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Coherence and Path Difference



Two-point source interference occurs when waves from one source meet up with waves from another source. If the source of waves produces circular waves, then the circular wavefronts will meet within the medium to produce a pattern. The pattern is characterized by a collection of nodes and antinodes that lie along nearly straight lines referred to as antinodal lines and nodal lines. If the wave sources have identical frequencies, then there will be an antinodal line in the exact center of the pattern and an alternating series of nodal and antinodal lines to the left and the right of the central antinodal line.

each line in the pattern is assigned a name (e.g., first antinodal line) and an order number (represented by the symbol m). A representative two-point source interference pattern with accompanying order numbers (m values) is shown below.



When waves are coherent and have a path difference that is a multiple of λ, then the interference is constructive.  However, if coherent waves have travelled n λ + λ/2, the interference results is 100% destructive.


To begin, consider the pattern shown in the animation below. Point A is a point located on the first antinodal line. This specific antinode is formed as the result of the interference of a crest from Source 1 (S1) meeting up with a crest from Source 2 (S2). The two wave crests are taking two different paths to the same location to constructively interfere to form the antinodal point.



The crest traveling from Source 1 (S1) travels a distance equivalent to 5 full waves; that is, point A is a distance of 5 wavelengths from Source 1 (S1). The crest traveling from Source 2 (S2) travels a distance equivalent to 6 full waves; point A is a distance of 6 wavelengths from Source 2 (S2). While the two wave crests are traveling a different distance from their sources, they meet at point A in such a way that a crest meets a crest. For this specific location on the pattern, the difference in distance traveled (known as the path difference and abbreviated as PD) is

PD = | S1A - S2A | = | 5λ - 6λ | = 1λ

 (Note the path difference or PD is the difference in distance traveled by the two waves from their respective sources to a given point on the pattern.)

For point A on the first antinodal line (m =1), the path difference is equivalent to 1 wavelength. But will all points on the first antinodal line have a path difference equivalent to 1 wavelength? And if all points on the first antinodal line have a path difference of 1 wavelength, then will all points on the second antinodal line have a path difference of 2 wavelengths? And what about the third antinodal line? And what about the nodal lines? These questions are investigated in the diagrams below through the analysis of the path difference for other points located on antinodal and nodal lines.

Point B in the diagram below is also located on the first antinodal line. The point is formed as a wave crest travels a distance of 3 wavelengths from point S1 and meets with a second wave crest that travels a distance 4 wavelengths from S2. The difference in distance traveled by the two waves from their sources to point B is

PD = | S1B - S2B | = | 3λ - 4λ | = 1λ




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