nature of coherent superimposed waves

Introduction

The concept of interference has been widely explored in physics, particularly wave physics that is, sound, optics and telecommunication physics. The term interference is used to describe a phenomenon where two waves of the same wavelength and amplitude are superimposed to form a single wave (Meyers, 2002). In most cases, wave interference can be likened to waves in water. In this description, when two waves in an ocean collide, the size of the waves either diminishes or enlarges in size. Similarly, in wave theory, it is expected that when two waves are superimposed, the amplitude either decreases or increases. However, for interference to result in a different wave pedigree, the waves must either be coherent or correlated. Coherent waves are two waves that have same period or frequency while correlated waves are waves from the same or similar source. Surface water is one of the best laboratory methods that have been employed in the past for explain wave interference.

General Interference Theorem

The concept of wave interference is based on a single principle.  The principle that defines the nature of coherent superimposed waves argues that point total displacement of two or more superimposed waves can be found by summing vector displacement of each wave (Zurek, 2003). That is to say, the total displacement of superimposed waves at a reference point can be found by summing individual vector displacements of each wave.

Therefore, if the crests or troughs of two coherent waves were to meet at a particular point, the resultant wave will be the sum of the two waves. Thus the new wave is expected to have higher amplitude and thus a higher energy. For the case of sound, the wave is expected to have higher volume and for the case of light the intensity of the light is expected to increase. In such a case, where the interference results in a bigger wave or larger displacement, the interference is referred to as constructive interference ( The Physics Classroom, 2012). In general therefore, a constructive interference occurs when the phase difference between two sinusoidal waves is either 2π or multiples of 2π.

On the other hand, if the crest of one wave was to meet the trough of the other coherent wave, the resulting waves is one which has no displacement. This is due to the fact that the displacement of the trough and that of the crest are opposite, thus when they are superimposed, the displacement cancel. Such an interference that results in reduced displacement due to superimposed trough and the crest is referred to as destructive wave interference. In general, a destructive interference exists when the phase difference between two sinusoidal waves is either π, 3π, 5π, 7π or odd multiples of π (Born & Wolf, 1999).

The two kinds of interference can be illustrated a shown below.

 

Interference of Light

Light, just as other waves, light waves experience interference and the results may either be destructive of constructive. As earlier mentioned, general wave interference theorem requires that the two waves have similar frequencies. Just as other waves, there are conditions that the two light waves must meet in order to have interference. First, the sources of the light waves must be monochromatic or coherent. This implies that two sources of light must emit light of same frequency. That is, the two different light waves must have same period and wavelength and phase coherence ( The Physics Classroom, 2012).

A second condition that must be attained is that the two sources of light must be very close to each other.  This allows the light waves to travel and superimpose within an expected region. At specified location, the path difference between the two waves can be ascertained and the excepted superimposed nature reviewed. At points where the path difference is an even multiple of λ/2, the expected interference is constructive interference (Zurek, 2003). On the other hand, a destructive interference is expected when the wavelengths difference is or odd intergers of λ (Zurek, 2003). In light interference, the reference to the distance from the monochromatic light source is due to the fact that the two waves have no phase difference. Thus the concept of interference is based on wavelength and distance from the source.

The third condition, which in a sense reinforces the second condition, is that the phase difference between the two sources should be zero. The two light waves that are expected to be superimposed should not have any phase difference as different fringes in light waves would not be identifiable.

However, most natural sources of light are not coherent and in most cases exhibit a high degree of polychromatic nature. Therefore, to obtain different monochromatic light waves is a rare fete to achieve. This is due to the fact that most light waves lie in the range of 10 15 Hz. Such high frequencies make it difficult for artificial generation of monochromatic light. Therefore ensuring that the sources of light are monochromatic may not be possible.

Experiment: the Young’s Double Slit Experiment

While attaining two sources of monochromatic light might be practically impossible, Thomas Young the great English physicists identified an experiment that would eliminate the need for a two coherent sources. The idea is to have a single source of light shining through two slits on an opaque surface. The experiment considers the distances in the experiment to be very significant.  That is, the distance between the two slits, d, and the distance between the slits and the opaque screen, D, should be significantly large (D>>d).

 

 

 

In the experiment above, the two slits behave like two monochromatic sources of light with their light wave superimposed on the opaque screen.  The result is fringe of dark and regions on the screen.  To locate a fringe on the screen, the angle q, and therefore, slit separation D and wavelength l can be used to determine the formula for the bright fringes in the screen

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According to the diagram, l1 is the distant travelled by top ray to reach at the screen, while l2 is the distance travelled by bottom-most ray to the reach the screen. The difference between the two distances has been identified as x (l1– l2= x) in the small triangle. According to Born & Wolf, (1999) the result of the experiment is:

Bright fringes are represented by

And dark fringes are identified by

Diffraction

A different form of light interference is diffraction, a phenomenon where by light waves bend upon encountering a different medium.  The most common practical example of diffraction is evident when light is seen underwater.  Diffraction is said to occur as interference due to relative changes in the medium with respect to wavelength of the light wave.  Similarly, light entering a lens is said to diffract at therefore bends light with regard to the nature of the lens.

 

 

 

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