\( g_{ w }=ge^{ iwt }=-g_{ o }e^{i (-\cfrac { x }{ r_{ e } } +wt) }\)
\(ix\) is a circular wave front parallel to the surface of Earth.
\(\cfrac{\partial\,g_{w}}{\partial\,t}=-iwge^{iwt}\)
which makes \(\cfrac{\partial\,g_{w}}{\partial\,t}\) also parallel to the surface of Earth. From which we formulate a gravitation potential term \(GPE\), the rate of change of gravitational potential,
\(\cfrac{d\,GPE}{d\,t}=m\cfrac{\partial\,g_{w}}{\partial\,t}h=-iwmghe^{iwt}\)
where \(h\) is the reference height along \(g\), and \(m\) the mass of a body.
Then we consider the same body in motion on the surface of Earth with speed \(v\),
\(KE=\cfrac{1}{2}mv^2\)
\(\cfrac{d\,KE}{d\,t}=\cfrac{1}{2}m\cfrac{d\,v^2}{d\,t}=mv.a\)
Since both KE and GPE, is in the same direction, because of the term \(i\), we can have resonance when,
\(\cfrac{d\,KE}{d\,t}=\cfrac{d\,GPE}{d\,t}\)
\(\cfrac{1}{2}m\cfrac{d\,v^2}{d\,t}=-iwmghe^{iwt}\)
Since, \(g=\cfrac{d^2\,h}{d\,t^2}\),
\(\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=-iwh\cfrac{d^2\,h}{d\,t^2}e^{iwt}\)
Multiplying \(i\) to both sides,
\(i.\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=wh\cfrac{d^2\,h}{d\,t^2}e^{iwt}\) ---(1)
\(i.v\cfrac{d\,v}{d\,t}=i.\cfrac{d\,x}{d\,t}\cfrac{d^2\,x}{d\,t^2}=wh\cfrac{d^2\,h}{d\,t^2}e^{iwt}\)
Because of the factor \(i\) that rotates \(\cfrac{d\,v^2}{d\,t}\) to the direction of \(h\), \(\cfrac{d\,v^2}{d\,t}\) acts like a driving force along \(h\). Changing the rate of change of \(v^2\) can drive the system (a car travelling perpendicular changing gravitational wave) to resonance.
We know that \(x\) is not oscillating; not a sinusoidal, Therefore, \(h(t)\) must have a sinusoidal time component to cancels the \(e^{iwt}\) factor in the differential equation (1). Consider,
\(h=h_o(t)e^{bt}\) then,
\(h^{'}=h_{ o }^{ ' }(t){ e }^{ bt }+bh_{ o }(t){ e }^{ bt }\)
\(h^{''}=h_{ o }^{ '' }(t){ e }^{ bt }+2bh_{ o }^{ ' }(t){ e }^{ bt }+b^{ 2 }h_{ o }(t){ e }^{ bt }=\left\{ h_{ o }^{ '' }(t)+2bh_{ o }^{ ' }(t)+b^{ 2 }h_{ o }(t) \right\} { e }^{ bt }\)
then,
\(wh\cfrac{d^2\,h}{d\,t^2}e^{iwt}=wh_o(t)\left\{ h_{ o }^{ '' }(t)+2bh_{ o }^{ ' }(t)+b^{ 2 }h_{ o }(t) \right\} { e }^{ 2bt }e^{iwt}\)
when,
\({ e }^{ 2bt }=e^{-iwt}\)
the sinusoidal time component in the differential equation cancels, ie,
\(h=h_o(t)e^{-i(w/2)t}\)
We can simplify further,
\(i.\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=wh\cfrac{d^2\,h}{d\,t^2}e^{iwt} \)
\(i.\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=wh_{ o }(t)\left\{ h_{ o }^{ '' }(t)+2(-i\cfrac { w }{ 2 } )h_{ o }^{ ' }(t)+(-i\cfrac { w }{ 2 } )^{ 2 }h_{ o }(t) \right\} \)
\(i.\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=wh_{ o }(t)\left\{ h_{ o }^{ '' }(t)-iwh_{ o }^{ ' }(t)-\cfrac { w^{ 2 } }{ 4 } h_{ o }(t) \right\} \) ---(*)
\(i.\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}\) is along real \(h\) and \(-iwh_{ o }^{ ' }(t)\) is along \(x\).
Equating the rotated parts,
From (*),
\(\cfrac{1}{2}\cfrac{d\,v^2}{d\,t} =wh_{ o }(t)h_{ o }^{ '' }(t)-\cfrac { w^{ 3 } }{ 4 } h^{ 2 }_{ o }(t)\) ---(2)
and
\(w^2h_{ o }(t)h_{ o }^{ ' }(t)=0 \)
\(\cfrac{1}{2}w^2\cfrac{d}{d\,t}\left\{h_{ o }(t)\right\}^2=0=\cfrac{d\,(-A^2)}{d\,t} \)
where \(-A^2\) is a contant
\(h^2_{ o }(t)=-\cfrac{2A^2}{w^2}\)
\(h_o(t)=i\cfrac{\sqrt{2}A}{w}\)
From (2), since, \(h_{ o }^{ ' }(t)=0\), \(h_{ o }^{ '' }(t)=0\)
\(\cfrac{1}{2}\cfrac{d\,v^2}{d\,t}=v\cfrac{d\,v}{d\,t}=v.a =0-\cfrac { w^{ 3 } }{ 4 }\cfrac{-2A^2}{w^2}\)
\(a=+\cfrac { wA^2 }{ 2v }\) this is the condition for resonance
And,
\(h=i\cfrac{\sqrt{2}A}{w}e^{-i(w/2)t}\)
\(\cfrac{A}{w}\) has the unit of m when \(A\) has the unit of ms-1, a velocity, which is why it is multiplied by the factor \(i\).
\(h=\cfrac{\sqrt{2}A}{w}e^{-i\left\{(w/2)t-\pi/2\right\}}\)
When \(t=0\), at the on set of resonance, considering magnitudes only,
\(h(t=0)=\cfrac{\sqrt{2}A}{w}\)
\(\cfrac{1}{2}mv^2=mgh(0)=mg\cfrac{\sqrt{2}A}{w}\)
\(A=\cfrac{w}{2g\sqrt{2}}v^2\), \(h=\cfrac { v^{ 2 } }{ 2g } e^{-i\left\{(w/2)t-\pi/2\right\}}\)
The condition for a car flip becomes,
\(a=\cfrac { wA^2 }{ 2v}=\cfrac { w\left\{ \cfrac { w }{ 2g\sqrt { 2 } } v^{ 2 } \right\} ^{ 2 } }{ 2v } =\cfrac { w^{ 3 } }{ 16g^{ 2 } } v^{ 3 }\)
So, when the acceleration \(a\) is,
\(a\)=(2*pi*7.489)^3/(16*9.087^2)*\(v^3\)=78.86\(v^3\) in ms-2
we will experience oscillation perpendicular to the direction of travel at a frequency of 7.489/2 = 3.745 Hz. Since, this is an exchange of KE and GPE, the greater the velocity the greater the amplitude of this resonance will be. This often happens at low speed, when the acceleration is pressed on hard.
More importantly and more frequently, this can also happen when,
\(a=-\cfrac { wA^2 }{ 2v }\)
ie. in sudden deceleration, the car can also flip when,
\(a\)=-78.86v^3
Note; With the initial condition of \(h=r_e\) when \(t=0\) at the onset of resonance, the result is a huge number that do not make sense. This is because \(h\) is the change in height not the absolute height.
