10.2.1 Two-Source Interference Pattern

All of us have seen those circular ripples before, in the bathtub, at the swimming pool, or the pond surface on a rainy day. Have you ever noticed the interference patterns?

It is probably easier if you had the ripple tank. The setup includes two dippers motorized to dip in and out of the water periodically to produce two concentric circular waves. When these two waves superpose, an unmistakable interference pattern is formed on the water surface.

Most strikingly, there are lines (on the surface) along which the water is now completely calm (despite being hit by two waves). These are the nodal lines. If you look very carefully, the locations where the maximum amplitude of oscillation occur also form lines (between each pair of nodal lines). These are the antinodal lines.

The easiest way to understand how this pattern is formed is to try to construct the pattern ourselves.

Let’s start by drawing concentric circles centred at two dippers A and B. These circles are meant to represent the wave crests (of the outward propagating waves). So the distance between two circular crests corresponds to one wavelength.

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Next we mark out the points where the crest lines intersect, and join those points up with smooth curves.

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These are the antinodal lines. Because the path difference must be $n\lambda$ at locations where “crest-meets-trough”.

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Now let’s add in the dashed circles that represent the wave troughs. This time, we look for points where a crest line intersects with a trough line. Join these points up with a smooth curve.

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These are the nodal lines. Because the path difference must be $\displaystyle (n+\frac{1}{2})\lambda$ at locations where “crest-meets-trough”.

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As you can see, there is always one nodal line in between two antinodal lines.

Now let’s figure out the details.

The middle antinodal line is called the 0^th order antinodal line (aka $n=0$ antinodal line), since these are locations with path difference of 0λ.

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On either side of the $n=0$ antinodal line, we have the $n=1$ , or 1st order antinodal lines, where the path difference is exactly 1l.

The other antinodal lines further out to the sides are the higher order antinodal lines.

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In between the nth order and (n+1)th order antinodal lines is the (n+1/2)th order nodal line, which joins up all the locations where the path difference is $\displaystyle (n+\frac{1}{2})\lambda$ .

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Voila. We have mapped out the antinodal and nodal lines, and they tally exactly with the pattern that is observed in the ripple tank. How satisfying!