On the computation of zeroes of $J_n(z)-iJ_{n+1}(z)=0$

The roots of the equation

$$J_n^2(z) + J_{n + 1}^2(z) = 0$$

, in which $n$ is a positive integer or zero, are of interest to the specialist in wave reflection from multi-sloped beaches [1]. This note shows how to obtain accurate roots of the equation when $n$ is not large.