Heat Wave Again

From the post "Temperature",

$\cfrac{\partial T}{\partial t}=\cfrac{L}{g_T}\cfrac{\partial^3 T}{\partial t^3}$

where $L$ is set large but does not change, and $T$ has a dependence on $x$.  ie $T\rightarrow T(x,t)$.  More importantly $T(L,t)=T_f$.  When $T_f=0$, then $L$ is at the limit beyond which the spot hot has no heat effect.

$\int  \partial T=\cfrac {L}{ g_{ T } } \int { \cfrac { \partial^{ 3 }T }{ \partial^{ 3 }t }  } dt$

$T=\cfrac {L }{ g_{ T } } (\cfrac {\partial^{ 2 }T }{ \partial t^{ 2 } } +C(x))$

Once again if we insist that $T$ propagate as a wave at velocity $v$,

$\cfrac { \partial^{ 2 }T }{ \partial t^{ 2 } }={ v }^{ 2 }\cfrac { \partial^{ 2 }T }{ \partial x^{ 2 } }$

From

$T=\cfrac { L }{ g_{ T } } (\cfrac { \partial ^{ 2 }T }{ \partial t^{ 2 } } +C(x))$

If we consider $T(x,t)$ to be of the form $T(x)T(t)$,  and $T(t)$ is without a DC value,

$C(x)=0$

$T(t)T(x)=\cfrac { L }{ g_{ T } } T(x)\cfrac { d^{ 2 }T(t) }{ dt^{ 2 } }$

$T(t)=\cfrac { L }{ g_{ T } } \cfrac { d^{ 2 }T(t) }{ dt^{ 2 } }$

Let $T(t)={ e }^{ \pm iwt }$,

$\cfrac { d^{ 2 }T(t) }{ dt^{ 2 } } ={ w }^{ 2 }T(t)$

$\cfrac { L }{ g_{ T } } =\cfrac { 1 }{ w^{ 2 } }$

$w=\sqrt { \cfrac { { g }_{ T } }{ L }  }$

From

$\cfrac { \partial ^{ 2 }T }{ \partial t^{ 2 } } ={ v }^{ 2 }\cfrac { \partial ^{ 2 }T }{ \partial x^{ 2 } }$

$\cfrac { d ^{ 2 }T(t) }{ d t^{ 2 } }T(x) ={ v }^{ 2 }\cfrac { d ^{ 2 }T(x) }{ d x^{ 2 } }T(t)$

${ w }^{ 2 }T(t)T(x)= { v }^{ 2 }\cfrac { d^{ 2 }T(x) }{ d x^{ 2 } } T(t)$

${ w }^{ 2 }T(x)= { v }^{ 2 }\cfrac { d^{ 2 }T(x) }{ dx^{ 2 } }$

$T(x)=\cfrac { L }{ g_{ T } } v^{ 2 }\cfrac { d^{ 2 }T(x) }{ dx^{ 2 } }$

Let    $T(x)={ e }^{ -iAx }$

$\cfrac { d^{ 2 }T(x) }{ dx^{ 2 } } ={ A }^{ 2 }T(x)$

$A=\cfrac { 1 }{ v }\sqrt{\cfrac { { g }_{ T } }{ L}}$

$T=T(x)T(t)={ e }^{i\left\{-\cfrac { 1 }{ v }\sqrt{\cfrac { { g }_{ T } }{ L}}x\pm \sqrt { \cfrac { { g }_{ T } }{ L }  } t \right\}}$

And in general,

 $T=T_f.{ e }^{ -i(Ax\pm wt )}$,  

where

$w=\sqrt { \cfrac { { g }_{ T } }{ L }  }$    and   $A=\cfrac { 1 }{ v }\sqrt{\cfrac { { g }_{ T } }{ L}}$

$v$ is the speed of propagation, $g_T$ is thermal gravity, $L$ is the point in space at which $T=T_f$

How does this wave look like?

Where    $g_{T1}$    and    $g_{T2}$  are constant around the circle and    $g_{T1}>g_{T2}$ .  $L$ is the distance at which $T=T_f$ constant around the circle.  In this derivation, $g_{T}$ is fixed and $L$ is also fixed both of which occurs as concentric circles around the hot spot.  $ix$ then is also around the same circle.