  Perturbations of a class of semiclassical strings known today as spiky
strings, are studied using the well-known Jacobi equations for small normal
deformations of an embedded timelike surface. It is shown that there exists
finite normal perturbations of the spiky string worldsheets embedded in a $2+1$
dimensional flat spacetime. Such perturbations lead to a rounding off of the
spikes, which, in a way, demonstrates the stable nature of the unperturbed
worldsheet. The same features appear for the dual spiky string solution and in
the spiky as well as their dual solutions in $3+1$ dimensional flat spacetime.
Our results are based on exact solutions of the corresponding Jacobi equations
which we obtain and use while constructing the profiles of the perturbed
configurations.
