As I had stated at the beginning of this post my skills in mathematics are from poor to non-existent. However, by careful examination of the equations I deemed to be relevant to my theory, I have selected a few possible openings:-
A wave can be described just like a field, namely as a function
F(x,t) where
x is a position and
t is a time. The value of
x is a point of space, specifically in the region where the wave is defined. In mathematical terms, it is usually a vector in the Cartesian three-dimensional space
R3.
The value of
F(x,t) can be any physical quantity of interest assigned to the point
x that may vary with time. For example, if
F represents the vibrations inside an elastic solid (e.g., the aether). The value of
F(x,t) is usually a vector that gives the current displacement from
x of the material particles that would be at the point
x in the absence of vibration.
In preparing a mathematical definition of my new theory of physics, I felt that a good place to start would be with a spherical wave in an
infinite three-dimensional medium. This is a wave that is independent of the angles
θ and
μ when it is described in terms of spherical coordinates. The wave function depends on
t and
r only, and it moves either toward larger values of
r --- an outgoing wave --- or toward smaller values of an incoming wave .
Writing the Laplacian operator in spherical coordinates. In spherical coordinates the Laplacian operator is given by:
∇2f=r21∂r∂(r2∂r∂f)+r2sinθ1∂θ∂(sinθ∂θ∂f)+r2sin2θ1∂ϕ2∂2f
Writing the Laplacian operator in spherical coordinates, and eliminating the angular derivatives, the wave equation becomes:
\(\displaystyle {\displaystyle 0 = \Box \psi = -\frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} +\frac{1}{r^2}\frac{\partial}{\partial r} \biggl( r^2 \frac{\partial
\psi}{\partial r} \biggr)}\)
∂ψ/∂r=r−1∂u/∂r−r−2u
so that
r2∂ψ/∂r=r∂u/∂r−u
After differentiation
∂r∂(r2∂r∂ψ)=r∂r2∂2u+∂r∂u−∂r∂u=r∂r2∂2u
Gives
r21∂r∂(r2∂r∂ψ)=r1∂r2∂2u
for
∇2ψ in spherical coordinates. On the other hand, differentiation with respect to
t
yields
∂t2∂2ψ=r1∂t2∂2u
and this gives
\(\displaystyle {\displaystyle \Box \psi = \frac{1}{r} \biggl( -\frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} + \frac{\partial^2
u}{\partial r^2} \biggr)}\)
for the wave operator acting on
ψ(t,r)
−v21∂t2∂2u+∂r2∂2u
This is recognized as the one-dimensional form of the wave equation, presented in a different notation, with {\displaystyle r}. The general solution to this equation is
ψ(t,r)=r1A(r−vt)+r1B(r+vt)
This represents the superposition of an outgoing wave described by
r−1A(r−vt) and an incoming wave described by
r−1B(r+vt).The functions
A and
B are arbitrary. Just as
in the case of the infinite string, and they can be the initial conditions imposed on the wave. The factor of
r in front of
A(r−vt) indicates that the wave's amplitude decreases as the wave expands toward
r=∞, the wave becomes increasingly attenuated, a familiar phenomenon in the context of sound waves. On the other hand, the factor of
r−1in front of
B(r+vt)reveals that the amplitude increases as the wave converges toward
r=0 .
While this concept is useful, it seems to lack in certain areas. For instance according to my theory, the ‘real’ photon travels through the ‘virtual’ photons of the aether, spreading our according to the inverse square law as it travels and transferring all of its energy to ‘virtual photons’ with which it comes into contact and being replaced by ‘real’ photon from the line or ray of aligned photons that the electron is emitting. . I therefore thought a mathematical description involving vector propagation might be more useful.
A perfect one-dimensional traveling wave follows the equation:
ψ(x,t)=Acos(kx−ωt+φ)ψ(x,t)=Acos(kx−ωt+φ)
where:
1)
x is position,
2)
t is time,
3)
ψ (a function of
xandt is the disturbance describing the wave (for example, for an ocean wave, {tex]{\displaystyle \psi }[/tex] would be the excess height of the water, or for a sound wave,
ψ would be the excess air pressure).
4)
A is the amplitude of the wave (the peak magnitude of the oscillation),
5)
φ is a phase offset describing how two waves can be out of sync with each other
6)
ω is the temporal angular frequency of the wave, describing how many oscillations it completes per unit of time, and related to the period
T by the equation
ω=2π/T,
7)
k is the spatial angular frequency (wavenumber) of the wave, describing how many oscillations it completes per unit of space, and related to the wavelength by the equation
k=2π/λ .
8)
k is the magnitude of the wave vector. In this one-dimensional example, the direction of the wave vector is trivial: this wave travels in the
+x direction with speed (more specifically, phase velocity) \(\displaystyle {\displaystyle \(\displaystyle {displaystyle\omega /k}\). In a multidimensional system, the scalar \(\displaystyle {\displaystyle kx}\) would be replaced by the vector dot product \(\displaystyle {\displaystyle {\mathbf {k} }\cdot {\mathbf {r} }}\), representing the wave vector and the position vector, respectively.
Still working on it……….\)
