What are relativistic effects on classical sine wave?

I am also interested and now studying SR and GR. From my viewpoint, in the ideal case(ignore the effect of gravity), the vibration equation of a string is $$begin{equation} begin{array}{ll} frac{partial^{2} u}{partial t^{2}}-a^{2} frac{partial^{2} u}{partial x^{2}}=0, & -infty<x<infty, t>0 left.u(x, t)right|_{t=0}=phi(x), & -infty<x<infty left.frac{partial u}{partial t}right|_{t=0}=psi(x), & -infty<x<infty end{array} end{equation}$$ notice that in equation there is $a^2$, where $a$ is the speed of wave, which implies that the information of a certain point in a string can not reach arbitrary point with infinite speed,to be specific,one can get the solution of $$begin{equation} u(x, t) =frac{1}{2}[phi(x-a t)+phi(x+a t)]+frac{1}{2 a} int_{x-a t}^{x+a t} psi(xi) mathrm{d} xi . end{equation}$$

$phi(x)$is the initial excited wave at t=0. the solution shows that vibration at x=x0 at t=0, then it will only effect the region:$[x-at,x+at]$ symmetrically. if you restric that the speed within the speed of light, then it obeys the SR.