# Why is there no exact formula for position of maxima in single slit interference?

On this website (http://www.ualberta.ca/~pogosyan/teaching/PHYS_130/FALL_2010/lectures/lect35/lecture35.html) it says that the formula $\sin(\theta_{max}) = (m+\frac{1}{2})\lambda$ derived for the maxima of a single slit interference is only approximate, while the formula for the minima $\sin(\theta_{max}) = m\lambda$ is exact (if we neglect other approximations like interference of parallel light waves). Why are the maxima not just in the middle of two minima?

We can clearly divide a light ray in three parts from which two cancel and thus get the mentioned formula for the maxima. Why is this formula not exact?

• There is no exact formula because that would require the solutions to equations which have both trigonometric and polynomial terms in them. In practice it doesn't matter. If, for some reason, you would need a precise solution, then you could get it numerically, but since it is very hard to determine a maximum experimentally with high precision, that necessity rarely arises. The vanishing of a signal, on the other hand, can be measured extremely well, and every high precision experiment will strive to reduce the actual measurement to the vanishing of a quantity. Oct 29, 2015 at 19:39
• Can you show me where I can find these equations or the derivation of these equations? I'm really interested in where the actual maxima are! Oct 29, 2015 at 22:07

The function describing the interference pattern is $$f(x)=(\sin (x)/x)^2$$, where $$x=\frac{\pi a}{\lambda} \sin(\theta).$$ To find the extremes you need to differentiate and equate to zero. For the minima it is easy, you find the condition $$\sin x=0$$, but for the maxima, you get $$\tan x=x$$ which has to be solved numerically.
• @YiyangZhi yes you are right the actual value of x is $\frac{\pi a}{\lambda} sin(\theta)$where theta depends on the position Nov 10, 2021 at 4:22