A sigle wave
Superposition of waves
© The scientific sentence. 2010

Equation of the harmonic wave
The relationship (3.3): y(x,t) = A sin[(2π/λ)(x  vt)]
is the harmonic wave function. From it, we will find the equation of
the harmonic wave.
Let's consider the partial derivate of this harmonic wave function:
. with respect to time t (holding x constant):
∂y/∂t =  v(2π/λ) A cos[(2π/λ)(x  vt)]
Wich gives the ycomponent of the velocity of an element.
The second derivative gives:
∂^{2}y/∂t^{2} =  v^{2}(2π/λ)^{2} A sin[(2π/λ)(x  vt)]
=  v^{2}(2π/λ)^{2} y(x,t) (4.1)
Wich gives the ycomponent of the acceration of an element.
Next:
. with respect to x (holding t constant):
∂y/∂x = A (2π/λ) cos[(2π/λ)(x  vt)]
This derivative gives the slope of the function at a time t
and point x.
The second derivative gives:
∂^{2}y/∂x^{2} =  A (2π/λ)^{2} sin[(2π/λ)(x  vt)]
=  (2π/λ)^{2} y(x,t) (4.2)
This second derivative gives the bending of the function.
While x is increasing, as it is shown in the figure 3, where "a" is the acceleration ∂^{2}y/∂x^{2}:
 If ∂^{2}y/∂x^{2} > 0, the slope increases, the function bends upward and
the acceleration is directed upward.
 If ∂^{2}y/∂x^{2} < 0, the slope decreases, the function bends downward and
the acceleration is directed downward.
 If ∂^{2}y/∂x^{2} = 0, the slope is constant, the function is straight at that point and
the acceleration is null.
Combining the equations (4.1) and (4.2), we have:
∂^{2}y/∂t^{2} / [ v^{2}(2π/λ)^{2}] = ∂^{2}y/∂x^{2}/[ (2π/λ)^{2}]
or:
∂^{2}y/∂t^{2} / v^{2} = ∂^{2}y/∂x^{2}
Rearranging, we obtain:
∂^{2}y/∂x^{2}  (1/v^{2})∂^{2}y/∂t^{2} = 0 (3)
∂^{2}y/∂x^{2}  (1/v^{2})∂^{2}y/∂t^{2} = 0
This differential equation is the wave equation.
Since we have obtained this equation from the derivatives of the wave function;
then the wave function satisfies the wave equation; and the solution of the wave
equation are, from now on, the wave functions. More generaly, any f(x  vt) or f(x + vt)
are also solution of the wave equation.
©: The scientificsentence.net. 2007.


