Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.
What are Two Identical Progressive Waves Moving in Opposite Direction?
Definition: When two progressive waves of the same amplitude, frequency and velocity travelling in opposite directions superpose in a region of space, the resultant wave is confined to that region and cannot progress through the medium. Such type of wave is called a stationary wave or standing wave.
Stationary wave keeps on repeating itself in a region and there is no transfer of energy along the medium in either direction. In can be explained as below.
Let a thin metallic wire AB is stretched between two rigid supports [Fig.]. The vire is struck on any arbitrary point in a normal direction. Two separate waves are produced and they travel towards the two ends of the wire. After getting reflected from the end support, each wave starts moving towards the opposite end. As a result, they superpose in the region between A and B.
The resultant wave is confined in the region AB and cannot travel like a progressive wave. Stationary waves in a stretched string are produced in this way. These waves are the sources of notes emitted from stringed instruments like sitar, violin, etc. Stationary waves are also produced in the air columns in instruments like flute, organ, etc.
Nodes and antinodes: A stationary wave remains confined in a region and cannot progress through the medium [Fig.]. As a result the waveform is such that in a few positions (like A, D, F, H, B), the particles in the medium remain stationary at all times, i.e., the wave amplitude at these positions is always zero. These positions are called nodes. On the other hand, the particles in a few positions (like C, E, G, I) continue to vibrate with maximum amplitude—these positions are the antinodes.
Definition: In a stationary wave, the positions where the particles of the medium always remain at rest are called nodes, and the positions where the particles vibrate with the maximum amplitude are called antinodes.
For vibrations of a stretched string, nodes and antinodes are formed at points on the string; they are nodal points and antinodal points, respectively. Similarly, during vibrations of stretched membranes, in instruments like tabla, drum, etc. nodal lines and antinodal lines are formed, while nodal surfaces and antinodal surfaces are formed in vibrating air columns.
Loop: The region between two consecutive nodes in a stationary wave is called a loop. For example, in Fig., each of AD, DF, FH and HB is a loop. If we consider any single loop, it is evident that all the particles are either in the equilibrium position or above or below that position at any instant. As the vibrations are usually very fast, the whole loop is visible.
Now, we consider two adjacent loops (say, AD and DF in Fig.). At any instant, if the particles in loop AD are below the equilibrium position along the line AB, the particles in loop DF would be above the line AB. So, the particles in adjacent loops of a stationary wave are in opposite phases, i.e., the phase difference is 180°.
Wavelength of a stationary wave: In Fig., the two antinodal points C and G are in the same phase because
1. when the displacement of the particle at C is maximum, the particle at G also goes to its maximum displaced position and
2. at every instant, the particles at C and G are on the same side relative to their equilibrium positions. Since C and G are consecutive points lying in the same phase, the distance between C and G is known as the wavelength (λ). It is to be noted that there is another antinodal point E between C and G, but E is in the opposite phase with respect to C and G. It is observed that the distance between the nodal point A and F is also equal to CG.
Definition: The distance between three consecutive nodes or three consecutive antinodes, is the wavelength (λ) of a stationary wave.
So, the length of a loop = distance between two consecutive nodes (say, AD) = \(\frac{\lambda}{2}\);
the distance between a node and the adjacent antinode (say, AC) = half of the length of a loop = \(\frac{\lambda}{4}\).
Resonant frequency: Let us consider a string of a guitar, is stretched between its two ends. Suppose a Continuous sinusoidal wave of a certain frequency is propagating along the string to the right. When the wave reaches the right end, it will reflect and begins to move towards the left. While moving towards the left end of the sting it must superpose with the wave that is still travelling towards the right. Similarly, after reaching the left end, the left going wave reflects and begins to travel to the right, which results in a superposition with the left and right going waves. In other words, within a very short time we may find many overlapping travelling waves, interfering with each other.
For certain frequency the interference produces a standing wave pattern accurately with nodes and antinodes. Such a standing wave is said to be produced at resonance and the string is said to a resonator at this certain resonant frequency. If the string is oscillated at some frequency other than its resonant frequency, a standing wave is not formed.
Explanation of the Formation of Stationary Waves by Graphical Method
Two identical, but oppositely directed progressive waves superpose in a region of an elastic medium to produce stationary waves. The formation of these stationary waves can be explained graphically.
Consider a progressive wave (wave 1, denoted by blue line) is moving towards right through a medium [Fig.]. Another progressive wave (wave 2, denoted by broken blue line) of the same amplitude, frequency and velocity is moving towards left. The relations T = \(\frac{1}{n}\) and λ = \(\frac{V}{n}\) confirm that their time periods and wavelengths are also equal. When the two waves superpose in the region AE of the medium, the following cases can occur:
i) At the beginning of a period (time, t = 0) the two waves are in opposite phases [Fig.(a)]. So, the resultant displacement is zero for every particle in the medium, i.e. every particle is in its equilibrium position. The graph of the resultant wave is the straight line AE.
ii) During the time t = \(\frac{T}{4}\), the 1st wave covers a distance \(\frac{\lambda}{4}\) towards right; the 2nd wave also travels the same distance towards left. So, at this instant, the two waves are in the same phase [Fig.(b)]. The resultant displacements of the points A, C, E become maximum, but the points B and D remain in equilibrium position. The graph of the resultant wave is shown as a red line.
iii) By the time t = \(\frac{T}{2}\), both the waves have progressed further towards their direction of propagation through a distance so they are again in opposite phases [Fig.(c)]. At this instant, every particle in the medium returns to its equilibrium position. So, the graph of the resultant wave becomes a straight line again.
iv) At time t = \(\frac{3 T}{4}\), the resultant displacement of every particle is equal but opposite to that at time t = \(\frac{T}{4}\) [Fig.(d)]. Here again, the points B and D remain in their equilibrium positions and the resultant displacements of the points A, C, E become maximum. The graph of the resultant wave is again shown as a red line.
v) Finally, at time t = T, each of the progressive waves completes a period and returns to its position as that at time t = 0 [Fig.(a)]. As a result, each particle in this medium is in its equilibrium position again. The graph of the resultant wave again becomes a straight line.
The above discussion, on the superposition of two identical but oppositely directed progressive waves, leads to the following inferences:
- The displacements of particles al B and D are zero at all times, i.e., they are always at rest. These points are called the nodal points. On the other hand, the particles at A, C, E vibrate with maximum amplitudes on the two sides of the equilibrium and are known as the antinodal points.
- The nodes and the antinodes do not travel through the medium. So the resultant wave is a stationary or standing wave.
- The two ends B and D of the portion BD remain stationary and the intermediate points vibrate periodically across the position of equilibrium. So, a loop is formed in the portion BD of the stationary wave.