Basic Definitions
Figure 2.1
Figure 2.2
Figure 2.3 Refer to Fig. 2.1 for the following definitions. Phase
Line: a continuous line division of space.
Cross Phase Field: the crossing of two or more phase fields. Wave dimensions and wave shapes are seen in Fig. 2.2 where w is 1/2 the basic wavelength W, and a is its amplitude  the three basic wave shapes assumed are the common wave A, the line wave B, and the concave wave C. Common Wave Space (CWS) is defined by a curvilinear coordinate system. It is a wave gnid pattern formed using an X or Y phase field orientation with one field a function of the other. This is what I call a functional geometric system and should not be confused with an overlay system where one phase field is randomly overlaid on another. Interphase Wave Space (IWS) is a wave pattern formed by plotting a second generation wave field on a CWS grid. Wave
Space Equation
where each
term is drawn sequentially, Y_{1} and X_{1}
indicate the primary wave field directions, the X_{1} field
being a function of the Y_{1} field to form a CWS grid configuration.
X_{2} is the secondary wave field taken as a function of, and interphasing
with the CWS coordinate grid. Interphasing occurs when one field track
intercepts another, going in the same direction, and returns without
crossing. The shape formed, I call a nodal
element, or node e_{n}. The other elemental shape
which is formed by two or more phase tracks crossing
each other, I call a common element e_{c},
as seen in Fig. 2.3. CWS is made of totally common elements;
IWS is the summation of common and nodal elements in a proportion dependent
on the interphasing geometries.
Figure 2.4 Common Wave Space (CWS)
Figure 2.5 Interphase Wave Space (IWS) An example of CWS is seen in Fig.2.4 where w_{Y}_{1}=6, a_{Y}_{1}=1, and w_{X}_{1}=6, a_{X}_{1}=2, or S_{W}=G(Y_{1}6/1X_{1}6/2), written G(Y_{1}6X_{1}6/2). An example of IWS is seen in Fig. 2.5 where the X_{2} field of w_{X}_{2}=4, a_{X}_{2}=2 interphases with the CWS coordinate system of Fig. 2.4 in the X_{1 }direction, the third term of S_{w}, The pq superscript in S_{w} indicates the phase position of the secondary wave on the primary wave and determines the rhythm of the interphase field as shown in Fig. 2.6.There are W possible starting positions on wavelength W; in our example, 7 out of 12 positions are shown in aG(X_{1}6) wave field interacting with a secondary X_{2}6 wave field. In Fig. 2.4 there are also 12 possible positions:W=2w_{X}_{1}=2×6=12. In example, Fig. 2.5, q=0 was chosen to establish the secondary X_{2}4/2 interphase wave field.Also note in the equation G[Y_{1}6(X_{1}6/2X_{2}4/2)] of Fig. 2.5 that the parenthetic bracketing indicates what is shown, and will be so in subsequent equations.
Figure 2.6 Phase Positions
Figure 2.7 Mock Axes The fourth
term of S_{w}, is what I call a mock
axes coordinate system; in other words, a IWS configuration used
as CWS grid. This is done by choosing an axes intersection point, i.e.,
an origin, with one coordinate starting in the primary field direction,
the other starting in the secondary field direction and at each node alternating
to the other field direction, giving the continuous, divergent x,y
mock axes X_{m} and Y_{m} as exemplified
in Fig. 2.7. (Notice X_{m} is drawn from
the upper left to lower right and Y_{m} from the lower left
to upper right). The "layered" position of X_{m} and Y_{m}
of S_{W}, means that an X_{m} interphase
field, orY_{m} interphase field,
or
both simultaneously can be plotted. Fig.2.7 shows the
single interphase track X_{m}2.
Figure 2.8 CWS
Figure 2.9 CWS
Figure 2.10 CWS
Figure 2.11 CWS The above examples are given to further demonstrate how Wave Space works. CWS Figs. 2.8, 2.9, 2.10, 2.11 show a progression in wave dynamics holding a_{X}_{1}, and Y_{1}2 constant with w_{X}_{1}, going from infinity to 2. Isolating
the mock axis field
X_{m} of Fig. 2.5,
we have Fig. 2.12. The semicolon in equations such
as X_{m}: G(...) and Y_{m}: G(...)
simply means X_{m} or Y_{m} of the specific
geometry G(...). Fig. 2.13 is an extension ofFig.
2.5 space in the mock
X axis direction by
X_{m}2.
Figure 2.12 IWS
Figure 2.13 IWS Figs. 2.14 and 2.15 is Fig. 2.13 separated into its X_{m} and Y_{m} phase fields. Fig. 2.16 is formed by extending Fig. 2.5 in both mock axes directions simultaneously.
Figure 2.14 IWS
Figure 2.15 IWS Figure 2.16 IWS Variation is further increased if we consider the shape of the wave w, or wavetrain W_{T}. Wave shape A was used in the previous examples. The other two basic shapes B and C, Fig. 2.3, should be noted. Figs. 2.17 and 2.18 show the B and C waves in the Fig. 2.4 CWS geometry; Figs. 2.19 and 2.20 show the B and C waves in the Fig. 2.5 IWS geometry. The symbol above the wave dimensions indicates the wave type and its orientation: ~ indicates a full wave; Ç È indicates a half wave oriented apex up or down respectively. Figure 2.17 CWS B Wave Figure 2.18 CWS C Wave Figure 2.19 IWS B Wave Figure 2.20 IWS C Wave Another approach is the compounding of wavelengths and/or wave shapes as seen in Figs. 2.21 and 2.22 in CWS, with a more complex CWS version in Fig. 2.23 and its IWS extension, Fig. 2.24. The various squiggles, as usedabove, show the specific makeup of a wave shape or wave train. Figure 2.21 CWS Compound Length Figure 2.22 CWS Compound Shape Figure 2.23 Compound CWS Figure 2.24 Compound M7S
