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h_v_ansari@yahoo.com 10-25-2006 01:38 PM

Cylindrical wave, wave equation, and mistakes
 
See the original text.

We use the following terminology in this article:
[] indicates subscript.
{} indicates superscript.
~A means the vector A.
^a means the unit vector a.
<four> means 4.
We indicate round d (as used in partial derivation) as <round>.

14 Cylindrical wave, the wave equation,
Ed 01.12.31 ------------------------------------
and mistakes
------------
Abstract
--------
It is shown that the wave equation cannot be solved for the general
spreading of the cylindrical wave using the method of separation of
variables. But an equation is presented in case of its solving the
above act will have occurred. Also using this equation the
above-mentioned
general spreading of the cylindrical wave for large distances is
obtained
which contrary to what is believed consists of arbitrary functions.

I. Introduction
---------------
The wave equation <round>{2}<xi>/<round>t{2}=v{2}<del>{2}<xi> is one of
the
most well-known equations in the classical physics. Particular
solutions to
this equation showing general spreading of plane and spherical waves
are
obtained easily using the method of guessing and trying (to these
solutions
is pointed in this article). But using this method finding the
particular
solution to this equation showing general spreading of a cylindrical
wave
has not been possible yet (we see this matter in the article).
Therefore,
for finding this particular solution some of the physicists resort to
another
method named as the separation of variables (eg see Optics by Hecht and
Zajak, Addison-Wesley, 1974, and Optics by Ajoy Ghatak, Tata McGraw
Hill,
1977) and using this method and the result obtained from it infer that
the
wave equation has been solved for the general spreading of the
cylindrical
wave excepting that contrary to the cases related to the plane and
spherical
waves there is no solution in terms of arbitrary functions in this case
and
eg for very far distances form of the wave function is restricted to
only
trigonometric functions (surprising that how a relation of wave motion
can
be restricted to only some particular functions).

In this article firstly it is shown that applying the method of
separation
of variables for obtaining the general spreading of the cylindrical
wave
from the wave equation is invalid, because with this act in fact only a
particular state of spreading of cylindrical waves arising from
interference
of waves (producing nodes and bulges) can be obtained (of course if the

boundary and initial conditions are satisfied) not general spreading of
the
cylindrical wave. Secondly, using the same methode of guessing and
trying we
obtain the general spreading of the cylindrical wave for far distances
from
the wave equation such that includes arbitrary functions (and therefore
there
won't be necessary that in this case, contrary to other ones, to limit
suddenly the arbitrary selection of functions for the wave function).

We implicitly get two results, one being that obtaining the general
spreading
of the cylindrical wave such that satisfies the wave equation appears
to be
an unsolved problem, and the other being that applying the method of
separation of variables in the case of wave equations does not yield
the
general spreading of the waves but with satisfying the boundary and
initial
conditions only can show a particular state of the interference of
waves
providing that also in the process of solving the prolem we don't
encounter
any contradictions, otherwise the problem can not be solved by this
method
at all. (This matter is important in general solving of the Schrodinger
wave equation in which the method of separation of variables is used
for
obtaining the general spreading of the wave.)

II. Invalidity of the separation of variables for obtaining
__________________________________________________ _________
cylindrical wave function from the wave equation
------------------------------------------------
Consider a stretched membrane fixed along its entire boundary in the
xy-plane. The tension per unit length T and the mass per unit area m
are
constant. The deflection z of the membrane, supposing that is
comparatively
small, should be obtained from the following equation (see eg
Differential
Equations by Simmons, McGraw-Hill, 1972 or Advanced Engineering
Mathematics
by Kreyszig, John Wiley & Sons, 1979):

<round>{2}z/<round>t{2} =
a{2}(<round>{2}z/<round>x{2}+<round>{2}z/<round>y{2}),a=(T/m){1/2} (1)

If this membrane is a circular one of radius <rho>=<rho>[0] and the
boundary
condition is z(<rho>[0],<theta>,t)=0 and the initial conditions are
z(<rho>,<theta>,0)=f(<rho>) and <round>z/<round>t|{t=0}=0 (ie it is
fixed
along its boundary and in t=0 it is motionless and has the symmetric
form
of f(<rho>)), then the solution of the equation (1) using the method of

separation of variables and considering these conditions results in the

following relation:

z=<summation from n=1 to
infinity>a[n]J[0]((<lambda>[n]/<rho>[0])<rho>)cos(
(<lambda>[n]/<rho>[0])at) ,
a[n]=2/(<rho>[0]{2}J[1](<lambda>[n]){2})<integ
ral from 0 to
<rho>[0]><rho>f(<rho>)J[0]((<lambda>[n]/<rho>[0])<rho>)d<rho>

Practically wherever the interference of waves and producing of
standing
waves are concerned, the method of separation of variables is
efficacious
for solving the wave equation. The reason of this matter can be seen
cursorily in the result of the interference of the progressive wave
sink(<rho>-vt) with the retrogressive wave cosk(<rho>+vt):

sink(<rho>-vt)+cosk(<rho>+vt)=(cosk<rho>+sink<rho>)(coskvt-sinkvt) (2)

As it is seen the variables are separated (the first parenthesis is a
function of only <rho>, and the second one is a function of only t, and
we
have obviously node situations). But while we are not faced by the
phenomenon
of interference of waves and the problem is only finding the relation
of
wave motion or in other words obtaining the general spreading of wave
by
proper particular solution of the wave equation, the method of
separation of
variables is wrong, because it is obvious that in this method we accept
the
existence of node situations implicitly, and anyway the relation of
wave
motion must have some arguments like <rho><+ or ->vt in order that it
can
demonstrate a wave motion and this is obviously contradictory to the
separation of variables.

For clearing the above-mentioned material we try to solve the wave
equation
<round>{2}<xi>/<round>t{2}=v{2}<del>{2}<xi> for the relation of
cylindrical
wave motion using the method of separation of variables and to see what
the
difficulty is. Suppose that source of the wave is the z-axis. Since the
wave
function <xi> is independent of <phi> and z, the wave equation takes
the
form of

<round>{2}<xi>/<round><rho>{2}+1/<rho><round><xi>/<round><rho>
= v{-2}<round>{2}<xi>/<round>t{2} . (3)

Suppose that the general solution of this equation is
<xi>(<rho>,t)=<summation over n>b[n]u[n](<rho>)w[n](t). Consider the
general
statement of this general solution,
<xi>[n](<rho>,t)=b[n]u[n](<rho>)w[n](t).
Applying this <xi>[n] in the equation (3) yields
d{2}u/d<rho>{2}u{-1}+<rho>{-1}du/d<rho>u{-1}=v{-2}d{2}w/dt{2}w{-1} each
side
of which must be equal to a unique constant. We show this separation
constant
as -<lambda>[n]{2} (with <lambda>[n]>0; it is easy to see why this
constant
cannot be non-negative). Therefore, the right side is solved as
w(t)=c[1]cos<lambda>[n]vt+c[2]sin<lambda>[n]vt,
and the left side results in Bessel's equation
<rho>{2}u"(<rho>)+<rho>u'(<rho>)+<lambda>[n]{2}<rho>{2}u(<rho>)=0 which
is
solved as
u(<rho>)=c[1]'J[0](<lambda>[n]<rho>)+c[2]'Y[0](<lambda>[n]<rho>).

Thus
<xi>[n](<rho>,t)=b[n](c[1]'J[0](<lambda>[n]<rho>)+c[2]'Y[0](<lambda>[n]<
rho>))(c[1]cos(<lambda>[n]vt)+c[2]sin(<lambda>[n]vt)) which for very
large
<rho>'s is reduced to <xi>[n](<rho>,t) being approximately equal to
b[n](2/(<lambda>[n]<pi><rho>)){1/2}(c[1]'cos(<lambda>[n]<rho>-<pi>/4)+c[2]'
sin(<lambda>[n]<rho>-<pi>/4))(c[1]cos(<lambda>[n]vt)+c[2]sin(<lambda>[n]vt)).
The first parenthesis is a vibrating function of only <rho>, and the
second
one is a vibrating function of only t, and obviously we have node
situations,
and the obtained form of <xi>[n] is rather similar to the cursory
example (2)
showing the result of interference of waves not spreading of a wave.

Maybe it is claimed hopelessly that although <xi>[n] does not show
spreading
of any wave (and can be result of some interference of waves),
summation
of all the <xi>[n]'s can probably demonstrate spreading of a wave. But
with
some contemplation it can be understood that a theorem which is not
true in
case of the components, can not be true in case of the whole; in other
words
each <xi>[n], as a particular solujtion, must demonstrate a physical
independent wave. Furthermore, even if this matter is probable, for
finding
all the constant coefficients, the initial and boundary conditions must
be
applied and for application of the initial conditions we must have form
of
the wave in a definite time beforehand, while our problem is just
finding
the very form of the wave! This vicious circle in addition to all other

material presented so far decisively shows that using the method of
separation of variables for obtaining the general spreading of the
cylindrical wave from the wave equation is invalid.

III. The way that the wave equation can be solved for the
---------------------------------------------------------
cylindrical wave
----------------
Equation <round>{2}<xi>/<round>t{2}=v{2}<del>{2}<xi> appears in physics

repeatedly wherever we know physically that the physical property <xi>
is
being propagated with the speed v. Therefore, it is named as wave
equation.
So far, the general solution of this equation has not been obtained
analytically such that generally it would have been proven that the
obntained general solution is the same relation of wave motion causing
propagation of the property <xi>. (Of course in the one-dimentional
case of
this equation
<round>{2}<xi>/<round>t{2}=v{2}<round>{2}<xi>/<round>x{2},
the general solution <xi>(x,t)=f[1](x-vt)+f[2](x+vt) with arbitrary
functions
f[1] and f[2] is obtained showing clearly a wave motion along the
x-axis
propagating the form of the arbitrary functions f[1] and f[2] with the
speed
v along this axis.) But since everywhere a wave motion is encountered
this
equation appears, we can be certain that the general solution to this
equation is really a wave motion relation and each wave motion relation

satisfies this equation. Therefore, eg we expect the motion relation of
a
plane wave propagating along the ^u with the speed v, ie
<xi>(~r,t)=f[1](^u.~r-vt)+f[2](^u.~r+vt), to satisfy the wave equation.

(f[1] and f[2] are arbitrary functions. It is clear that in a plane
wave
the wave amplitude is constant.) Considering the direction cosines of
the
constant unit vector ^u and Cartesian components of ~r, it can easily
be
seen that this relation satisfies the wave equation.

We also expect that a spherical or cylindrical wave (or other forms of
wave
eg an ellipsoid one) to satisfy the wave equation. A spherical wave has
the
form of

<xi>[1](~r,t) = g[1](r,t)(f[1](r-v[1]t)+f[2](r+v[1]t)) , (4)

and a cylindrical wave has the form of

<xi>[2](~r,t)=g[2](<rho>,t)(f[3](<rho>-v[2]t)+f[4](<rho>+v[2]t)), (5)

in which f[1], f[2], f[3] and f[4] are arbitrary functions, and g[1]
and
g[2] are amplitude coefficients (because it is clear that with wave
spreading its amplitude decreases and probably is periodic in time in
terms
of the form of the wave).

We accept the general validity of the wave equation and apply it to the

relations (4) and (5) in order that the amplitude coefficients will be
obtained; then we shall justify the form of the waves obtained with
these
amplitudes physically.

If the independent variables in the spherical and cylindrical polar
coordinates are (r,<theta>,<phi>) and (<rho>,<phi>,z) respectively,
because
of the independence of <xi>[1] from <theta> and <phi> and of <xi>[2]
from
<phi> and z we shall have the following equations (using the Laplacian
in
its proper form in each coordinates):

<round>{2}<xi>[1]/<round>t{2}
= v[1]{2}r{-2}<round>/<round>r(r{2}<round><xi>[1]/<round>r) (6)

being the wave equation for the spherical wave <xi>[1], and

<round>{2}<xi>[2]/<round>t{2} =
v[2]{2}<rho>{-1}<round>/<round><rho>(<rho><round><xi>[2]/<round><rho>)
(7)

being the wave equation for the cylindrical wave <xi>[2].

After some simple differentiations and algebraic operations, we shall
have

<round>{2}g[1]/<round>t{2}+v[1]A[1]<round>g[1]/<round>t=v[1]{2}(<round

and

<round>{2}g[2]/<round>t{2}+v[2]A[2]<round>g[2]/<round>t
= v[2]{2}(<round>{2}g[2]/<round><rho>{2}+(2B[2]+1/<rho>)<round>g[2]/<
round><rho>+(B[2]/<rho>)g[2]) (9)

for the spherical and cylindrical waves respectively, in which A[1]=
A[1](r,t)=2(-df[1]/d(r-v[1]t)+df[2]/d(r+v[1]t))/(f[1]+f[2]), B[1]=
B[1](r,t)=2(df[1]/d(r-v[1]t)+df[2]/d(r+v[1]t))/(f[1]+f[2]) , A[2]=
A[2](<rho>,t)=2(-df[3]/d(<rho>-v[2]t)+df[4]/d(<rho>+v[2]t))/(f[3]+f[4])
and
B[2]=B[2](<rho>,t)=(df[3]/d(<rho>-v[2]t)+df[4]/d(<rho>+v[2]t))/(f[3]+f[4]).

To obtain g[1] and g[2] the partial differential equations (8) and (9)
must
be solved. It is obvious that these equations can not be solved by the
method of separation of variables. We shall now solve the equation (8)
easily
and also solve the equation (9) for when <rho> approaches infinity, but
its
general solution should be found by interested physicists or
mathematicians.

In order to solve (8) we try a solution that is independent of time
(causing
the left side of the equation to be zero) and its dependence on r is
such
that the terms including B[1] cancel each other, ie some g[1] that
satisfies
the equation

B[1]<round>g[1]/<round>r+(B[1]/r)g[1] = 0 . (10)

Then for finding out that this solution is acceptable or not we must
try it
for other terms (excluding B[1]) of the right side. If sum of them is
zero,
g[1] will be the acceptable solution of (8).

Thus first of all we solve the equation (10). Its solution,
considering being independent of time, is

g[1] = 1/r . (11)

Trying of this solution shows that sum of all the terms of the right
side
and also each term of the left side is zero. Then (11) is really the
solution
of (8). Therefore, the spherical wave has the form of
r{-1}(f[1](r-vt)+f[2](r+vt)), and this is quite natural physically,
because
the conservation law of energy necessitates that since the sphere
surface
is proportional to r{2} causing the proportion of the surface density
of
energy to 1/r{2}, the wave amplitude is proportional to (1/r{2}){1/2}
or 1/r.

Trying to solve the equation (9) we try the same method used for
solving
the equation (8). Then we solve the equation
2B[2]<round>g[2]/<round><rho>+(B[2]/<rho>)g[2]=0 and obtain the
solution

g[2] = 1/<rho>{1/2} . (12)

But trying of this solution yields the expression

v{2}(1/4)<rho>{-5/2} (13)

for the right side of (9), while the left side will be zero. Then
generally
(12) is not an acceptable solution to (9), but when <rho> approaches
infinity, (13) approaches zero ie approaches being equal to the left
side
being zero. Thus for infinite <rho>'s the solution of (9) is (12). In
other
words the cylindrical wave, when <rho> approaches infinity, has the
form of

<xi> = <rho>{-1/2}(f[1](<rho>-vt)+f[2](<rho>+vt)) , (14)

and this is also natural physically, because the conservation law of
energy
necessitates that since the lateral area of the cylinder is
proportional
to <rho> causing the proportion of the surface density of energy to
1/<rho>,
the wave amplitude is proportional to 1/<rho>{1/2}. But, why is this
physical justification true only for very large <rho>'s? Because
generally
all the energy produced from the cylinder axis is not propagated
through the
lateral surface, but some of it propagates through the two bases of the

cylinder which this itself does not allow the coefficient to be
1/<rho>{1/2}
exactly. (Visualize the wavelets produced from each point of the axis
which
necessarily spread through the bases.) Furthermore, it is
comprehensible
that the part of the energy that passes through the bases depends also
on
the form (or shape) of the wave which this itself justifys the
dependence
of the amplitude on the time which probably we shall observe after
obtaining the general solution of (9).

But when <rho> increases, the bases area increases proportional to
<rho>{2},
while the lateral area increases proportional to <rho>. If we suppose
that
the corner wavelets (at the circumferences of the bases) transmit the
same
energy through the bases as through the lateral surface, then we
conclude
that when <rho> increases the surface density of the energy of the
waves
passing through the bases decreases proportional to 1/<rho>{2}, while
the
surface density of the energy of the same waves passing through the
lateral
surface decreases proportional to 1/<rho>. It is obvious that when
<rho>
increases very much the importance of the wave energy passing through
the
bases decreases very much in comparison with the one passing through
the
lateral surface.

The other similar manner to obtain (14) as the cylindrical wave for
infinite
<rho>'s is trying (14) in each side of the wave equation
<round>{2}<xi>/<round>t{2}=v{2}<del>{2}<xi>. For the left and right
sides
the following expressions are obtained respectively:

<round>{2}<xi>/<round>t{2} = v{2}<rho>{-1/2}(d{2}f[1]/d(<rho>-vt){2}+
d{2}f[2]/d(<rho>+vt){2}) (15)

v{2}<del>{2}<xi> = v{2}<rho>{-1/2}(d{2}f[1]/d(<rho>-vt){2}+
d{2}f[2]/d(<rho>+vt){2})+(v{2}/4)<rho>{-5/2}(f[1]+f[2]) (16)

It is clear that when <rho> increases very much the importance of the
second
term of the right side of the relation (16) decreases in comparison
with
the first term, and when <rho> approaches infinity we can relinquish
it,
and then deduce from (15) and (16) that (14) satisfies the wave
equation
for infinite <rho>'s.

Hamid V. Ansari


The contents of the book "Great Mistakes of the Physicists":

0 Physics without Modern Physics
1 Geomagnetic field reason
2 Compton effect is a Doppler effect
3 Deviation of light by Sun is optical
4 Stellar aberration with ether drag
5 Stern-Gerlach experiment is not quantized
6 Electrostatics mistakes; Capacitance independence from dielectric
7 Surface tension theory; Glaring mistakes
8 Logical justification of the Hall effect
9 Actuality of the electric current
10 Photoelectric effect is not quantized
11 Wrong construing of the Boltzmann factor; E=h<nu> is wrong
12 Wavy behavior of electron beams is classical
13 Electromagnetic theory without relativity
14 Cylindrical wave, wave equation, and mistakes
15 Definitions of mass and force; A critique
16 Franck-Hertz experiment is not quantized
17 A wave-based polishing theory
18 What the electric conductor is
19 Why torque on stationary bodies is zero
A1 Solution to four-color problem
A2 A proof for Goldbach's conjecture

My email addresses: hamidvansari<at>yahoo<dot>com or
hvansari<at>gmail<dot>com
To see all the articles send an email to one of my above-mentioned
email addresses.

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Salmon Egg 10-25-2006 09:07 PM

Cylindrical wave, wave equation, and mistakes
 
On 10/25/06 6:38 AM, in article
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"[Only registered and activated users can see links. Click Here To Register...]" <[Only registered and activated users can see links. Click Here To Register...]> wrote:


Boring. Probably wrong. Who cares!
-- Fermez le Bush



Eric Gisse 10-26-2006 01:47 AM

Cylindrical wave, wave equation, and mistakes
 

[Only registered and activated users can see links. Click Here To Register...] wrote:

[...]

Never thought I'd see a crank arguing about separation of variables....



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