It can be shown that the radial distance of a light ray travelling along the z axiz of an optical fiber is described by the following equiation. d2r/dz2=(1/nr)*(dn/dr)
for the case where the index of refraction n=n0(1-ar2) and ar2,,1 find a solution of the differential equiation for r(z).

Your target variable is δf, the magnitude of the difference in frequency between the waves emitted from the sonar device and the waves received by the device after reflecting off the whale. write an expression for δf in terms of the relevant frequencies using the subscript notation introduced above.