Graceli functions of infinitesimal waves.


Graceli elements for a variable and infinitesimal geometry and trigonometry.

Author : Luiz Ancelmo Graceli .

For harmonic waves of the same amplitude the superposition principle is in the form.




Using the relationship.

y(x,t) = y₁(x,t) + y₂(x,t) =  y_m [sen(kx - wt + f) + sen(kx - wt )]                

sen a + sen b = 2 sen [(a+b)/2] cos [(a-b)/2]             

y(x,t) =  [2y_m cos(f/2)] sen(kx - wt + f/2)   

Thus, if the phase f = 0 , the interference is constructive

whereas if the phase f = p , the interference is destructive

If the amplitudes are different interference is partial .


For the wave variables Graceli variables .



[ logx / x ... n ] + [ x * Rposit . Rnegati , 0 =

y(x,t) =  [2y_m cos(f/[a/t[logx/x n…] + [x * Rposit. Rnegati, 0 = A])] sen(kx - wt + f/[a/t + [logx/x n…] + [x * Rposit. Rnegati, 0 = A])     

 f = [a/t + [logx/x n…] + [x * Rposit. Rnegati, 0 = A].

whereas if the phase  f= [a / t + [ logx / ... xn ] + [x * Rposit . Rnegati , 0 = ] .


that is, the waves have Graceli infinitesimal variable and discontinuous variations, and movements and rotations over time.


And by measuring sides of triangles , spheres, cubes , and other geometric elements , and we have a geometry and variable, dynamic and infinitesimal trigonometry.