# Hilbert Book Model Project/Slide F6

The solutions of ${\displaystyle {\color {white}\boxdot \ \psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi =0}}$
 We focus on warp shock fronts of the form ${\displaystyle {\color {white}{\vec {\psi }}=f_{r}(c\tau -{\vec {r}})\,{\vec {i}}}}$ The real scalar ${\displaystyle {\color {white}c}}$ equals the speed at which the shock front travels The vector ${\displaystyle {\color {white}{\vec {r}}}}$ is the distance from the trigger location to the shock front Vector ${\displaystyle {\color {white}{\vec {i}}}}$ represents the amplitude of the front, which directs perpendicular to ${\displaystyle {\color {white}{\vec {r}}}}$ The function ${\displaystyle {\color {white}f_{r}(q)}}$ is a real function that describes the shape of the front in the direction of ${\displaystyle {\color {white}{\vec {r}}}}$ In the string the direction of vector ${\displaystyle {\color {white}{\vec {i}}}}$ may rotate as a function of the sequence number of the warp. This gives the string a circular polarization Circular polarization of warp string. The arrows represent the amplitudes of the warps that constitute the string. The red line is a help line. No wave is involved!