# Hilbert Book Model Project/Slide F3

hbmp
Hilbert Book Model Project
F3

The quaternionic field equations offer two second order partial differential equations

${\color {white}\chi =\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=}$ ${\color {white}\boxdot \ \psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$ This equation splits into two first order partial differential equations

${\color {white}\chi =\nabla ^{*}\varphi }$ ${\color {white}\varphi =\nabla \psi }$ ${\color {white}\zeta =\Box \ \psi =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }$ is the quaternionic wave equation

${\color {white}\Box }$ is the quaternionic version of d'Alembert's operator

The homogeneous versions of both second order differential equations
offer shock fronts as part of their solutions
These shock fronts occur in odd numbers of participating dimensions
Separate shock fronts cannot be observed

The homogeneous d'Alembert's equation offers waves as part of its solutions

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