Recent content by the_dialogue

Of course! My mistake. Thanks fzero for the prompt response.

Homework Statement \int_{0}^{b} \int_{0}^{2\pi} C_{k,m}(r)^2 \left{\begin{array}{cc}cos(m\theta)^2\\sin(m\theta)^2 \end{array}\right} r dr d\theta Homework Equations See above The Attempt at a Solution Ignoring the 'r' integral for a second, the solution that I see written...

In: \newcommand{\colv}[2] {\left(\begin{array}{c} #1 \\ #2 \end{array}\right)} u_{k,m}(r,\phi)=J_{m}(\lambda_{k,m}r){\colv{cos(m\phi}{sin (m\phi)}, (k=1,2...; m=0,1,2...) But what does it mean for u to be = to a column vector? Shouldn't the phi solution just be cos(m\phi)+sin(m\phi)?

Oh I see. So then would this be correct? \newcommand{\colv}[2] {\left(\begin{array}{c} #1 \\ #2 \end{array}\right)} \u_{k,m}(r,\phi)=J_{m}(\lambda_{k,m}r){\colv{cos(m\phi}{sin (m\phi)}, (k=1,2...; m=0,1,2...) Would \lambda_{k,m} be determined by calculating J_{m}(\lambda_{k,m}b)=0?

Now if I have (the Bessel equation): (\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+(\lambda^2 -n^2\frac{1}{r^2})u_{k,m}(r,\phi)=0, (0<r<b,0<=\phi<=2\pi) \phi_{k,m}(r,\phi)=0, (r=b, 0<=\phi<=2\pi) The solution would be: u_{k,m}(r,\phi)=AJ_n(\lambda r)+BY_n(\lambda r)...

I apologize -- I was in error in a few of my previous posts. 'fzero' was of course correct in saying that the slope cannot be zero at the boundaries. I have a membrane that is clamped at the outside edge and is otherwise a full/regular membrane. What is the corresponding Sturm-Liouville...

Hmm I see. Unfortunately that has been my question from the beginning. Could you help me with this particular problem? Also, is "derivative 0 at b" equivalent to "dr/dn=0"? If not, how does one describe this condition?

Yes. But from the geometry of my application, I know the slope must be zero at the edges. If this is so, how can the above Sturm system be solved? Thank you for your help!

Thanks for the response. If the edge @ r=b is clamped, then the slope should of course be zero. In 1-dimension, this would be e.g. dr/dx=0 -- would this be equivalent to dr/dn=0?

Homework Statement If I have the following Sturm-Liouville system: (\frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2}+\lambda_{k,m}^2)\phi_{k,m}(r,\phi)=0, (a<r<b,0<=\phi<=2\pi \phi_{k,m}(r,\phi)=0, (r=a,0<=\phi<=2\pi...

Thanks LCKurtz. Refer to the bottom of the 1st page ("Since..."). How can we use an expression and plug it into the problem, when that expression is taken from an equation (2nd expression on 1st page) that is not equivalent to the original differential equation? The author says this is...

Well all the author does is solve for 'y_n' (or rather a normalized form of y_n) and then use that term to substitute into 3.21. In other words the author is basing his future derivation on the presumption that the 2nd differential equation and the 1st (original) have as their base y or y_n...

I apologize. The author states that we may express equation 3.21 with the equation that follows, since the y_n and y satisfy the same boundary conditions. To me, however, it seems like they are 2 different equations. How can we say that they satisfy the same boundary conditions, in order to...

Homework Statement I'm learning Sturm-Liouville theory and currently studying Green's function. In the following image, the author states that y_n(x) satisfy the "same" boundary conditions as y(x). Homework Equations http://img2.pict.com/e9/87/37/3735148/0/1277836655.jpg The Attempt at a...

Thanks poormystic. Is it reasonable to suggest that some kind of interference can happen depending on the geometry of the chamber? In other words, can we draw the analogy from light scattering in a conical mirror and apply it sound waves? I would think not, but nonetheless I seem to think...