- #1
Dustinsfl
- 2,281
- 5
$u_{tt} = c^2\nabla^2u$ where $\nabla^2 = \frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2}$.
Suppose $u(r,\theta,t) = G(r,\theta)e^{i\omega t}$.
Then
$$
-\omega^2Ge^{i\omega t} = c^2\nabla^2Ge^{i\omega t}
$$
which leads to the Helmholtz equation $(\nabla^2G + k^2G = 0)$ where $k^2 = \frac{\omega^2}{c^2}$.
Let $G(r,\theta) = R(r)e^{\pm im\theta}$.
Then
$$
\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial}{\partial r}\right]Re^{im\theta} + k^2Re^{im\theta} = 0
$$
which leads to
$$
r^2R'' + rR' + (k^2r^2 - m^2)R = 0\quad \text{(Bessel's equation)}.
$$
Therefore, $R(r) = \mathcal{J}_m(kr) = \sum\limits_{j = 0}^{\infty}\frac{(-1)^j}{(j!)^2(k + m)!}\left(\frac{kr}{2}\right)^{2j + m}$.
The circular membrane is of radius $a$ whose edges are fixed.
That is, $R(a) = \mathcal{J}_{mn}(k_{mn}a) = 0$.
Let $z_{mn}$ be the zeros of $J_{mn}$ and $z_{mn} = k_{mn}a\iff k_{mn} = \frac{z_{mn}}{a}$.
Then the form of the general solution is
\begin{alignat*}{3}
u(r,\theta,t) & = & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[A_{mn}\cos m\theta + B_{mn}\sin m\theta\right] \cos\left(z_{mn}\frac{ct}{a}\right)\\
& + & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[C_{mn}\cos m\theta + D_{mn}\sin m\theta\right] \sin\left(z_{mn}\frac{ct}{a}\right)
\end{alignat*}
Using the first initial condition, we have\begin{alignat*}{5}
u(r,\theta,0) & = & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[A_{mn}\cos m\theta + B_{mn}\sin m\theta\right] & = & 0
\end{alignat*}
How can I create a table of the eigenvalues for the Bessel equation in Mathematica?
Would it be
\begin{alignat*}{5}
u(r,\theta,0) & = & 0 & & \\
u_t(r,\theta,0) & = & \delta(\mathbf{x} - \mathbf{x}_0) & = & \delta(r - r_0, \theta - \theta_0) \end{alignat*}
$$
\int_A\delta(\mathbf{x} - \mathbf{x}_0)f(r,\theta)dA = \int_0^{2\pi}\int_0^a\delta(r - r_0, \theta - \theta_0)f(r,\theta)rdrd\theta = f(\mathbf{x}_0)
$$
Suppose $u(r,\theta,t) = G(r,\theta)e^{i\omega t}$.
Then
$$
-\omega^2Ge^{i\omega t} = c^2\nabla^2Ge^{i\omega t}
$$
which leads to the Helmholtz equation $(\nabla^2G + k^2G = 0)$ where $k^2 = \frac{\omega^2}{c^2}$.
Let $G(r,\theta) = R(r)e^{\pm im\theta}$.
Then
$$
\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial}{\partial r}\right]Re^{im\theta} + k^2Re^{im\theta} = 0
$$
which leads to
$$
r^2R'' + rR' + (k^2r^2 - m^2)R = 0\quad \text{(Bessel's equation)}.
$$
Therefore, $R(r) = \mathcal{J}_m(kr) = \sum\limits_{j = 0}^{\infty}\frac{(-1)^j}{(j!)^2(k + m)!}\left(\frac{kr}{2}\right)^{2j + m}$.
The circular membrane is of radius $a$ whose edges are fixed.
That is, $R(a) = \mathcal{J}_{mn}(k_{mn}a) = 0$.
Let $z_{mn}$ be the zeros of $J_{mn}$ and $z_{mn} = k_{mn}a\iff k_{mn} = \frac{z_{mn}}{a}$.
Then the form of the general solution is
\begin{alignat*}{3}
u(r,\theta,t) & = & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[A_{mn}\cos m\theta + B_{mn}\sin m\theta\right] \cos\left(z_{mn}\frac{ct}{a}\right)\\
& + & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[C_{mn}\cos m\theta + D_{mn}\sin m\theta\right] \sin\left(z_{mn}\frac{ct}{a}\right)
\end{alignat*}
Using the first initial condition, we have\begin{alignat*}{5}
u(r,\theta,0) & = & \sum_{n = 1}^{\infty} \sum_{m = 0}^{ \infty} \mathcal{J}_{mn} \left(z_{mn}\frac{r}{a}\right) \left[A_{mn}\cos m\theta + B_{mn}\sin m\theta\right] & = & 0
\end{alignat*}
How can I create a table of the eigenvalues for the Bessel equation in Mathematica?
Would it be
Code:
ClearAll["Global`*"]z = Table[N[BesselJZero[m, n]], {n, 1, 20}, {m, 0, 5}];
z // TableForm
{
{2.40483, 3.83171, 5.13562, 6.38016, 7.58834, 8.77148},
{5.52008, 7.01559, 8.41724, 9.76102, 11.0647, 12.3386},
{8.65373, 10.1735, 11.6198, 13.0152, 14.3725, 15.7002},
{11.7915, 13.3237, 14.796, 16.2235, 17.616, 18.9801},
{14.9309, 16.4706, 17.9598, 19.4094, 20.8269, 22.2178},
{18.0711, 19.6159, 21.117, 22.5827, 24.019, 25.4303},
{21.2116, 22.7601, 24.2701, 25.7482, 27.1991, 28.6266},
{24.3525, 25.9037, 27.4206, 28.9084, 30.371, 31.8117},
{27.4935, 29.0468, 30.5692, 32.0649, 33.5371, 34.9888},
{30.6346, 32.1897, 33.7165, 35.2187, 36.699, 38.1599},
{33.7758, 35.3323, 36.8629, 38.3705, 39.8576, 41.3264},
{36.9171, 38.4748, 40.0084, 41.5207, 43.0137, 44.4893},
{40.0584, 41.6171, 43.1535, 44.6697, 46.1679, 47.6494},
{43.1998, 44.7593, 46.298, 47.8178, 49.3204, 50.8072},
{46.3412, 47.9015, 49.4422, 50.965, 52.4716, 53.963},
{49.4826, 51.0435, 52.586, 54.1116, 55.6217, 57.1173},
{52.6241, 54.1856, 55.7296, 57.2577, 58.7708, 60.2702},
{55.7655, 57.3275, 58.873, 60.4032, 61.9192, 63.4221},
{58.907, 60.4695, 62.0162, 63.5484, 65.067, 66.5729},
{62.0485, 63.6114, 65.1593, 66.6932, 68.2142, 69.7229}}u(r,\theta,0) & = & 0 & & \\
u_t(r,\theta,0) & = & \delta(\mathbf{x} - \mathbf{x}_0) & = & \delta(r - r_0, \theta - \theta_0) \end{alignat*}
$$
\int_A\delta(\mathbf{x} - \mathbf{x}_0)f(r,\theta)dA = \int_0^{2\pi}\int_0^a\delta(r - r_0, \theta - \theta_0)f(r,\theta)rdrd\theta = f(\mathbf{x}_0)
$$
Last edited: