Intuitively, consider the displacement long the helical path to be of two components. One component \(\Delta a_l\) to be along the circumference of the orbit and \(\Delta a_t\) to be transverse to the circumference.
\((\Delta a)^2=(\Delta a_l)^2+(\Delta a_t)^2\)
So,
\(\sum{(\Delta a)^2}=\sum{(\Delta a_l)^2}+\sum{(\Delta a_t)^2}\) --- (*)
where the summation is taken once along the orbit. We know that, when \(\Delta a\), \(\Delta a_l\) and \(\Delta a_t\) are small, and all positive,
\(\sum{\Delta a}=L_{or}\)
\(\sum{\Delta a_l}=L_{cir}\)
\(\sum{\Delta a_t}=n2\pi r_p\)
where \(L_{or}\) is the total length of the orbit, \(L_{cir}\) is the circumference of the orbit, \(n\) is the number of turns along the helical path and \(r_p\) is the helical radius.
As,
\(\sum{(\Delta a_l)^2}\propto\sum{\Delta a_l}\) which implies,
\(\sum{(\Delta a_l)^2}\propto L_{cir}\)
and
\(\sum{(\Delta a_t)^2}\propto \sum{\Delta a_t}\) which implies,
\(\sum{(\Delta a_t)^2}\propto n2\pi r_p\)
When \(L_{or}\) is fixed so is \(\sum{(\Delta a)^2}\), increasing \(L_{cir}\) increases \(\sum{\Delta a_l}\) and \(\sum{(\Delta a_l)^2}\) that decreases \(\sum{(\Delta a_t)^2}\) as the expression (*) shows. A decrease in \(\sum{(\Delta a_t)^2}\) in turns decreases \(r_p\) as \(n\) is also fixed. So, an increase in \(L_{cir}\) leads to a decrease in \(r_p\).
\(r_p\propto \cfrac{1}{L_{cir}}\)
This can be verified by stretching a helical spring coil with a fixed number of turns. The small radius of the helix becomes smaller as the length of the coil increases.
Furthermore,
\(r_p\propto \cfrac{1}{f}\)
So,
\(f\propto L_{cir}\)
Frequency increases with increasing orbital circumference/radius. We also have,
\(c=f\lambda_{or}\)
where \(\lambda_c\) has been redefined as the distance traveled along the circumference of the orbit after a time, \(T_p\).
\(\lambda_{or}\propto \cfrac{1}{L_{cir}}\)
This wavelength decreases with increasing orbital circumference/radius.
The period, \(T_p\) was previously taken to be \(\cfrac{1}{f}\), the reciprocal of the frequency of the particle wave, \(\psi\) formulated in the posts "Maybe Not", "Still Not" and "Particle Wave Duality, Seriously MP". This is the familiar frequency, \(f\) we associate with wave in a straight path,
\(f=\cfrac{1}{T_{p}}\) where \(f\lambda=c\)
The following diagram shows how a frequency component, \(f\) might develop on the passing of a particle in helical motion. From the post "Double-Bell, Balls",
The constant component, \(B_c\),
\(B_c=\left( \cfrac { \mu_omc^{ 2 } }{ 6\pi } \right) ^{ 1/2 }\cfrac { 1 }{ x^{ 3/2 } }\)
where \(x\) is measured from the center of the helical circular motion.
\(B=\left( \cfrac { \mu _{ o }mc^{ 2 } }{ 6\pi } \right) ^{ 1/2 }\left\{ \cfrac { 1 }{ (x+r_{ p }e^{ i2\pi ft })^{ 3/2 } } \right\} \)
\(B=\left( \cfrac { \mu _{ o }mc^{ 2 } }{ 6\pi } \right) ^{ 1/2 }\left\{ \cfrac { 1 }{ x^{ 3/2 }(1+\cfrac { r_{ p } }{ x } e^{ i2\pi ft })^{ 3/2 } } \right\} =B_{ c }\cfrac { 1 }{ (1+\cfrac { r_{ p } }{ x } e^{ i2\pi ft })^{ 3/2 } } \)
where \(\cfrac{r_p}{x}\lt1\) and we use a binomial expansion,
\(\cfrac{1}{(1+x)^{3/2}}=1-\cfrac { 3 }{ 2 } x+\cfrac { 15 }{ 8 } x^{ 2 }-\cfrac { 35 }{ 16 } x^{ 3 }+\cfrac { 315 }{ 128 } x^{ 4 }+O(x^5)\)
We obtain,
\(B=B_c\left\{1-\cfrac { 3 }{ 2 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) +\cfrac { 15 }{ 8 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) ^{ 2 }-\cfrac { 35 }{ 16 } \left( \cfrac { r_{ p } }{ x } e^{ i2\pi ft } \right) ^{ 3 }+...\right\}\)
Surprisingly, we find that there are other frequency components in the measured frequency,
\(f_m=f+2f+3f+...nf+...\)
\(n=1,2,3,...\)
where \(f_m\) is the measured frequency in \(B\) as the wave passes and \(f\), the frequency of the helical circular motion which is also the frequency of the wave. The harmonic frequencies decrease with an envelop of,
\(env=\cfrac { 1 }{ n! } \cfrac {- 3 }{ 2 } (\cfrac { -3 }{ 2 } -1)(\cfrac { -3 }{ 2 } -2)..(\cfrac {- 3 }{ 2 } -(n-1))(\cfrac{r_p}{x})^n\)
where \(n-1\) the nth harmonic and \(n=1\) being the fundamental at \(f\). Given \(r_p\lt x\), at high values of \(x\) only the fundamental component, \(f\) is prominent.
However, in this derivation, the wave bent into a torus need not have an integer multiple of turns or equivalently the length of the orbit be of an integer multiple of wavelength, \(\lambda_{or}\).
\(2\pi r_{or}\ne n_z\lambda_{or}\)
where \(n_z=1,2,3...\) and \(r_{or}\) is the orbital radius,
At any cross-section of the torus, the phase of the wave can change on each pass. There is simply no constrain that the phase of the wave along its orbit be fixed; that the wave be a standing wave. As long as \(n\) is a constant, not necessarily an integer we have,
\(f\propto{L_{cir}}\)
What about the standing wave postulated by Neil Bohr? What is the constraint necessary?
