Interference fringes

Parametric equations

Two sources stand a distance $d$ apart. The path difference from a point to the two sources decides, in the formula, whether the waves meet in phase or out of phase. The figure draws the in-phase curves, the bright fringes $\Delta r=n\lambda$. For $n\ne 0$ those curves are hyperbolas:

• d is the distance between the sources. λ is the wavelength and sets how far apart the orders sit. An order is drawn only when $|n|\lambda <d$.
• 2a = nλ is the path difference, with $n$ the order of a bright fringe.
• b = √(c² − a²) and c = d/2. The order $n=0$ is the perpendicular bisector between the sources.

Interaction

• Source distance d: change the distance between the sources. The hyperbolas open or close, the number of orders changes, and the curves grow out again from the middle.
• Wavelength λ: change the spacing of the orders. The curves then grow out again from the middle.
• Time speed ω: advance the phase. The drawn curve oscillates about the fixed curve of that order.

What to notice

• The bright fringes of the formula are the curves $\Delta r=n\lambda$, including the straight order $n=0$ on the perpendicular bisector.
• Source distance d and wavelength λ together decide which orders fit, since an order is drawn only when $|n|\lambda <d$.
• With the sources symmetric, the pattern is symmetric about the perpendicular bisector.

Related

• Superposition of trigonometric functions and wave interference
• Stationary wave
• Chladni figures

Further reading

• Wave interference (Wikipedia)