Summarizing Previously Learned Concepts
After having studied the properties of a wave, use this lesson plan to introduce your class to the concept of standing waves.
Begin by summarising the properties of waves previously learnt as:
- A wave is a disturbance that travels from one point to another point away from the source without the transport of material or medium itself.
- Transverse and longitudinal waves
- Draw a picture of the wave on the board. Ask students to point out the wavelength, amplitude, frequency, crests and troughs of the wave one by one.
- Lastly, the wave formula
v = f.λ
v = velocity of the wave; f = frequency of the wave**;** λ = wavelength
Boundary Behavior - Reflection of Waves
Demonstration: Tie a rope to a pole. Hold the other end such that the rope is outstretched and parallel to the ground. Introduce a small pulse at one end. It is observed that the pulse returns, inverted.
Explain to the students that the end tied to the pole is a fixed end. The initial medium of travel for the pulse was the rope. At the end, the medium was the pole. The pole has very little vibration and is hence treated as a fixed end. The interface between the pole and the rope is called the boundary.
The wave is reflected at the fixed end. Now explain, as follows: the phenomenon of reflection of the pulse (emphasis on the words in bold). When the disturbance carried by the pulse reaches the fixed end, some of its energy is lost due to vibrations in the pole. This is the transmitted energy. The rest of it is reflected back.
- On one side of the boundary is the last particle of medium A (the rope) and on the other side is the first particle of medium B (the pole).
- When a crest reaches the end of a medium A, the last particle of the medium A has an upward displacement.
- This particle is attached to the first particle of the second medium at the boundary. This upward displaced particle of medium A exerts an upward pull on the first particle of medium B.
- Owing to the Newton’s law of action and reaction, the first particle of medium B pulls downwards on the last particle of medium A.
- The upward pull on the first particle of medium B has little effect upon this particle due to the large mass of the pole. But the effect of the downward pull on the last particle of medium A (a pull which is in turn transmitted to the other particles) results in causing the upward displacement to become a downward displacement.
The upward displaced incident pulse thus returns as a downward displaced reflected pulse.
Write down these characteristics of the pulse reflected on the board -
- It has the same velocity as the incident pulse (discuss why - since velocity is a function of the medium of travel.)
- The wavelength of the pulse is also the same as the incident pulse.
- The amplitude of the reflected pulse is less than the amplitude of the incident pulse (discuss why, since some of the energy of the pulse was transmitted into the pole at the boundary. )
The concept of the traveling wave can be explained as follows -
A wave is a disturbance that keeps traveling as the particles of the medium vibrate and pull on to the next particle. The wave pattern of crests and troughs continues to move uninterrupted until it encounters another wave along the medium or a boundary with another medium. Such a wave is called a traveling wave.
Superposition of Waves
Discuss the following with your class -
What do you think will happen when two waves in a medium meet? Will they kill each other? Or bounce upon each other? How will they interact?
Draw the diagrams above to explain the phenomenon. These are two square waves moving towards each other. Tell them that this is what happens when the waves meet. Ask them to explain what could have happened.
When these two meet, the waves add up to give the resultant wave over the region of overlap. This is called the principle of superposition, which can be stated as (stress upon the bolded words):
When two waves overlap, the resulting displacement of the medium at any location is the algebraic sum of the displacements of the individual waves at that same location.
The overlapping of waves is called interference of waves and the waves are said to be interfering with each other.
Now ask this question to the class - what will happen if two pulses of equal amplitude and wavelength, but inverted with respect to each other, meet? You may draw a diagram on the board to make things clearer. The result will be no displacement at all at the points where the two waves exactly overlap.
Observing Standing Waves
Modify the first demonstration used to explain reflection of waves so that now the free end is instead tied to a tuning fork. Set the tuning fork in vibrations. The length of the string should be such that it is half the wavelength of the waves produced by the tuning fork.
It is observed that after some time, the oscillations form a regular pattern, which gives the impression that waves are not traveling anywhere, they are in fact “standing waves”. Explain this phenomenon as below.
- Since the fork is rigid, it also acts as a fixed end. So both the ends are fixed ends and one end continuously supplies waves at a frequency f.
- It produces a wave of some amplitude, say A. This wave gets reflected at the fixed end, gets inverted and again gets reflected at the fork. It is reinverted here. This wave has travelled a distance of 2L which is equal to the wavelength. Hence, the next wave is produced by the fork only when the first wave reaches back to the fork (discuss why - because frequency is the reciprocal of time period.)
- The wave produced at this instant and the already reflected wave interfere and add up by the principle of superposition. So a wave of 2A amplitude travels to the fixed end this time. Similarly, next time round, a wave of amplitude 3A would travel.
- In practice, due to energy losses an equilibrium is reached, where the energy lost each time equals the energy supplied by the tuning fork while producing each wave. Hence, a fixed amount of energy is confined between that region.
- In such a steady state, the waves traveling in both directions have equal amplitudes and it does not change with time anymore. This produces standing waves, where the wave doesn’t seem to travel because it is confined in a region. Their interference produces a pattern such that some points do not move at all and some points show a maximum displacement.
Properties of Standing Waves
Summarise the properties of standing waves as below. Remember to stress upon the bolded words.
- Traveling waves of equal frequency and amplitude traveling in opposite directions interfere to form standing waves.
- Energy is confined between two fixed ends in the case of standing waves unlike traveling waves where energy travels.
- Standing waves are characterized by points which do not move at all. These points are called nodes. There are also antinodes, the points which oscillate the maximum. There can be a number of nodes and antinodes in a standing wave pattern depending upon the frequency and the length of the string used. The example considered above had two nodes and only one antinode.
- For a particular length of string, only some specific frequencies are capable of producing standing waves. The lowest such frequency (which has two nodes and one antinode) is called the fundamental frequency of oscillation. Higher frequencies produce a greater number of nodes and antinodes.
This post is part of the series: Waves and Sound
This series covers lesson plans covering the concepts of wave motion and sound.