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

Abstract

The roots of the equation \[ J n 2 ( z ) + J n + 1 2 ( z ) = 0 J_n^2(z) + J_{n + 1}^2(z) = 0 \] , in which n 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 n is not large.