Thread: bessel roots from wave equation

In cylindrical polar coordinates ), the Hemholtz equation is

attachment 1 (cylind wave)

From above, bessel function is obtained to solve the function in terms of r. With proper boundary condition, equation below is obtained.

attachment 2 (bessel char)

I just need to get the roots k_mn for n=1,2,3... and m=0, with a=0.306, b=0.179

It can be done numerically but i don't know how,

In Matlab, i run this codes (it give same answer as in literature) but the paper could be wrong since it didnt tell how it was obtained. It suppose to be bessel function of order 0 (m=0). But in this codes, m is 4 and n is 2.34. In another paper, the root given by this code is same as for m=0, n=4. In codes below, from my understanding nu2 and nu3 suppose to be 0 since m is 0, but it doesnt give answer.

nu2=4; nu3=2.34;

z = (0:0.2:100)';
a = besselj(nu2,z);
da = diff(a);

b = bessely(nu2,z);
db = diff(b);

c = besselj(nu3,z);
dc = diff(c);

d = bessely(nu3,z);
dd = diff(d);

f=da.*dd;
g=db.*dc;
h=f-g;

plot(f);
hold all
plot(g);
hold all
plot(h);
hold all

axis([10 25 0 .01])

Hi !

The equation in attachment 2 looks strange. The two terms are identical. So the roots are undefined.
Moreover, what does mean the sign ' on the right side of J ? If it is the notation for derivative, then would you specify what exacly is the variable for the derivation.

thanks for your reply.

it does seems weird, but it is really the characteristic equation for waves in cylinder (torus) in radial direction once we apply
the boundary condition sucha as the normal velocity at inner radius equals to that at the outer radius.

It reported the same in other phd thesis, but it does not tell how to solve the equation, only stating it can be solve numerically.

The symbol ' is the derivative of the bessel funct. The variable is k_mn*a or k_mn*b, a and b is provided, so i need the roots or k_mn for m=0 (when m=0, it is caller bessel funct or order 0).

thanks

Sorry, I cannot hepl you as long as the equation in attachment 2 is contradictory to the code. In the code, they are to different parameters nu2 and nu3. In the equation, there is only one parameter m. The writting of the equation is a nonsense.

The code is not supposed to be right. But it gives the same answer as reported in Phd thesis from Sweden.

The code suppose to be:

m=0;

z = (0:0.2:100)';
a = besselj(m,z);
da = diff(a);

b = bessely(m,z);
db = diff(b);

c = besselj(m,z);
dc = diff(c);

d = bessely(m,z);
dd = diff(d);

f=da.*dd;
g=db.*dc;
h=f-g;

plot(f);
hold all
plot(g);
hold all
plot(h);
hold all

axis([10 25 0 .01])


but it doesnt give the right roots. Forget about the codes, i just want to evaluate the roots of the equation in attachment 2. Any help is appreciated.

Rgds
zm

Hi !

I don't worry about the code. I worry about the equation which isn't correctly written.You use the same symbol m for two differents functions J_m in the equation. Please, use diferent symbols for each one instead of the same symbol for both. Before any work on the question of solving the equation, the equation has to be correctly written in order to avoid confusion and to lose time.
But, it seems that the discussion is turning to a dialogue of the deaf. That is the last time that I suggest you to correct the writting of the equation of attachment 2.

Question about your attachment 2 :

geez, u dont have to make a derogatory comment. I dont know how to do it, that is why i ask if someone in this forum knows about it. The equation is correct. I attach a published work from Molisani (2004).

Well, this confirm that your equation is wrong.
The equation from Molisani's work is an equation which can be solved, while your equation cannot.
At first sight, I couldn't guess where exactly was the mistake. In fact, it was not the index m as I supposed first. You inverted the functions J and Y on the right term.
So, it's clear. Nevertheless, it should be fair that you rewrite your equation in attachment 2.
now, I have a correct starting point to look at your problem.

u're right, i mistyped the eq. thanks

I just had a look at the code.
Again, there are some ambiguities, which could be causes of missunderstanding and mistakes.
The worse is that the notations in the code are not consistant to the notations in the equation.
For example, in the code, "a" means BesselJ and "b" means BesselY, while in the equation "a" and "b" are coefficients of z.
Where those coefficients "a" and "b" from the equation appear into the code ? Is it supposed that the coefficients are equal to 1 ?
Would you mind provide a numerical example with data :
n= ?
m= ?
a= ?
b= ?
and the result :
root given by the code z= ?
expected root z= ?
Please, chose an example so that the result obtained is different than expected, since it is the main object of your question, if I well understand.

Hi !

In order to close this question without wasting too much time, I send you a numerical example (m=0, a=0.306, b=0.179)

Attachment 1 shows the equation to be numerically solved.
Note that solving is impossible in case of a=b. So, the coefficients a and b must appear into the code and the numerical values must not be equal.
Note that the equation do not includes the parameter n (which appears a bit strange). As a consequence, the roots of the equation don’t depend on n. If the equation is not correct, you will have to do again all the work by yourself.

I do not have matlab under the hand, So I did the job with Mathematica.
But, the programming isn’t much different, so you certainly can do the same with matlab.
Attachment 2 shows the code and the graph of the function y(x).
From the graph, rough approximate of the roots can already be obtained.

Attachment 3 shows the more accurate values of some of the roots.
Usually, if an equation has several roots, the software requires to specify a starting value not too far from a root in order to compute an accurate value of this root (and not made a confusion with another root). I used “FindRoot” from Mathematica. With matlab, you will have to use a similar tool, such as “Solve…”.
The roots are 25.0005 ; 49.6105 ; 74.3026 ; 99.0168 ; etc.

In addition, solving can be carried out with WolframAlpha :
Solve BesselJ[0,0.306*x]'*BesselY[0,0.179*x]'-BesselY[0,0.306*x]'*BesselJ[0,0.179*x]'=0 - Wolfram|Alpha