Oscillations and waves

Interference of mechanical waves

Coherent waves superpose to form stable reinforcement and cancellation; for in-phase sources, path difference determines the pattern.

Drop two pebbles into water at once: their circular ripples meet. Crest meeting crest gives a large oscillation; crest meeting trough can cancel.

Delta r=r_2-r_1;quad ext{maxima: }Delta r=mlambda,qquad ext{minima: }Delta r=(m+ frac12)lambda

Definition: Interference conditions

Coherent sources have the same frequency and a constant phase difference. For in-phase sources, maxima occur at $Delta r=mlambda$ and minima at $Delta r=(m+1/2)lambda$ (integer $m$ ). An initial phase offset modifies these conditions.

See circular ripples superpose with an amplitude map; bright and dark regions indicate strong and weak oscillation.

Maxima and minima

Loci of constant path difference are hyperbolas in a plane. On the perpendicular bisector, $r_1=r_2$ , so in-phase sources produce a central maximum. The resultant amplitude varies with position; energy is redistributed rather than destroyed at nodes.

Example: Example

Two in-phase sources are 0.30 m apart and emit waves with $lambda=0.10$ m. Is point P, with path difference 0.20 m, a maximum or minimum?

Solution

$Delta r/lambda=0.20/0.10=2$ , an integer, so P is a maximum.

Quick check

A stable interference pattern requires sources of the same frequency and constant phase difference. For equal component amplitudes, the resultant amplitude at a maximum is twice that of either wave; because intensity is proportional to amplitude squared, it is four times the single-source intensity in the ideal model. At a minimum, opposite-phase oscillations cancel.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.

For two in-phase sources, what kind of fringe occurs at $Delta r=3lambda/2$ ?

What is needed for a stable interference pattern?

References

Young, Freedman (2019). University Physics