Preface to the lecture, 1

568

 

Derivation of the wave equation

 

•  Starting-point: the fundamental field equation

 

 

•  with a magnetic flux density    B = B(r(t)).

 

1

st

 condition   for eq. 27.26*: 

the special case, if  =0     and    

.    (27.16*)

 

The remaining vortex term is transformed by applying already 

used relations  (eq. 27.10    and eq. 27.17):

 

 

If the velocity of propagation:   v = (v

x

, v

y

=0, v

z

=0) ;   v =  dx/dt  , 

then the simplified field equation (if the coordinates are orientated 

on the vector of velocity) results in the general wave equation 

(involved with the x-component) in the form:

 

 

2

nd

 condition   for eq. 27.28: 

v = c     .

 

The wave equation in the usual notation (= inhomogeneous 

Laplace equation,  = purely a special case!) now reads:

 

 

(27.28)

 

Fig. 27.13: Derivation of the wave equations 

(inhomogeneous 

Laplace equation) as a special case of the equations 

of transformation of the electromagnetic field. 

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Faraday versus Maxwell ________________________________________________ 569

 

27.13 Derivation of the wave equation

 

The first wave description, model for the light theory of Maxwell, was the inhomogeneous 

Laplace equation:

 

 

There are asked some questions:

 

•

 

Can also this mathematical wave description be derived from the new approach? 

•

 

Is it only a special case and how do the boundary conditions read? 

•

 

In this case how should it be interpreted physically? 

•

 

Are new properties present, which can lead to new technologies? 

Starting-point is the fundamental field equation (27.26). We thereby should remember the 

interchangeability of the field pointers, that the equation doesn't change its form, if it is 

derived for H, for B, for D or any other field factor instead of for the E-field pointer. This 

time we write it down for the magnetic induction B and consider the special case, that we 

are located in a badly conducting medium, as is usual for the wave propagation in air. But 

with the electric conductivity also 

 tends towards zero (eq. 27.16*). With that

 

the eddy currents and their damping and other properties disappear from the field 

equation, what also makes sense. There remains the potential vortex term (1/ )*dB/dt , 

which using the already introduced relations (eq. 27.10 and 27.17) involved with an in x- 

direction propagating wave (v = (v

x

, v

y

 = 0, v

z

 = 0)) can be transformed directly into:

 

 

The divergence of a field vector (here B) mathematically seen is a scalar, for which reason 

this term as part of the wave equation founds so-called ,,scalar waves" and that means that 

potential vortices, as far as they exist, will appear as a scalar wave. We at this point tacitly 

anticipate chapter 28, which provides the reason for the speed of light losing its vectorial 

nature, if it is correlated with itself. This insight however is valid in general for all 

velocities (v = dr/dt), so that in the same way a scalar descriptive factor can be used for 

the velocity (v = dx/dt) as for c.

 

From the simplified field equation (27.26*) the general wave equation (27.27) can be won 

in the shown way, divided into longitudinal and transverse wave parts, which however can 

propagate with different velocity.

 

Physically seen the vortices have particle nature as a consequence of their structure 

forming property. With that they carry momentum, which puts them in a position to form 

a longitudinal shock wave similar to a sound wave. If the propagation of the light one time 

takes place as a wave and another time as a particle, then this simply and solely is a 

consequence of the wave equation. Light quanta should be interpreted as evidence for the 

existence of scalar waves. Here however also occurs the restriction that light always 

propagates with the speed of light. It concerns the special case v = c. With that the derived 

wave equation (27.27) changes into the inhomogeneous Laplace equation (27.28). 

The electromagnetic wave in both cases is propagating with c. As a transverse wave the 

field vectors are standing perpendicular to the direction of propagation. The velocity of 

propagation therefore is decoupled and constant. Completely different is the case for the 

longitudinal wave. Here the propagation takes place in the direction of an oscillating field 

pointer, so that the phase velocity permanently is changing and merely an average group 

velocity can be given for the propagation. There exists no restriction for v and v = c only 

describes a special case.

 

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570

 

The new field approach in synopsis

 

 

•  From the dual field-

 

From Maxwell's field 

equations can be

theoretical approach

 

are derived:

 

derived:

 

=> Maxwell's

field equations

 

=> 0 

=> the wave equation

 

(with transverse and

longitudinal parts) 

=> only transverse waves

 

(no longitudinal waves) 

=> scalar waves

 

(Tesla-/neutrino radiation) 

=> 0 

(no scalar waves) 

=> vortex and anti-vortex

 

(current eddy and potential

vortex ) 

=> only eddy currents

 

=> Schrodinger equation

 

(basic equation of chemistry) 

=>0

 

=> Klein-Gordon equation

 

(basic eq. of nuclear physics) 

 

=>0

 

Fig. 27.14:   Comparison of the efficiency of both approaches. 

(as  an interim result, if it concerns the question, which 

approach of the two is the more efficient one and which one 

better should be discarded. The final balance is made in 

chapter 28). 

It here concerns partial aspects of the following theories: 

=> theory of objectivity

 

=> theory of relativity

 

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