\(\cfrac{\partial T}{\partial t}=\cfrac{L}{g_T}\cfrac{\partial^3 T}{\partial t^3}\)
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.
