on the wave equation

Enrique Cantera del Río

On the wave equation

ON THE WAVE EQUATION Enrique Cantera del Río 1- Experimental introduction. 2- The solution of the wave equation in one dimension. 3- Physical interpretation of the wave equation solution. 4- Stationary Waves. 5- The Fourier theorem and the linearity of the wave equation. 6- The wave equation in three dimensions. 7- Spherical waves. 8- Relationship between the Laplace’s and wave equation. 9- The Huygens principle and diffraction. 10- Stationary waves in three dimensions. 11- Guided waves. 12- Waves on a guitar string. 13- Continuity, Reflection and Refraction. 14- Waves on electric transmission lines. 15- Waves and special relativity. Doppler effect. 16- Mathematical Appendix. The spring without mass. Linear differential equations. Coupled oscillations and linear systems. Analysis model for waves in gases. Separation of variables in spherical coordinates from the Laplace equation. Recursive formula for the Legendre polynomials. Generalization of the formula for Fraunhoffer diffraction. Fresnel diffraction Fresnel areas and Telecommunications. The Fourier series. The Parseval Theoreme. The Fourier transform. The Fourier series and the Dirac delta. Green functions for linear differential equations. Fourier transform of vector fields: longitudinal and transversal components. The delayed potentials of a point charge. Introduction to Bessel’s functions.

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1- Experimental introduction. For this experience the reader needs a vertical panel with cork on which he can draw and nail thumbtacks. Several equal elastic bands, the rectangles drawing in gray, and several weights or other objects with the same known mass, the ellipse drawing in gray. It is convenient that the bands are of negligible x mass with respect to the weights applied. Take an elastic band and extend it, without forcing it, on the x-axis from the origin of coordinates, marking with a pen, in said band and on the x-axis itself, several reference points. Then we place the band on the y-axis, subjecting it to tension with a known weight as it appears in the drawing. We mark the points of the band on the y-axis. If we join the corresponding points as if they were the values of a function y(x) in y Cartesian coordinates, we will see that, while the elastic behavior is maintained, the function y(x) corresponds to a straight line that passes through the origin. We can express this function like this: y( x)  x   ( x)  x tan( a)   ( x)  (tan( a)  1) x

where ξ(x) represents the displacement experienced by the point x of the band not subjected to tension. In differential analysis language, we eliminate the need for a specific coordinate origin:  ( x)  A  tan( a)  1 x

The next step is to determine the dependence of the A value, constant for a given experience. We can proceed like this: If we take as weight the double of the previous case, we will have that the displacement ξ(x) will be greater and therefore A will increase in value; then A depends directly on the force applied to the band. On the other hand, if the double weight is applied to a set of two equal bands, the displacements ξ(x) will be the same as in the case of a unit weight and a single band. For this to be true, the cross sections of each band must support the same force; The force per unit area or pressure must be homogeneous throughout the cross section. So we have Y

 ( x) F  x S

where the constant Y, known as Young's modulus, is a constant that adjusts the units of the equation and depends on the material from which the band is made and temperature. Hooke's law can be obtained from the previous formula F  YS

 ( x) YS YS   0  k 0 ; x0 l banda; k  x x0 lbanda



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where, given static equilibrium conditions, the derivative is constant and independent of x, we have calculated the derivative by means of the length of the band in the absence of tension (Δx0) and the corresponding displacement at the end of the band (Δξ0). Classical mechanics allows an extension of these static results to a dynamic case. For this we can start from the mechanical model of a tight rope. Strings without mass, used extensively in the problems of basic mechanics, are characterized by a unique value of the stress module in the whole string; or in its case in the part of it that does not transfer energy, to a pulley or other object. But if we consider the mass of the string, then the modulus of tension is no longer constant and varies along the string. The mechanical model is the one in the attached drawing, where T1 dm dm the rope is broken down into elements of mass dm dl dl T2=T1+dT connected by ideal strings without mass of constant length dm (does not vary in time) dl and subjected to a stress that dl T3=T2+dT’ varies from one section to another. The difference of tensions between the two sides of an element dm will cause dm the acceleration of said element according to Newton's second law. In the case of our elastic band the lengths dl do not remain constant in a dynamic situation, unlike the case of the rope that maintains its length constant; and the tensions between elements correspond to Hooke's law. If we extend the approach to a non-static one-dimensional situation, the displacement (elongation) associated with each point of the band will also depend on the time ξ(x,t). Let us take at instant t two contiguous band segments, as small as we want, centered on the static points x and x+dx. The previous equilibrium analysis considered that the mass of the band is essentially negligible, so we must introduce a point mass dm that connects our elementary segments, which we will also consider of elementary section dS. In a nonstatic situation we must accept that the force F is not the same at any point in the band; so that at the ends of the elementary segments the external force, which connects one element with another, is not compensated and the acceleration of the corresponding point mass must occur. With the dynamic model described, we can use the previous static result for our two contiguous differential segments:  ( x, t ) F ( x, t )  x dS  ( x  dx , t ) F ( x  dx , t ) F ( x, t )  dF Y   x dS dS Y

subtracting and thinking that, at a determined time t, the non-compensated force dF must accelerate the elementary mass between the segments, and that such acceleration can be expressed with the function ξ(x,t) Y

 2 ( x, t ) dF dm  2 ( x, t )  2 ( x, t )  0  2 ( x, t ) dm dx     ; 0  2 2 2 2 x dS dS t x Y t dSdx

taking dm as the mass contained in the dx static segment of the non-tensioned band, then ρ0 represents the static density in volume of the elastic band not subjected to tension. The result corresponds to a longitudinal wave equation associated with the displacement of each band point with respect to its position of static equilibrium with a wave velocity of value


On the wave equation c

Y

4

0

It must be said that the deduced wave equation, and therefore the wave velocity, is associated with an inertial observer for which the band, when not under tension, is completely at rest. In fact, this band in static equilibrium (rest) is confused with the observer's coordinate system. The mechanical model used can be easily extended to the case of waves in gases, as shown in the appendix.

2-La The solution of the wave equation in one dimension.  2 ( x, t ) 1  2 ( x, t )  2 x 2 c t 2

This wave equation can be factorized so   1    1        ( x, t )  0  x c t  x c t 

Which suggests solutions like F(x+ct)+G(x-ct). It is possible to prove that the above expression is the general solution for this equation. If we make the following change of variables   x  ct ;   x  ct

Partial derivatives transform by the chain rule like this

        (  , )     x  x  x              (  , )    c  t  t  t    

second derivative is           (  , )      x  x              (  , )  c 2   t  t   

          

             

            

and replacing this in the wave equation  2 1  2      0       x 2 c 2 t 2

whose general solution is, as we said before   f (  );   ( x, t )   f (  )d  G( )  F ( x  ct)  G( x  ct) 



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3- Physical interpretation of the wave equation solution. We can observe one of the components, F or G, at the instant t=0 in the plane (ξ, x) as an arbitrary function, for example

For any other instant t and any other position x we can make the relation with coordinates x at the instant t=0 for functions F and G as well x0F  x  ct  F ( x0F ,0)  F ( x, t ) x0G  x  ct  G ( x0G ,0)  G ( x, t )

What allows to visualize the first component F as the displacement of the "object" F(x,0) on the x axis at speed –c : x = -ct + x0. The second component G corresponds to the displacement of the "object" G(x,0) on the x-axis at speed c : x=ct + x0.

In this way we see that the two components are traveling waves that represent the movement of a disturbance capable of maintaining a recognizable shape. Take the case that we start with an elastic band from which a weight hangs. Parting from a resting situation, we can slightly stretch down the weight. If we stop the force the weight will experience an oscillating movement equal to the classic case of the movement of the spring. According to our results during this motion, the wave equation is applicable to the band (or spring). But if we try one of the possible functions as a solution, either F(x+ct) o G(x-ct), we will see that they can not represent the problem's physics by themselves. If we look at the point of contact of the band with the roof, the stretch ξ is always zero for any moment of time. This is not what is expected as a traveling wave that passes through that point: it can be canceled in some instants or in some time intervals, but not always. According to the logic, if one of the functions F, G can not give us a solution then we must resort to the complete solution F(x+ct)+G(x-ct). If the point of contact with the ceiling corresponds to the coordinate x=0, the functions F and G must verify the following functional relationship F(u)+G(-u)=0. What indicates that, in our geometric vision, the functions F and G are rotated 180 degrees in the plane (ξ,x) with respect to the origin of coordinates and they are moving, at the same speed and in opposite directions. The above representation shows that a solution is possible in which the stretch, that is, the sum F+G, is null permanently at x=0. Evidently, to the left of the vertical axis there is a wall



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(or roof), or in general a rigid object of very high mass and there is no wave in this part of space. However, the effect of this very massive object with respect to the band is to absorb the incident wave and transform it into a reflected wave on the same band. The massive and rigid character of the wall(roof) means that the incident wave can not cause internal displacements in said wall and the energy transported by the wave must be reflected as a bounced wave towards the same band. The reader can check this behavior with a thick rope. If we tie the rope to a fixed end and shake the other end we can see the generated wave or pulse moving above the line of the rope. When the wave reaches the fixed end, the wave bounces in the form of a pulse that moves below the line of the string. In this case we have a transverse wave instead of a longitudinal wave as in the case of the elastic band, but the wave equation is equally applicable. Following the case of the elastic band, the oscillatory movement of the system can be maintained indefinitely if we were able to eliminate the energy losses by friction or internal heating. According to the usual literature, the motion of the end mass can be described as F m

d 2x d 2  m  k  x  xi   max cos(t ) dt 2 dt 2

   x  xi   max sin( t ) ;   k / m

That is, the position of the mass is the sum of the initial position xi of the mass m at the static equilibrium state of the band and the relative elongation of the elastic band from this initial position, where ξmax is the maximum elongation in xi. If no further explanation is given, it is evident that the previous solution(corresponding to a second order differential equation as discussed in appendix)) is not correct, since for m=0 an infinite frequency is predicted for the oscillation of the free band or spring; which obviously does not happen. On the other hand, in the previous result we have combined the Hooke’s law, essentially static, with Newton's 2nd law, essentially dynamic. The result found is valid, as we shall see, in a margin where the mass of the band is negligible compared to the mass of the ball (m); and for the endpoint of the band that this ball occupies. However note the reader that we can analyze the motion of a point in the band other than the endpoint using this same result if we assume that k(x) remains constant and the mass of the band is negligible. In this way we can write a general solution for the elongation of the band like this  ( x, t )  max ( x) cos(t )

Where, for the case of the point of the band in permanent contact with the wall, the elongation of said point must always be zero and therefore ξmax(0)=0; since that point can not move and maintain contact with the wall. Retrieving the previous wave analysis, the following equations must be compatible



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 ( x, t )  F ( x  ct )  G( x  ct )   max ( x) cos(t ) ;  max (0)  0 ; F (u)  G(u )  0

If we choose for F and G a sine or cosine function of the same amplitude A (G=F=sin() for ξ moduled by cos(ωt) or G=cos() and F=-cos() for ξ moduled by sin(ωt)), we can quickly see, using the sine / cosine formulas of a sum / difference, the following  ( x, t )  A sin( K ( x  ct ))  A sin( K ( x  ct ))  2 A sin( Kx ) cos(t ) ;   Kc   max ( x)  2 A sin( Kx )

The K (capital) value is introduced so that the argument of the sine is dimensionless (radians), and therefore has inverse units of the length. K is defined completely from the frequency of oscillation and the speed of propagation of the waves: ω=Kc. It is evident that this parameter corresponds to the wave vector of a sine wave: K=2π/λ, where λ is the wavelength. But the previous solution is an exact solution of the wave equation and, in principle, it can correspond to a real case. However, in Newton's dynamic equation we have used a quasi-static approximation so that the value of the elastic parameter k can be considered constant. Mathematically the first approximation is the constant value of the following derivative  max ( x)  max (lbanda )   cte x lbanda

If we apply this condition to the solution obtained of the wave equation we have  max ( x)  2 AK cos( Kx) x

Supossed A and K constants, to approximate the last derivative to a constant, it must be Kx≈0 for all points x of the static non-stressed band, so that the cosine value is near to 1, that is to say Kx  0 

2



lbanda  0    lbanda

The wavelength must be much greater than the length of the rest band. (This approximation is analogous to the case of alternating current circuits and electromagnetic waves-see appendix) If we use the definition of K we have Klband 

 c

lband 

k m

0 Y

lband 

YS 0 lband  mlband Y

0 Slband m



mband m

And therefore the approximation considered is equivalent to mband >lband If we analyze the solution found in a greater range, we arrive at the behavior described in the following images. If we assume that the red wave and the blue wave are sine waves with the same amplitude and move in opposite directions, the sum of the two corresponds to  ( x, t )  2 A sin( Kx ) cos(t )   x ( x)t (t )

We see immediately that the points x that verify Kx=nπ (n=0,1,2,3,4...) remain at rest in their position, since for them ξ(x,t)=0. These characteristic points are called nodes. In the same way, as we have seen, there are also instants of time in which the speed of all points affected by the wave process is null. These characteristics inform us that the wave process that is happen corresponds to a standing wave. These standing waves normally occur under conditions of spatial confinement of the wave process. In our case it is evident that the elastic waves are produced in a bounded area of space, since the elastic medium is not infinite and can not be extended or compressed arbitrarily; at the risk of losing its elastic properties or breaking. In the development that we have made, we start from a general solution F(x+ct)+G(x-ct) that in principle is not bounded. Choosing a sinusoidal function for F and G we have seen that we can reproduce the behavior of a node, that is, of a fixed point. However we see that there are also multiple fixed points in the band, so that we could connect the band to a rigid wall in one of these points and the solution would still be valid. In this way the waves bounce off the two walls to which the elastic band or spring is connected. Since the wave equation does not specify a certain function as a solution, but rather a functional form, we have been using synodal functions with the advantage of knowing all its properties. However, the wave functions in the spring or elastic band may be different. Imagine a spring anchored between the two walls indicated above. From the resting state we take the spring midpoint and move it slightly to the right. The displacement field generated +ξ +ξ will have the form of the attached drawing. To the left of the point where we apply the force there is a stretch and to the right there is a -ξ compression of the spring or elastic band. If we release the spring, it will be the functional form drawn that moves to the left, bounce on the wall, continue to the right and bounce to the opposite wall, and so periodically if there are no energy losses. In the initial position there will be a certain amount of accumulated energy as potential energy. When releasing the spring, it is transformed



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into kinetic energy at the beginning and later the speed decreases and the potential energy accumulates until a limit position symmetrical to the initial one regarding the midpoint; as shown by the displacement field -ξ dotted line. In the previous example it is specified that the midpoint of the spring is stretched and this has its importance. If the spring is homogeneous, its midpoint separates two substantially equal sub-springs. From the point of view of Hooke's Law this means that the tension of the spring on both sides of the stretched point will be the same; and this is true also at the moment when the spring is free. However, if we stretch another different point, the sub-springs will be different and Hooke's Law indicates that the initial tensions will not be the same in both sub-springs. This will also be true if the stretch point separates two springs made of different material. In this case, after the release there is a transient process that generates a wave pulse until the tensions at each point of the spring are equalized. This means that the displacement field ξ does not correspond to the previous drawing.

5- The Fourier theorem and the linearity of the wave equation.

2L

L

0

-L

3L

Consider a function ξ(x) of period L, otherwise arbitrary, as seen in the attached drawing. For this function the Fourier Theorem is valid, according to which this function can be approximated, as much as we want, by means of an infinite series of sums of sines and cosines of period L 

 n 1 

 ( x)  a0   an sin(

1 2 2n 2 2n  ( x)dx ; an    ( x) sin( x)dx ; bn    ( x) cos( x)dx  L0 L0 L L0 L L

a0 

2n 2n  x)  bn cos( x) L L 

L

L

We can apply this to the case of the disturbance in the field of displacements that we have created in the spring, where L is the fixed distance between the two walls in which the ends of the string are anchored. The Fourier theorem gives us an approximation to the form of the function in the interval [0,L] in terms of a sum of synodal waves that we already know and an additional constant a0 corresponding to the average value of the function analyzed in the period [0,L]. Depending on the shape of the wave, more or less terms of the Fourier series will be necessary to get an approximation with a mean square error smaller than a given value. In particular, if the function has a vertex as in the case we have seen before, then ξmax many more terms in the series are necessary to get a given l1 l2 approximation. L



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In the last part of the previous section we saw the case of slightly shifting the center of the spring. Note the reader that the point where we apply the force represents a vertex in the field of displacements in which the derivative ∂ξ/∂x is discontinuous because the slope on one side and another of this point is different; and the second derivative ∂2ξ/∂x2 is not well defined at the same point; whereby the wave equation would not be fully applicable. However, the Fourier series allows us to bring near ξ(x) as much as we want (in terms of least squares curve fitting) by means of an infinite series of sines and cosines; that is, by successive approximations of continuous and derivable in any order functions. The more terms of the series we have the approximation will be more precise and there is a index value "n" for which the contribution of the other terms (infinite in number terms) can be considered negligible. This is analogous to the concept of bandwidth in the transmission of communication signals. In this way the Fourier series allows to approximate, in a bounded interval of the independent variable, arbitrary functions that present some kind of non-essential discontinuity (that does not take infinite values) in the value of the function or its derivatives. These approximations are carried out through analytical functions, continuous, derivable and with continuous derivatives in any order. This approach of the Fourier series is very different from that of the Taylor series. However, we should note a L discrepancy between the Fourier analysis of ξ(x) and the standing wave dynamics proposed in section 4. The functional components of the Fourier theorem are spatial functions sines/cosines of period L/n with n=1, 2,3 ... but in section 4 we saw that there are standing waves of period 2L; or in other words semiperiod L. For the Fourier series to be useful in this case we must expand the field ξ(x) according to the attached figure and calculate the Fourier series of the extended function of period 2L. We recognize in the left part of the extended displacement field the contribution of the bounced wave. It is easy to see that the coefficients a0 and bn are canceled in this case because the field of displacements is an odd function ξ(-x)=-ξ(x); so only the sine components remain. The coefficients of the Fourier series can be calculated in our case as  x ; 0  x  l1   ( x)   ( L  x) ; l1  x  L  (  x)   ( x) 

an 

L  2  l1  2 n 2 n a n  2    x sin( x) dx    ( L  x) sin( x) dx   2L 2L  2 L  0 l1  

l1 L  2 n n x) dx    ( L  x) sin( x) dx ;   x sin( L  0 L L l1 



 ( x)   a n sin( n 1

n x) L

Where α and β can be calculated geometrically as α=ξmax/l1 , β =ξmax/l2; but they depend physically on the initial tensions on the both sides l1 and l2 of the spring; forces that are equal regarding the Hooke’s law for a spring at rest. The spatial problem that we have proposed offers us the ξmax(x) value of each wave component and can be completed with the temporal part. For this we remember that the rebound process performed by each wave component is independent of the rest, since there is no energy exchange between components. Each component must also verify the wave equation; therefore the field ξ(x,t) will be of the form

Enrique Cantera del Río 

 ( x, t )   an sin( n 1


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a   n n n n n   n  x) cos( ct) ; an sin( x) cos( ct)  n sin  ( x  ct)   sin  ( x  ct)  L L L L 2   L   L 

The function ξ(x) corresponds to an initial perturbation that we can cause in the displacement field of the spring and that can be arbitrary. Specifically if we choose the disturbance  ( x)  A sin( Kx ) ; K 

 L

We can verify that it corresponds to the term n=1 of the Fourier series, so that the coefficient a1 does not cancel out and the rest of the coefficients is zero. In fact this functional form is the one we used in the analysis of the previous section where we talk about standing waves. In this way, if we initially excite the spring with the anterior sinusoidal displacement field, this form moves in the spring so that the rebounds of the wave will cause the appearance of the nodes typical of the standing wave. The appendix includes an ampliation on the Fourier series. The importance of choosing the midpoint of the spring for stretching was explained in section 4; so there are no discontinuities in the spring tension. However, we can model the case of an initial discontinuity of tension in the spring by taking a displacement field with the shape of the attached drawing. In this image an equivalent displacement is represented so that we can ideally consider the two sub-springs identical. The discrepancy of displacements at the stretch point corresponds to the different initial tensions. Of Course, the Fourier analysis can be made too on this displacement field.

6- The wave equation in three dimensions. The image represents an elastic band or spring in the direction l. The band is represented as a set of elastic fibers. The curve represents a common elastic Z displacement field of all fibers. A Cartesian coordinate system is also represented in its l X,Y,Z axes. The one-dimensional wave equation that we have seen corresponds in this context to Y X  2 (l , t ) 1  2 (l , t )  2 l 2 c t 2

That is, the relevant spatial variable corresponds to measurements made on the direction l and the displacement field is the same for all the fibers. This approach is not in conflict with the experimental approach at the beginning of this work. It is evident that given a point (x,y,z) it identifies a single fiber and a value of l(x,y,z) within said fiber. Without loss of generality, we can assume that the origin of the central fiber of the band coincides with the origin of coordinates. In this case, if k now designates a dimensionless unit vector in the direction of the fibers, it is verified





13



l ( x, y , z )  k x , k y , k z   x , y , z   k  r

That is, given a value (x,y,z) within the band, the value of l for the corresponding fiber is given by the previous expression. Since the displacement field is the same for all fibers, that is, it is the same for any two fibers at two points with the same l value, we can introduce the field ξ(l(x,y,z),t). Starting from this field we can express the wave equation in three dimensions using the chain rule of derivatives:  (l ( x, y, z ), t )  (l , t ) l ( x, y, z )  (l , t )   kx x l x l

As kx is constant, the second derivative is  2 (l ( x, y , z ), t )    (l , t )  l ( x, y , z )  2 (l , t ) 2   kx   kx l  l x x 2 l 2 

Repeating this for the coordinates y,z and adding all we have





 2  2  2  2 (l , t ) 2  2 (l , t ) 2 2    k  k  k  x y z x 2 y 2 z 2 l 2 l 2

As k is a unit vector. Finally the wave equation in three dimensions is  2  2  2 1  2 1  2 2        ;  ( x, y , z , t ) x 2 y 2 z 2 c 2 t 2 c 2 t 2

The wave process that we have assumed in the volume of the elastic band to deduce this equation is called a plane wave, since at a certain instant all the points of the band in a plane, perpendicular to the band in this case, have the same value of ξ. However, the final equation is valid for an arbitrary field ξ(x,y,z,t), not just for a plane wave. Note the reader that the differential equation obtained is valid in any coordinate system. We can generalize the solution of the one-dimensional case to the three-dimensional case, but only for plane waves  ( x, y, z , t )  F (k  r  ct )  G (k  r  ct ) ; k unit vector

7- Spherical waves.



14

The image corresponds to the fall of a water drop on the surface of a pond at rest. The pressure changes produced are transmitted in the form of a circular wave from the center of impact. On the water surface appear ridges and valleys that follow a pattern similar to that of the graph on the right taken on a radial line (x) that represents the initial level of water; so that as the distance to the center increases the ridges and valleys become less pronounced with respect to the initial water level. Over time the energy is distributed in larger and larger areas, so that the displacement caused by the wave, which propagates homogeneously in all directions on the water surface from the center of impact, is being smaller time until it becomes negligible. In this case no boundary conditions or standing waves appear. There is no physical limit in which water is forced to remain at rest in front of the impacting wave. This is easy to see if we generate the wave in a homemade container: the water at the edges also oscillates and there is no easy way to make it stay at rest. According this arguments, the wave pattern would be a function like ξ(r,t), where ξ represents the height of the water above the initial level and r the distance to the center of impact. All the directions with respect to the center behave symmetrically. The case described is a wave phenomenon in two dimensions that will allow us to make a test of the wave equation in three dimensions of the previous section. Given the functional dependence, we can express the Laplacian operator of the wave equation in spherical coordinates centered on the point of impact and apply the wave equation in this 1    2   1  2   r   r 2  r  r   c 2 t 2

2  2  1  2    2   2 2 c t  r r r 

If we multiply the whole expression by the r coordinate, according to the properties of the partial derivatives we have   2   2 r  1  2 r   2  r 2    2  r r  r 2 c t 2 

With which we obtain a one-dimensional wave equation for the field rξ and therefore the general solution of this equation is r (r , t )  F (r  ct )  G (r  ct )   (r , t ) 

1 F (r  ct )  G(r  ct ) r

We see quickly that for a wave that moves away from the center of impact is G(r+ct)=0, and we can take a sinusoidal function for F(r-ct) to reproduce approximately the wave pattern, characterized by a decreasing amplitude as the distance r increases to the center of impact. Obviously the solution diverges for r=0 and it would be necessary to consider the appropriate boundary conditions. The case of an impact of two objects inside the water corresponds to three-dimensional spherical waves that propagate from a center. According to the Huygens principle, every point of a system affected by a wave process emits spherical secondary waves.



15

8- Relationship between the Laplace equation and the wave equation. The Laplace differential equation for a magnitude ξ(x,y) in two dimensions looks like this  2  2  0 x 2 y 2

many physical phenomena with planar symmetry follow this equation; thus for example: potential currents in a fluid or the electrostatic potential of a charge near a conductive plate. This Laplace equation in two dimensions is formally (mathematically) very similar to the wave equation. Mathematically we can take the parameter c, corresponding to the wave speed, be the imaginary unit i. After all, it is a number that admits the same operations as any real number and with the property i2=-1; this transforms the wave equation as i2

 2  2  2  2    0 x 2 t 2 x 2 t 2

expression that formally is equal to the Laplace equation, although the temporal coordinate must now be interpreted physically as a spatial coordinate. Following the mathematical formalism, the general solution of the Laplace equation in two dimensions is the same as that of the wave equation, but considering the imaginary velocity:  ( x, y)  F ( x  iy)  G( x  iy)

that is, any function whose variable is a complex number turns out to be a solution of the Laplace equation in two dimensions. Of course the solutions with physical sense must be real, which can be achieved by applying the properties of the conjugate of a complex number in this way 1 ( x, y)  2 Re[F ( x  iy )]  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy ) i 2 ( x, y)  2 Im[F ( x  iy )]  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy )  F ( x  iy )

following the formalism, the previous relationships are similar to the case of incident and bounced waves that generated standing waves and therefore we can look for solutions like ξ(x,y)= ξx(x)ξy(y) for the Laplace equation in two dimensions. Similarly, the Laplace equation in three dimensions formally is a wave equation (z → time)  2  2 2  2  i 0 x 2 y 2 z 2

the solutions of plane wave moving in the direction of the unit vector u will therefore be as  ( x, y, z )  F (u  r  iz )  G (u  r  iz ); r  ( x, y ), u  (u x , u y )

On the other hand this analogy induces to think also in the search of solutions like ξ(x,y,z)= ξx(x) ξy(y) ξz(z) for the Laplace equation in three dimensions; something that we will use next. It is possible that ξ also represents a physically real wave if we include



16

time as a variable: ξ(x,y,z,t); although the Laplacian does not operate on the variable time. In this case, the previous solution takes the following form  ( x, y, z , t )  F (u  r  ct  iz )  G (u  r  ct  iz ); r  ( x, y ), u  (u x , u y )

where c is the speed of the wave. A real solution included in this group has been used in the section on surface waves in water of the work on fluid mechanics with F(a)=G(-a) = eia. Evidently there are solutions of the Laplace equation other than this group, as is the case of gravitational potential in empty space regions. We have seen for the case of a spherical wave a solution of the form ξ(r,t) = f(r-ct)/r. We can take this expression to the limit by choosing f(r-ct) = constant; which actually cancels the wave behavior. Taking the constant value as 1 and without loss of generality, the wave equation takes the form of the 3-dimensional Laplace equation in this way 2

1  1       0 r  r

where we have used the properties of the gradient operator, as it was introduced in the work on fluid mechanics. Using the gradient in Cartesian coordinates we can check the previous result 

1    r  x

, y,z

 y

, x, z

 z

r 2  x2  y2  z2   

  1    x r 

, y,z

 y

, x, z

 z

1   1  r2 x, y  r

r x

 y,z

x r ; r y

 r   x 

, y,z

 x, z

r y

, x, z

y r ; r z

r z

y,x

 ;  x, y 

z  ; r

1 1 r   3  x, y , z    3 r r r

 1  3 3( x 2  y 2  z 2 )   3 3    x , y , z    3    3  3   0 5 r3   r r r   r x, y    

which justifies at first sight the Laplace equation for non-zero r values. For r=0 the result should be considered indeterminate. However the Laplacian of 1/r reserves a surprise for us. If we apply the divergence integral theorem on a sphere centered on the origin (0,0,0) and of non-null arbitrary radius we have



2

1 r  1  1 dv        dv       d S    3  d S  4 r r  r  r

where the surface integral corresponds to the definition of solid angle which justifies the non-zero value of the integral. We therefore have a mathematical object that turns out to be null for all the points of the space except for one and to any integral of volume that contains this point we must assign it a non-zero value. This new mathematical species is distinguished from the concept of function by calling it a generalized function or distribution, in particular it is a distribution delta Dirac δ(r) type. By entering a constant q/ε, we can write the result as



2

17

q q    (r ) ;   (r )dv  1 4 r  volumeincluding r  0

the distribution δ(r) is defined as null for r other than zero and the volume integral of δ(r) is equal to 1 if the volume includes the point r=0. The units of δ(r) are those of the inverse of a volume and the reader can clearly see the similarity of the previous equation and the Poisson equation for the potential of an electrostatic (or gravitational) field, where the density corresponds to a charge or point mass located at r=0. Obviously the point charge does not have to be in r=0 and it can be located in an arbitrary point r', in this case the previous equation is like this q

2

 4q (r  r ')

r  r'

where the gradient operator is applied over the coordinates r of the field observation point, not over the location of charge r'. From this result and from Green's theorem a general solution can be established for the Poisson equation. Green's theorem is a consequence of the integral divergence theorem (see mathematical section of the introduction to fluid mechanics work). If we take arbitrary derivable functions {φ,g} and defining the vector function A from them like this A  g  g    A   2 g    g  g    g 2   2 g  g 2

   Adv  

2



g  g 2 dv   A  d S   g  g   d S (teorema de Green)

We choose the functions {φ,g} in the following way

 (r ) (ec. de Poisson )   2 g   (r  r ')

 2  

and applying the Green theorem we have 

  (r  r ' )  g 

 (r )   dv  g  g  d S  

from the preceding results we see that we can choose g 

  (r  r ' )dv  

1

 (r )

4 r  r '

dv 

1 4 r  r '

and then

  1  1 1 dS       4   r  r' r  r'  

if we consider r' a fixed point in the field and r a point that varies throughout the space over which the integration is made, the result is an integrated value for the potential function of the Poisson equation



 (r ') 

 (r )

1

4  r  r '

dv 

18

  1  1 1 dS       4   r  r' r  r'  

The surface integral corresponds to boundary conditions on the value of the potential function and its gradient at points on a closed surface. The development followed here to solve the Poisson equation can be generalized in the Green function method. Given a linear differential operator D such as the Laplace operator, the wave equation, the diffusion equation, the heat propagation equation, the Schrödinger equation, etc. It is about finding the Green function g solution of a differential equation of the form D( g )   (r  r´) . From here you can find a solution for the differential equation D(f)=u(r), where u is an arbitrary function. The method of the separation of variables is also applicable to find solutions to the Laplacian. The case of spherical coordinates with axial symmetry is described in the appendix. Another set of solutions of the Laplace equation is found in the terms of the multipolar expansion of the Coulombian/Newtonian potential V(r). Each term of the multipolar expansion is a solution of the Laplace equation in areas far from the masses/charges. Using the algebra of the gradient operator developed in the work on fluid mechanics and for the first two terms of the expansion we have 4 V (r ) 

M 1  P    ....; M , P consantes,  r r

   1 1 1 1     P     P        P     P      r r r r     1 1 1        2     P        P          P     0  r r  r  1 1        P    P     r r   2









M 0 r





9- The Huygens principle and diffraction. In 1678 Christiaan Huygens proposed the following heuristic principle to explain the wave propagation mechanism of a wave: Any point affected by a wave can be considered as a source of secondary spherical waves that extend in all directions with the same speed, frequency and wavelength as the wavefront from which they come. Probably the reader remembers models that use this heuristic principle to visualize the processes of reflection and refraction of a wave. However, the statement of the principle presents problems of physical interpretation. Normally a wave has a propagation direction associated with it: it goes from point A to point B. But according to the principle, secondary spherical waves propagate in the same way in all directions, there is no privileged direction. Take the case of the circular wave generated by a stone that falls on the surface of a lake at rest. A point distant from the center of impact becomes affected by the wave and according to the principle becomes secondary



19

emitter of spherical waves. The sum of all secondary contributions generates in principle both a disturbance that advances and one that recedes with respect to the initial motion of the wave. However, the heuristic principle serves as the basis for a more precise mathematical approach. According to the principle we can establish that the functional form of a wave will be of this type  (r , t )   Ai (k , r ', r ) k ,r '

sin( kR  wt ) ; R

R  r  r'

where we have taken a sinusoidal wave for the function f(r-ct) of the spherical wave. The point r' corresponds to the point considered as a secondary emitter and r is the point of observation of the wave. The factors Ai adjust the amplitude of the secondary wave and we consider that, for a stationary situation, they only depend on the wave vector k, on the secondary emitter point r' considered and on the observation point of the field r. For the case of a monochromatic wave characterized by a single wavelength (k=constant), the principle indicates that the secondary waves have that same value of k. We can use Euler's formula eix=cos(x)+i*sin(x) and express the previous relationship so e i ( kR  wt )  e ikR  i ( wt )  c (r , t )   Ai (k , r ', r )   Ai (k , r ', r )  c (r )c (t ) ;  (r , t )  Re( c (r , t )) e R R k ,r ' k , r '   where the wave amplitude corresponds to the real part of the complex expression ϕc. We see that the expression admits a factorization in part spatial and part temporal. If we apply the wave equation to the form that we have obtained we have

c (t )  2c (r ) 

c (r )  2c (t ) c2

t 2

 c (t )

w2 c ( r )  c2

 2c (r )  k 2c (r )  0

The previous differential equation, known as the Helmholtz equation, represents the set of all solutions compatible with the Huygens principle for the case of monochromatic waves. We can find the integrals of the Helmholtz equation by using the Green function described above. For this we need to find the Green function g corresponding to the Helmholtz equation  2 g (r )  k 2 g (r )   (r  r ' )

the next candidate turns out to be valid for the Green function corresponding to the Helmholtz equation, and it can be shown that he is the only candidate g (r ) 

 e ikR   R    e ikR       R

 1 e ikR ; 4 R

 ik R ikR R ikR 1  e  3 e  e ikR 1  ikR ;  R R R R 





     e ikR 1  ikR 1    e ikR 1  ikR   1  e ikR 1  ikR 2 1 ;   R  R R  





20



 e ikR 1  ikR  ikeikR 1  ikRR  e ikRikR  e ikR k 2 RR    e ikR       R

ikR    e ikR k 2 RR  R  4 ( R)e ikR 1  ikR  k 2 e  4 ( R)   R R3 

  1 e ikR 2   4 R 

ikR     k 2  1 e   4 R  

    ( R)   (r  r ' )  

where the factor to the right of δ(R) is eliminated since as R tends to zero this value tends to 1 (see appendix for Dirac’s delta). If we now apply Green's theorem by varying r throughout the integration volume and keeping r' at a fixed point

 c



 1 e ikR ;  2c (r )  k 2c (r )  0 4 R   2 1 e ikR    1 e ikR  e ikR  e ikR 1  2      k   ( r  r ' )   k  ( r ) dv    c   d S     c c    c  4 R   4 R  4    R  R     2

g  g 2c dv   c g  gc   d S ;



 c (r  r ' )dv  c (r ' ) 

g



  e ikR  e ikR 1      c   d S ;  c    4    R  R

R  r  r'

resulting that the value of the field at a point: ϕc(r) ,depends on a surface integral enclosing said point r'. Difraction We can apply the previous result to the case in which a wave passes S3 through a small opening made in a plate. For light waves, the geometric optics tells us that when the light S2 passes through the opening, there is R a shadow zone in which there is no S1 light. But experience indicates that such a geometric shadow occurs if d the size of the aperture is larger than Observer (r’) the wavelength, but for comparable S2 dimensions it is possible to detect Focus light in the supossed geometric shadow areas. The drawing represents a focus emitter of spherical monochromatic waves, and a fixed material plate of large dimensions with a small opening. The dotted circle corresponds to a spherical surface of very large radius that we will use to calculate the surface integral of the previous mathematical result on an observation point r’. We decompose the integration surface into three parts: Part S3 : It is the spherical surface part that does not touch the plate or the opening. Parts S2 : they are the parts of the spherical surface touching the plate. Part S1 : it is the part of the spherical surface with the opening.



21

If ϕ(r) is the wave function, to the left of the plate we assume that the plate does not reflect the waves emitted from the focus and we assume that the wave function c 

corresponds to a spherical wave

Aeikd d

The integral that we must calculate can be divided into three parts according to the surfaces that we have described

c ( r ' ) 

  1   e ikR  e ikR   c   d S i c    i 1, 2, 3 4   R  R 

Where dS is a vector pointing to the outside of the spherical integration surface. We consider the following 1-The integral on S3 is essentially a secondary wave in backward motion, so the value of this contribution must be canceled, since there is no reflective object. 2- The integrals on S2 correspond to a zone of geometric shadow and, although strictly speaking it is not true due to the phenomenon of diffraction, we will assume that ϕ(r) takes a negligible value in this area so that the integrals are negligible. 3- In S1 the value of the wave function ϕc(r’) is the same as if there were no plate. Therefore c ( r ' ) 

 A  e ikd  e ikR  e ikR  e ikd   A  e ikd ikR 1 e ikR ikd 1    d S1    e 1  ikR  e 1  ikd    d S1        4  d  R  R  d  4  d R R d  A  e ik ( R  d )  1 1   R1  ikR  d 1  ikd    d S1   4  Rd  R d 

4- One more approximation: we assume the wavelength much smaller than the distances R and d: kR>>1, kd>>1 and therefore c (r ' )  ik

A  e ik ( R  d )  2 1 1  A 2  R   d    d S1   ik   4  Rd  R d  4 ik



 e ik ( R  d )  R d    Rd  R  d   d S1    



A e ik ( R  d ) cos(d , d S 1 )  cos(R, d S 1 ) dS1 4  Rd

Additional approximations can be added to this result to explain the cases of Fresnel diffraction (observer close to plate-see appendix) and Fraunhoffer (observer away from the plate). In the Fraunhoffer case we assume the focus on infinity and therefore what reaches the plate is a plane wave of constant amplitude. To simplify the model we assume that the rays arrive parallel to a plane perpendicular to the plate that passes through the geometric center of the slit, plane which we will call "soil"; in the previous drawing it may be the same plane of the paper. Due to the angle (d, dS1) between the incident plane wave and the plane of the slit there will be a geometric phase shift δd in the value of the phase plane wave exp(ikd) at the points of the slit. An observer (r') very far from the plate will only perceive a component of the wave refracted in the slit, corresponding to the plane wave whose phase plane is perpendicular to R. Due to the



22

angle (R, dS1) between said diffracted plane wave and the plane of the slit there will be a geometric phase shift δR in the value of the phase plane wave exp(ikR) at the points of the slit; the lens, if any, does not introduce any additional phase shift. With these two approaches the previous formula is like this c (r ' )  ik

A eik ( R0  d 0 ) cos( d )  cos( R ) eik ( R  d ) dS1  d  d  d S 1 , R  Rd S 1 4 R0 d 0

The values R0, d0 are constants associated with the incident and refracted plane wave front. For the simple case of a rectangular slit the calculation of the offsets δ can be carried out using the image of an open door. The hole in the door is the slit, where we have to determine the values of the phase, and the plane of the door corresponds to the wave front of the plane wave. With the help of the attached drawing δr θR we can easily calculate the phase shift x δd  d   R  k x sen ( R )  sen ( d ) 

h

c (r ' )  ik

A e ik ( R0  d 0 ) cos( d )  cos( R )h  eik x sen( R ) sen( d )  dx 4 R0 d 0 0 D

Where R0,d0 are constant values corresponding to x=0 (the door frame) and we have taken as reference of angles the x direction of the slit, which affects its sign (θR>0, θd0. If the end oscillator does not oscillate with a pure frequency, but a combination or spectrum of frequencies according to the corresponding temporal Fourier analysis of signals, then only the corresponding frequencies above the cutoff will be transmitted through the guide. In this way the waveguide is a filter that eliminates frequencies below the cutoff frequency. In the case of a cylindrical waveguide it will be kr2+ kg2=ω2/c2 and kr is determined by the boundary conditions of the type kr R =x0n, where R is the radius of the cylinder and x0n is a zero of the corresponding Bessel function for n=kθ. Again, only frequencies higher than the cutting frequency can be propagated in the cylindrical guide. Because of this we notice alteration when listening to the voice of a person through a tube.



28

12- Waves on a guitar string. Consider the pulsation of a guitar string. When the string is at rest we know that it has defined a constant mechanical tension associated with the pitch of its tuning. Any point on the string divides it into two parts. The force with which both parts are attracted corresponds to the tension of the string and it is easy to see, according to Newton's laws, that it must be the same at any point of the string at rest; since there is no part of the string subjected to acceleration. We tighten the rope with a finger, only very slightly, bringing the finger to a height h. This causes the length of the string to change and, in the margin of Hooke's law, this assumes that the string stores a certain amount of elastic potential energy. When releasing the string h uy this potential energy becomes kinetic and therefore L1 L2 ux the length of the string begins to decrease. Because of this the points of the string move in principle in the two directions of the plane: since they move downwards, because the string is shortening as the elastic potential energy decreases. We can limit the displacement of the points of the string in the direction since this displacement can not be greater than the maximum modification in the length of the string  h2 x  L12  h 2  L22  h 2  ( L1  L2 )  L1 1  2  2L 1 

2     L2 1  h   2 L2 2  

 h h h    x      ( L1  L2 )   2 L L  2   1     y  h

Obviously, around the midpoint we can easily achieve a greater height h in the pulsation of the string, so that if it is verified that the height is much less than the length of the string: h0 2-Second order equations. Here we can use the previous result on first order equations t  s  0  t d  d   b (t ) b ( x ) ( x  s)ea ( x ) a ( s )      (t )    (t )  y (t )   (t  s)  G (t , s)   dx e t       b ( t )  b ( x ) a ( x )  a ( s ) 0 t  s  dx  ( x  s ) e e  dt  dt    0  

This integral is complex in general, so we simplify the case in which the differential equation is of constant coefficients. Taking outside the integral factors not depending on x and integrating by parts we have t

0 dx ( x  s) e  ( x  s)

e    x  

t

   x   ( x  s ) dx e    x



t

t

 dx  ( x  s ) dx e    x     0 0



t  e    x  e    x  dx  ( x  s )   ( x  s )        0



t



0

0

e    s dx  ( x  s )   0 t







68

and including the remainder factors G (t , s) 

 t  s    (t  s) t s    t    s  e  t  s   e  e e e   (t  s) ; s0         





Note the reader that for the case of equations with constant coefficients it turns out to be G(t,s)=G(t-s) and that the particular solution of the differential equation corresponds to a convolution integral 

yp 

 G(t  s) f (s)ds  G * f (t )



Convolution The convolution integral of two functions f(x), g(x) is defined as 

 f * g ( x)   f ( x  u )g (u)du 

making the change x-u = u' the order of integration is reversed and we have 







 f * g ( x)   f (u ' )g ( x  u ' ) du'   f (u ' )g ( x  u ' )du'  g * f ( x) so that the convolution operation is commutative. It is also shown that the Fourier transform of the convolution function is equivalent to the product of the Fourier transforms of f(x) and g(x). However, this is also true for the Laplace transform, which is a generalization of the Fourier transform. We obtain the Laplace transform by making the change ik→z in the Fourier transform, so that z = (a + ib) is a complex number with real and imaginary parts. The factor 1/√2π of the Fourier transform is also abstracted 



dx e  zx  f * g ( x) 











dx du e  z  x u  f ( x  u ) e  zu g (u ) 





  du  dx e  z  x u  f ( x  u ) e  zu g (u )    





where, since x,u vary independently, we have grouped the dependent terms of the variable x in the integral of the bracket. We can see that the value of this integral is independent of the value of u and that it coincides with the Laplace transform of f(x), since within this integral d(x)=d(x-u) being constant and the limits of integration [-∞, ∞] eliminate the reference to the variable u. Therefore we can factorize in this way 



 





 



 zx  z  x u  f ( x  u )   du e  zu g (u )   dx e  f * g ( x)    d x  u e  





 zx  dx e  f * g ( x)  fˆ ( z ) gˆ ( z )



where f(z), g(z) they are now the Laplace transforms of the functions f(x), g(x). Linking this result with the previous one, we have that the Laplace transform of yp will be Yˆp ( z )  Gˆ ( z ) Fˆ ( z )



69

We can see in F(z) a representation of the input signal to a system and in Yp(z) a representation of the output signal; G(z) corresponds to the transfer function that transforms the input into the output and describes the system as a black box, without going into physical details. For this interpretation we must be able to force the input signal or setpoint according to a certain function, which can be a change of electrical potential. If the output signal can also be maintained in this way, it can become an input to another system; even for the producer system itself. The classical theory of automatic control develops mathematical models of connectable systems like this. The discovery of the transistor was a step forward allowing the simple design of connectable electrical sub-systems so that the output of one is the input of another. The inverse of the Laplace transform would be something like

 fˆ ( z)e

zx

dz whose value

and limits of integration would have to be elucidated in the mathematical context of the complex variable analysis.

Fourier transform of vector fields: longitudinal and transversal components. The Fourier transform is a compound operation of multiplication by a scalar and sum (integral). Therefore the Fourier transform is an essentially linear operation and it makes sense, if the integral is convergent, on any object with linear algebra: vectors, matrices, tensors and even differential mathematical operators. In the case of a vector field δ(x,y,z,t) the Fourier transform will be, abstracting constant factors    3 ; i k r ˆ  (r , t )  dk x dk y dk z z (k , t )e   K3

ˆ(k , t )   dxdydz (r , t )e i k r



r  ( x, y, z )  k  (k x , k y , k z )

where the integral extends to the whole wave space (K3) with wave vector k or geometric space (R3) with position vector r. Therefore, the Fourier transform of a vector field is another vector with the same Euclidean linear algebra. In this way we can propose a decomposition of the vector field of the Fourier transform δ(kx,ky,kz,t) in the parallel (k=) and perpendicular (k┴) components to the vector k. This is achieved by the formula of the double vector product in this way  k  ˆ ˆ k  ˆ ˆ  ;    k  k2 k   ˆ  k   k 2 ˆ   k  ˆ  k  ˆ(k , t )  ˆ (k , t )k  k  Â (k , t ) ;      k  ˆ ˆ  k ˆ ˆ  A  2 ;   k   A k 

in this way the inverse Fourier transform is formulated in two components. The component corresponding to δΦ can be rewritten using the gradient operator in this way

 dk x dk y dkzˆ (k , t )ke K3

i k r

 i  dk x dk y dk zˆ (k , t )ei k r    ;   i  dk x dk y dk zˆ (k , t )e i k r K3

K3



70

The component corresponding to δA can be rewritten using the algebra of the rotational operator, developed in the work on fluid mechanics, in this way





   Â e i k r   e i k r  Â  iei k r k  Â   

 dk x dk y dk z k  Â e K3

i k r

 i



 dk x dk y dk z    Âe K3

i k r

   A ; A  i dk dk dk ˆ e i k r  x y z A  K3



In short we have that a vector field δ(x,y,z,t) can be represented in this way

 (r , t )   K3

dk x dk y dk z ˆ(k , t )e i k r

 k  ˆ i k r    i dk x dk y dk z e 2 k  K3      A ;   ˆ  k i k r  A  i dk x dk y dk z 2 e k  K3





The result found is the same as the known Helmholtz decomposition of a vector field, which is presented in the work on fluid mechanics. If it is the case of a vector field whose spectrum only includes a wave vector k0, as it may be the case in a waveguide; then the previous result indicates the existence of two components of the vector field, one parallel to k0 corresponding to a longitudinal wave and another perpendicular to k0 or transverse wave. The delayed potentials of a point charge Let us suppose that we have a point charge Q at rest in our origin of coordinates and at a distance D from the origin an observer maintains a much smaller charge q at rest by applying the corresponding force to balance the coulombian interaction. The system remains unchanged, but at time t=tr’ the charge Q moves abruptly. What effects does this action have on the observer? According to electromagnetism, the action supposes a modification of the field of the charge Q and this modification of the field follows the wave equation for a wave that propagates, in a vacuum, at the speed of light c; so for the electric and magnetic fields outside the charge Q the wave equation is verified 2 E 

1 2 E 1 2 B 2 ;  B  c 2 t 2 c 2 t 2

Following the Huygens principle, the observer will perceive the consequences of the field modification only when the first spherical wave, starting from the origin of coordinates in t=tr’, covers the distance D to the observer; that is, the observer will perceive changes at a later time to= tr’ +D/c. The method of the delayed potentials allows to calculate the electric (V) and magnetic (A) potentials modified by the wave propagation effect of the electromagnetic field. In the case of the electric potential V, in a static case and in an empty space, free of boundary conditions we have



V (r o )  k 

 (r ' ) r o  r'

71

dv '

where ro indicates the point of observation and r' numbers the charge elements ρ(r’)dv’ distributed in volume that create the field. The retarded potential starts from the previous expression, but now the charge density at a point r' varies with time and the we have to know the charge element in r' at an earlier time tr’ with respect to the observation time to ; these times being related by the propagation of a spherical wave at the speed c V ( r o , to )  k 

 (r ' , tr ' ) r o  r'

dv ' ; t r '  to 

1 r o  r' c

Given observation coordinates to,ro , for each possible source-point r' there is a determined time delay tr’(to,ro,r’); where it can even be ρ(r’,tr’)=0 and not contribute to the delayed potential. Using the Dirac delta we can functionally factor ρ(r’,tr’) in this way 

 (r ' , t r ' )    (r ' , t ) (t  t r ' )dt 

so that in the right part of the equation we now have an authentic density function ρ(r’,t) of independent variables r' and t. if the field is created by a point charge, our charge distribution can also be represented by the Dirac delta by introducing rQ(t), which represents the point charge trajectory Q:  (r ' , t )  Q (r ' r Q (t ))  







 (r ' , t r ' )    (r ' , t ) (t  t r ' )dt   Q (r 'r Q (t )) (t  t r ' )dt

Introducing this result in the delayed electric potential and changing the order of integration we have k  dv '







dt

Q (r ' r Q (t )) (t  t r ' ) r o  r'



k





dt  dv '

Q (r ' r Q (t )) (t  t r ' ) r o  r'



 kQ  dt 

 (t  t r '(Q ) ) r o  r Q (t )

Since we have an integral in the time variable, we must consider the argument of the Dirac delta as a function f(t) of the time variable, so that we can apply the result calculated in the previous section on the Dirac delta f (t )  t  t o 

 (t  t 0i ) i 1 r o  r Q (t ) ;  ( f (t ))   ; t0  t r ' c i df / dt t i 0

where i numbers the instants of time in which t-tr’=0 , that is, the delayed times; with the condition that, fixed the time and position of the observer to, ro and the position of the source r' the corresponding time of delay tr’ is unique, as can be demonstrated. Therefore, the previous sum is reduced to a single term


df dt

 f 0




 1 1 ro  rQ   1  t  to  r o  r Q  t  c c ro  rQ  t t r '





r Q t

t t r '

 1 t t r '

r

o

 r Q (t r ' )

r o  r Q (t r ' )

72

 v

Q (t r ' )

c

Since, according to relativity, it must always be vQ < c we can dispense with the exterior module since the argument will always be positive; and substituted in the expression of the retarded potential produces 

kQ  dt 

 (t  t r ' ) r o  r Q (t )



 (t  t f 0 )



v Q (t r ' ) r o  r Q (t )  r o  r Q (t r ' )  c

 kQ  dt







kQ

V (r o , to ) 





r o  r Q (t r ' )  r o  r Q (t r ' ) 

v Q (t r ' ) c

Returning to the case with which we initiate this section, we see that the first modification of the potential with respect to the static case that the observer perceives is due to the velocity vQ of the charge. Furthermore, if the velocity is much less than c, the modification can be neglected, although the potential V must still be evaluated in the delayed time tr’. We can perceive the time delay by imagining that the system evolves backwards in time starting from the instant to. In this case we imagine that at to a spherical wave is emitted at speed c from the observer (ro) that progresses in space as time decreases, and the particle Q moves recovering previous positions of its trajectory. Since in the empty space nothing can move above the speed of light3 (c), we intuitively see that there will be a certain moment when the spherical wave reaches Q; that moment is the delayed time. The retarded potentials correspond in electromagnetism to the Lorenz gauge. Introduction to Bessel’s functions. In telecommunications, the frequency modulation of a carrier signal is a method of transmitting signals used, for example, in FM radio stations. If ω is the instantaneous signal frequency and ω0 the constant frequency of the carrier wave, the frequency modulation produces this result  (t )   0  2A m(t ) m(t )  1

where the function m(t) encodes the message (normalized to the unit) and A is the maximum message amplitude. The complete physical signal x(t) can be expressed by t t    x(t )  cos  (u )du   cos 0t  2A m(u )du      0 0   





for the simple case m(t)=cos(ωmt) we have 3

In the Cerenkov effect appear charged subatomic particles moving in a dielectric (crystal, water ...) above the speed of light in this medium. The medium is not the vacuum and the speed of light (or speed of propagation of changes in electromagnetic fields) corresponds to the refractive index in said medium; where these particles move faster than their own electric field.



x(t )  cos0t  x sin(mt ) 



73



1 i (0t  x sin( m t )) e  e i (0t  x sin( m t )) ; x  2A / m 2

The previous signal difficult analysis comes from the factor associated with the exponential of the sinus, which we can simplify as eix sin( ) . Obviously this function have a period of 2π in the variable α and therefore admits a Fourier series representation 

e ix sin( )   J n e in ; J n ( x)  

1 2



 d e

i ( n  x sin( ))



the coefficients Jn(x) dependent on the variable x correspond to the Bessel functions of the first kind. It is easy to show that these coefficients are real values since they coincide with their complex conjugate: Jn(x)=J*n(x), changing the sign of the integration variable 







     J n ( x)    d ei ( n  x sin( ))   d ei ( n  x sin( ))  J n* ( x)  J n ( x) Real

in the previous deduction, we have abstracted the factor 1/2π and we will do it henceforth to simplify the results. On the other hand, if we change the variable as α = π-β we have sin(α)=sin(β) and if we make this change of variable in the definition of J-n(x) we have 0

J  n ( x)    d e i (  n (   )  x sin( ))  ein

2

2

 d e

i ( n  x sin( ))

 J  n ( x)  (1) n J n ( x)

0

The Bessel functions can be analyzed in their properties using the Euler formula for sines and cosines depending on the complex exponential and the integration by parts. To begin with we will analyze the derivative dJn/dx 







dJ n 1  i  d e i ( n  x sin( )) sin( )   d e i ( n  x sin( )) e i  e i  dx 2   dJ n 1  J n 1  J n 1  dx 2

next we analyze an integral, I, complementary to the previous one in which, instead of the sine factor, we have a cosine factor 

I

i ( n  x sin( )) cos( )   d e









1 1 d e i ( n  x sin( )) e i  e i  J n 1  J n 1   2  2

Integral I can also be calculated integrating by parts



U  e in ; dV  d e ix sin( ) cos( ) ; dU   d inein ;V  

I   UdV  UV 



 

  VdU  

1 ix sin( ) in e e ix



74

1 ix sin( ) e  ix



 

n d e i ( n  x sin( )) x 

matching this result and the previous one we have 1 J n1  J n1   n J n 2 x

Let's go now with the second derivative d2Jn/dx2 d 2Jn dx 2



   d e i ( n  x sin( )) sin 2 ( )   J n  



 d e

i ( n  x sin( ))

cos2 ( )



integrating by parts properly and using the complex exponential for the cosine we have





1 i ( n 1) e  e i ( n 1) ; dV  d e ix sin( ) cos( ) ; 2 i 1 dU  d (n  1)e i ( n 1)  (n  1)e i ( n 1) ;V  e ix sin( ) 2 ix

U  e in cos( ) 







UdV  UV



 





 VdU  



1 in e cos( )e ix sin( ) ix d 2Jn dx

2

 J n 





 

 



1 d (n  1)e i ( n 1)  (n  1)e i ( n 1) e ix sin( )  2 x 

1 n  1J n 1  n  1J n 1  2x

Finally, if we recover the previous results, written in the following way, we immediately deduce the differential equation that Bessel functions must verify    2  dJ 2 d Jn  x n  x2  n2 J n  0  x 2 dx dx  2  d Jn 1 nJ n1  J n1   ( J n1  J n1 )  Jn  2 2x dx 

dJ n dx n J n 1  J n 1  2 J n x J n 1  J n 1  2





which is the same differential equation found in the text for the problem of standing waves with cylindrical symmetry; in this case dependent on the radial variable r and therefore r>0. There are functions Gn(x) with similar properties as Jn(x) but in terms of real functions that include hyperbolic sines and cosines that the reader can analyze with the same techniques used here







Gn ( x)  d e ( n  x sinh( )) ; x  0 0

75

 dGn 1  Gn 1  Gn 1   2  dx n 1 1  Gn  Gn 1  Gn 1   2 x x  d 2 Gn 1 n  1Gn1  n  1Gn1   1  Gn   2 2 x x  dx

We see that the properties of these functions include additive terms dependent on 1/x. These terms can be canceled linearly if we consider the following function Fn; which is the same as Jn but integrated into a semi-period





Fn ( x)  d e i ( n  x sin( )) ; x  0 0

 dFn 1  Fn 1  Fn 1   2  dx n 1 i n  Fn  Fn 1  Fn 1    1  1 x 2 x   d 2 Fn 1 n  1Fn 1  n  1Fn 1   i  1n 1  1  2  Fn  2 x x  dx









We see that the properties of the imaginary part Im(Fn(x)) contain terms in 1/x that can be canceled in a linear combination with the properties of Gn(x); however, other terms appear depending on powers of -1. We can eliminate these terms if we consider the properties of a new function Hn(x) defined from Gn(x)  dH n 1  dx  2 H n 1  H n 1   n  1n 1 n H n ( x)  (1) G n ( x)   H n  H n 1  H n 1   2 x x 2 d Hn  1n 1        H  n  1 H  n  1 H   n n 1 n 1 2 2x x  dx

if we now consider the following real function Yn defined as an algebraic linear combination from the previous functions F,G,H so that the additive terms in 1/x are canceled, we recover the properties that lead to the Bessel differential equation



Yn ( x)  Im( Fn )  Gn   1n G n

  dYn 1  Yn 1  Yn 1      dx 2 2  n dY 1 2 d Yn   Yn  Yn 1  Yn 1   x n  x 2  n 2 Yn  0  x 2 2 dx dx x   d 2Yn  1 n  1Yn1  n  1Yn1   2  Yn  2x  dx 





Yn also verifies the Bessel differential equation and is clearly a solution linearly independent of Jn. Therefore the general solution of the Bessel differential equation can be described as a linear combination of the form aJn + bYn. Note the reader that, because of the Gn component, the second-species Bessel functions Yn diverge at x=0. The reader can verify the next property





76

Y n ( x)  (1) n Yn ( x)

In waveguides with cylindrical symmetry, the functions Jn are associated with the allowed waves and the functions Yn with the waves rejected. For the analysis of Jn(x) in Taylor series with respect to x=0 it is interesting to calculate the derivatives dmJn/dxm in x=0. From the integral form of the function Jn, the formula of Euler for the sine and the formula of Newton for the power m of a binomial we have d mJn dx

m x 0

im  2



 d e

in





 ei   e i   2i 



m

1 1  m  2 2 

r m 

 m

  d  1r  r ei(mn2r ) ; m, n, r  0 r 0 

We see that if m-n-2r ≠ 0 the corresponding integral is null and only the integrals with m-n-2r = 0 are not null. Therefore the derivatives in x=0 verify

d mJn dx m

x 0

  1r  m    si  r  0 / m  n  2r    2 m  r   0 si no

in particular, the first not null derivative of Jn in x=0 is precisely the n derivative : dnJn/dxn that verifies the previous conditions with r=0. From the index n, the derivatives that do not have the same parity as n are null and those that have the same parity as n are not null; since m-n must be an even number if the corresponding m derivative is not null. Therefore it is verified

.........

d n2 J n dx n  2

 0; x 0

d n 1 J n dx n 1

 0;

d nJn

x 0

dx n

d n2k J n 

dx n  2 k

d

n 2k

dx

 x 0

 x 0

Jn



n 2k

x 0

1  n   10 d n 1 J n   n ; 2 n  0  2 dx n 1

 1k  n  2k  ;

2 n  2 k  k d

 

n  2 k 1

dx

Jn n  2 k 1

0; x 0

d n  2 k 1 J n dx n  2 k 1

0 x 0

x 0

d n2 J n dx n  2

  0 k  0    k  0  

and then related Taylor’s series are 

 1k

 n  2k  x     k  2  k 0 n  2k ! 

J n ( x)  

n 2k

 1k  x  n2k   k 0 k!n  k ! 2  



; n0

 x 0

 11  n  2  ; ....... 2 n  2  1 