GEOMETRICAL QUATERNIONIC COUPLING FOR THREE DIMENSIONAL WAVE EQUATIONS

The present work has the scope to show the relationship between four three-dimensional waves. This fact will be made in the form of coupling, using for it the Cauchy-Riemann conditions for quaternionic functions [1], through certain Laplace’s equation in [2]. The coupling will relate those functions that determine the wave as well as their respective propagation speeds. AMS Subject Classification: 30G99, 30E99


Introduction
The study of the three-dimensional wave patterns in major physical problems Received: December 13, 2013 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author is such that the coupling of these waves can bring physical insights not yet detected.Let us consider that waves propagate with different velocities v 1 , v 2 , v 3 and v 4 , and wave function given by F 1 , F 2 , F 3 and F 4 .
To fix ideas, it will be considered a quaternionic function denoted by F (q) with q = t + xi + yj + zk, and given by F (q) = F 1 + F 2 i + F 3 j + F 4 k where the functions F 1 , F 2 , F 3 and F 4 , are functions of the variables t, x, y and z.The following theorem establishes the Cauchy-Riemann conditions for quaternionic functions: Theorem 1.For any pair pontis a and b and any path joining them simply conect subdomain of the four-dimmensional space, the integral b a f dq is independent form the given path if and only if there is a function , and satisfying the following relations: Proof.The proof of this theorem can be analyzed in greater detail in [1].
What will be done now is the derivation of each of the above relations on each one of the variables of the problem, t, x, y and z.Then: (5) Thus, the following equations are obtained: and

Wave Equations
The wave equations presents in Mathematical Physics are in the format: for the one-dimensional case.The three-dimensional case, is written as: where c is the speed of wave propagation, or

Coupling Equations
Considering the equations: and Assuming that the set of equations described in (9), (10), ( 11) and ( 12) have no physical sense, that will be reached by considering the transformation below: where F i is function of variables t ′ , x, y and z.The transformation (19) makes the set of equations ( 5), ( 6), ( 7) and (8) to be rewritten as follows:

Concluding Remarks
Taking only the equations that depend on time in the sets of equations ( 20) -( 23), follows that: ∂t ′ ∂z obtaining the equation: Proceeding similarly, we have that: where, adding plots and establishing equalities, we have that: Following the set of equations ( 22).We obtain that: (iii) which gives us: Finally, we have that: (iv) again adding and making the equalities, we have: The above equations determine when a coupling between the wave equations may be considered.This coupling is done at variable time.
Considering c 1 = c 2 = c 3 = c 4 = c then the following general time depending coupling equation is obtained: ) . (28)

Conclusion
The work succeeded in establishing in a single equation the coupling between three-dimensional waves.The problem posed in this paper can be applied in the following areas of physics: (i) Quantum Mechanics; (ii) Electromagnetism (in the treatment of electromagnetic waves); Therefore, we believe the formula (28) is suitable for coupling waves in space.