  We study dispersion properties of linear surface gravity waves propagating in
an arbitrary direction atop a current profile of depth-varying magnitude using
a piecewise linear approximation, and develop a robust numerical framework for
practical calculation. The method has been much used in the past for the case
of waves propagating along the same axis as the background current, and we
herein extend and apply it to problems with an arbitrary angle between the wave
propagation and current directions. Being valid for all wavelengths without
loss of accuracy, the scheme is particularly well suited to solve problems
involving a broad range of wave vectors, such as ship waves and Cauchy-Poisson
initial value problems for example. We examine the group and phase velocities
over different wavelength regimes and current profiles, highlighting
characteristics due to the depth-variable vorticity. We show an example
application to ship waves on an arbitrary current profile, and demonstrate
qualitative differences in the wake patterns between concave down and concave
up profiles when compared to a constant shear profile with equal depth-averaged
vorticity. We also discuss the nature of additional solutions to the dispersion
relation when using the piecewise-linear model. These are vorticity waves,
drifting vortical structures which are artifacts of the piecewise model. They
are absent for a smooth profile and are spurious in the present context.
