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| | </math> | | </math> |
| | </center> | | </center> |
| − | Substituting this into the equation for <math>\pi</math> yields | + | Substituting this into the equation for <math>\phi</math> yields |
| | <center> | | <center> |
| | <math> | | <math> |
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| | r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2} | | r} \left( r \frac{\partial Y}{\partial r} \right) + \frac{1}{r^2} |
| | \frac{\partial^2 Y}{\partial \theta^2} \right] = - \frac{1}{Z(z)} | | \frac{\partial^2 Y}{\partial \theta^2} \right] = - \frac{1}{Z(z)} |
| − | \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = \eta^2. | + | \frac{\mathrm{d}^2 Z}{\mathrm{d} z^2} = k^2. |
| | </math> | | </math> |
| | </center> | | </center> |
| − | The possible separation constants <math>\eta</math> will be determined by the | + | The possible separation constants <math>k</math> will be determined by the |
| | free surface condition and the bed condition. | | free surface condition and the bed condition. |
| − |
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| − | {{separation of variables for a free surface}}
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| − |
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| − | For the solution of
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| − | <center>
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| − | <math>
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| − | \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial
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| − | Y}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 Y}{\partial
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| − | \theta^2} = k_m^2 Y(r,\theta),
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| − | </math>
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| − | </center>
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| − | another separation will be used,
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| − | <center>
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| − | <math>
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| − | Y(r,\theta) =: R(r) \Theta(\theta).
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| − | </math>
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| − | </center>
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| − | Substituting this into Laplace's equation yields
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| − | <center>
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| − | <math>
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| − | \frac{r^2}{R(r)} \left[ \frac{1}{r} \frac{\mathrm{d}}{\mathrm{d}r} \left( r
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| − | \frac{\mathrm{d} R}{\mathrm{d}r} \right) - k_m^2 R(r) \right] = -
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| − | \frac{1}{\Theta (\theta)} \frac{\mathrm{d}^2 \Theta}{\mathrm{d}
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| − | \theta^2} = \eta^2,
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| − | </math>
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| − | </center>
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| − | where the separation constant <math>\eta</math> must be an integer, say <math>\nu</math>,
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| − | in order for the potential to be continuous. <math>\Theta
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| − | (\theta)</math> can therefore be expressed as
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| − | <center>
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| − | <math>
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| − | \Theta (\theta) = C \, \mathrm{e}^{\mathrm{i} \nu \theta}, \quad \nu \in \mathbb{Z}.
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| − | </math>
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| − | </center>
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| − | We also obtain the following expression
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| − | <center>
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| − | <math>
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| − | r \frac{\mathrm{d}}{\mathrm{d}r} \left( r \frac{\mathrm{d}
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| − | R}{\mathrm{d} r} \right) - (\nu^2 + k_m^2 r^2) R(r) = 0, \quad \nu \in
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| − | \mathbb{Z}.
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| − | </math>
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| − | </center>
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| − | Substituting <math>\tilde{r}:=k_m r</math> and writing <math>\tilde{R} (\tilde{r}) :=
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| − | R(\tilde{r}/k_m) = R(r)</math>, this can be rewritten as
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| − | <center>
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| − | <math>
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| − | \tilde{r}^2 \frac{\mathrm{d}^2 \tilde{R}}{\mathrm{d} \tilde{r}^2}
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| − | + \tilde{r} \frac{\mathrm{d} \tilde{R}}{\mathrm{d} \tilde{r}}
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| − | - (\nu^2 + \tilde{r}^2)\, \tilde{R} = 0, \quad \nu \in \mathbb{Z},
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| − | </math>
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| − | </center>
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| − | which is the modified version of Bessel's equation. Substituting back,
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| − | the general solution is given by
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| − | <center>
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| − | <math>
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| − | R(r) = D \, I_\nu(k_m r) + E \, K_\nu(k_m r), \quad m \in
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| − | \mathbb{N},\ \nu \in \mathbb{Z},
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| − | </math>
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| − | </center>
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| − | where <math>I_\nu</math> and <math>K_\nu</math> are the modified
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| − | [http://en.wikipedia.org/wiki/Bessel_function Bessel functions] of the first
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| − | and second kind, respectively, of order <math>\nu</math>.
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| − |
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| − | The potential <math>\phi</math> can thus be expressed in local cylindrical
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| − | coordinates as
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| − | <center>
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| − | <math>
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| − | \phi (r,\theta,z) = \sum_{m = 0}^{\infty} Z_m(z) \sum_{\nu = -
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| − | \infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m
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| − | r) \right] \mathrm{e}^{\mathrm{i} \nu \theta},
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| − | </math>
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| − | </center>
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| − | where <math>Z_m(z)</math> is given by equation \eqref{sol_Z_fin}. Substituting <math>Z_m</math>
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| − | back as well as noting that <math>k_0=-\mathrm{i} k</math> yields
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| − |
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| − | <center><math>
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| − | \phi (r,\theta,z)
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| − | = F_0\cos(-\mathrm{i} k (z+d)) \sum_{\nu = - \infty}^{\infty}
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| − | \left[ D_{0\nu} I_\nu (-\mathrm{i} k r) + E_{0\nu} K_\nu (-\mathrm{i} k r)\right]
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| − | \mathrm{e}^{\mathrm{i} \nu \theta}
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| − | </math></center>
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| − | <center><math>
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| − | + \sum_{m = 1}^{\infty} F_m\cos(k_m(z+d)) \sum_{\nu = -
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| − | \infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m
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| − | r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.
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| − | </math></center>
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| − |
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| − | Noting that <math>\cos \mathrm{i} x = \cosh x</math> is an even function and the
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| − | relations <math>I_\nu(-\mathrm{i} x) = (-\mathrm{i})^{\nu} J_\nu(x)</math> where <math>J_\nu</math> is the Bessel
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| − | function of the first kind of order <math>\nu</math> and <math>K_\nu (-\mathrm{i} x) = \pi / 2\,\,
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| − | \mathrm{i}^{\nu+1} H_\nu^{(1)}(x)</math> with <math>H_\nu^{(1)}</math> denoting
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| − | the Hankel function of the first kind of order <math>\nu</math>, it follows that
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| − |
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| − | <center><math>
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| − | \phi (r,\theta,z)
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| − | = F_0\cosh(k (z+d)) \sum_{\nu = - \infty}^{\infty}
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| − | \left[ D_{0\nu}' J_\nu (k r) + E_{0\nu}' H_\nu^{(1)} (k r)\right]
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| − | \mathrm{e}^{\mathrm{i} \nu \theta}
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| − | </math></center>
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| − | <center><math>
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| − | + \sum_{m = 1}^{\infty} F_m \cos(k_m(z+d)) \sum_{\nu = -
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| − | \infty}^{\infty} \left[ D_{m\nu}' I_\nu (k_m r) + E_{m\nu}' K_\nu (k_m
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| − | r) \right] \mathrm{e}^{\mathrm{i} \nu \theta}.
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| − | </math></center>
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| − |
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| − | However, <math>J_\nu</math> does not satisfy the [[Sommerfeld Radiation Condition]]
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| − | and neither does <math>I_\nu</math>
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| − | since it becomes unbounded for increasing real argument. These
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| − | two solutions represent incoming waves which will also be
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| − | required later.
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| − |
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| − | Therefore, the solution of the problem requires <math>D_{m\nu}'=0</math>
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| − | for all <math>m,\nu</math>. Therefore, the
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| − | eigenfunction expansion of the water velocity potential in
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| − | cylindrical outgoing waves with coefficients <math>A_{m\nu}</math> is given by
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| − | <center><math>
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| − | \phi (r,\theta,z) = \frac{\cosh(k (z+d))}{\cosh kd} \sum_{\nu = -
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| − | \infty}^{\infty} A_{0\nu} H_\nu^{(1)} (k r) \mathrm{e}^{\mathrm{i} \nu \theta} + \sum_{m = 1}^{\infty} \frac{\cos(k_m(z+d))}{\cos k_m d}
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| − | \sum_{\nu = - \infty}^{\infty} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i} \nu \theta}.
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| − | </math></center>
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| − | (where we have set the parameters <math>F_m</math> so that our vertical
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| − | eigenfunctions are unity at the free surface <math>z=0</math>).
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| − | The two terms describe the propagating and the decaying wavefields
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| − | respectively.
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| − |
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| − | We can write this expression in compact notation as
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| − |
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| − | <center><math>
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| − | \phi (r,\theta,z) = \sum_{m = 0}^{\infty} f_m(z)
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| − | \sum_{\nu = - \infty}^{\infty} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i} \nu \theta}.
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| − | </math></center>
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| − | where
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| − | <center><math>
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| − | f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.
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| − | </math></center>
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