all around us. In the phenomenon of echo, the reflection of waves also occurs when the sound is reflected from a distant object and reaches the listener with some delay. In this article, we will see what happens when a pulse or a travelling wave encounters a fixed and boundary. We will also see how under such a situation, the pulse or the wave gets reflected.

Reflection of Waves

What happens when a ball is thrown against a hard wall? Yes, it bounces back. This can be termed as “reflection“. This aspect can be obtained in terms of energy and momentum conservation. In another case, if the collision between the ball and wall is effortlessly elastic, the ball will bounce back at the same speed because the incident energy and the momentum will return. Again, if the collision is inelastic, the ball would not bounce back at the same speed because the wall (or ball) would absorb a share of the momentum and incident energy. When a wave incident on any surface, then some part of it is reflected, and some part is transmitted into the second medium, and this transmitted wave is known as a refracted wave, as shown in the below figure.

There are two types of surfaces: one can be fixed, and the other can be, from which the reflection of a wave or a pulse can occur is discussed below.

Fixed End Reflection

Let us consider an elastic rope that is stretched from end to end. In which, attached one end of the rope to a pole on a lab bench while holding the other end of the rope in hand to introduce pulses into the medium, as shown in the below figure. When the disturbance reaches the last particle of the rope, it will be unable to move because the right end of the rope is attached to a pole. This end of the rope is called a fixed end. This introduced pulse at the left end of the rope will travel through the rope towards the right end of the medium. Since this pulse is approaching the boundary with the pole so this is called the incident pulse. Two things may occur when the incident pulse reaches the boundary that is given below:

The pulse carries some portion of the energy, which gets reflected and returns towards the left end of the rope, and the disturbance that returns to the left end of the rope after bouncing off the pole is called the reflected pulse.
The pulse carries some portion of the energy gets transmitted to the pole, causing the pole to vibrate.

When the other end is fixed during this phenomenon, we observe that the reflected pulse is inverted. Upward displaced pulse returns as a downward displaced pulse, and downward displaced pulse returns as an upward displaced pulse when incident towards a fixed end boundary.

Some other characteristics of the reflected pulse are:

The speed of the reflected pulse and the speed of the incident pulse are the same.
The wavelength of the reflected pulse and the wavelength of the incident pulse are the same.
The reflected pulse amplitude is less than the amplitude of the incident pulse.

End Reflection

Let us suppose another case in which the end of the rope is allowed to movely with the help of a ring that loosely fits around the pole. When the disturbance reaches the right end of the rope, the last particle of the rope will move because the right end of the rope is to move. This end is the end of the rope.

Again, if we introduce a pulse at the left end of the rope, as shown in the below figure, it will travel through the rope towards the right end of the medium. In this case, the last particle of the rope can no longer interact with the first particle of the pole when the incident pulse reaches the end of the medium. And they will slide past each other because the rope and pole are no longer attached and interconnected. So, the last particle of the rope receives the same upward displacement when a crest reaches the end of the rope; but there is no adjoining particle to pull down upon the last particle of the rope. So the reflected pulse is not inverted. After reflection, an upward displaced pulse returns as an upward displaced pulse, and a downward displaced pulse returns as a downward displaced pulse when incident upon a end. In-end reflection, inversion is not observed.

Equations for Reflection at the Rigid Boundary

Consider a string that is fixed to the wall at one end. When an incident wave hits a wall, it will exert a force on the wall. Then, the wall exerts an equal and opposite force of equal magnitude on the string by Newton’s third law. Since the wall is rigid, the wall won’t move; therefore, no wave is generated at the boundary. This implies that the amplitude at the boundary is \(0\) as both the reflected wave and incident wave are completely out of phase.

Therefore \(\varphi = \pi \)

Therefore, \({y_i}(x,\,t) = a\,\sin (kx\, – \,\omega t)\)

\({y_r}(x,\,t)\, = \,a\,\sin (kx + \omega t + \pi ) = – a\,\sin (kx + \,\omega t)\)

By superposition principle, \(y = {y_i} + {y_r} = 0\)

With a phase reversal of \(\pi \,or\,{180^ \circ }\) the reflection at the rigid body will take place.

Summary

When a wave travels through a medium, it will often reach the end of the medium and encounter an obstacle, or it could travel to another medium. Some of the well-known wave phenomena, such as echoes, result from the reflection of waves. Furthermore, the generation of a standing wave is also a result of this wave reflection.

When we introduce a pulse on the string, it reaches the right end of the rope that is clamped at the wall. The element present at the right end of the rope exerts a force on the clamp, which exerts equal and opposite force on the element. Thus the incoming wave gets reflected from the fixed end, which is inverted to the original wave. When a wave pulse is sent on the string whose right end is attached to a light frictionless ring that can movely, the element is acted on the force from the left to go up as the waves reach the right end. No inversion takes place in the reflection of the wave by the end.

Q.1. What is a reflection of a wave?
Ans: When an incident wave strikes the boundary between different mediums, there is an abrupt change in the wave’s propagation direction. This phenomenon is called the reflection of a wave.

Q.2. Give some examples of a reflecting wave?
Ans: The phenomenon of echo is an example of the reflection of waves. When waves bounce back from a surface they cannot pass through, then reflection occurs. In any wave, not just sound waves, reflection can happen. Light waves can also be reflected.

Q.3. How does the reflection of waves occur on a fixed and end on a string?
Ans: When a transverse wave is produced on a string fixed at the endpoint, the reflected wave we get is inverted from the incident wave. Still, when a transverse wave is produced on a string at the endpoint, the reflected wave is not inverted from the incident wave.

Q.4. What are the different types of waves?
Ans: Waves come in two kinds, longitudinal and transverse. For example, transverse waves occur on the water, with the water’s surface going up and down, and longitudinal waves occur in sound waves, consisting of alternating compressions and rarefactions in a medium.

Q.5. What are the standing waves?
Ans: Standing waves can be transverse or longitudinal. It is also known as stationary waves. From one particle to the adjoining particle, there is no forward motion of the disturbance and so on beyond this particular region.

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