The main idea behind the Burton-Miler method is regularize the standard integral equation (which for exterior problems might suffer from so-called irregular frequencies). The first step is to take the normal derivative with respect to the position of the original integral equation
\[ \frac{\partial}{\partial n(\mathbf{y})}\left( \beta c(\mathbf{y})p(\mathbf{y}) - \int_\Gamma\frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma G(\alpha,\beta,\mathbf{x},\mathbf{y})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x}\right) = 0. \]However, one needs to think of when this operation is valid. First of all the pressure must be differentialable at the position \(\mathbf{y}\). In practice this means that only discontinuous elements can be applied. As a result of the discontinuity the value of \(c(\mathbf{y})\) is constant with respect to the position \(\mathbf{y}\). Therefore the above reduces to the equation
\[ \beta c(\mathbf{y})\frac{\partial p(\mathbf{y})}{\partial n(\mathbf{y})} - \int_\Gamma\frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma \frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})}v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} = 0. \]In order to ease notation it is common to define
\[ \begin{aligned} I_1(\alpha,\beta,\mathbf{y}) &= \frac{\partial}{\partial n(\mathbf{y})}\left( \beta c(\mathbf{y})p(\mathbf{y}) - \int_\Gamma\frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma G(\alpha,\beta,\mathbf{x},\mathbf{y})v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x}\right),\\ I_2(\alpha,\beta,\mathbf{y}) &= \beta c(\mathbf{y})\frac{\partial p(\mathbf{y})}{\partial n(\mathbf{y})} - \int_\Gamma\frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})\partial n(\mathbf{x})}p(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x} + s(\alpha)k\int_\Gamma \frac{\partial G(\alpha,\beta,\mathbf{x},\mathbf{y})}{\partial n(\mathbf{y})}v_n(\mathbf{x})\ \mathrm{d}\Gamma_\mathbf{x}. \end{aligned} \]The Burton-Miller method then couples the two by
\[ I_1(\alpha,\beta,\mathbf{y}) + \eta I_2(\alpha,\beta,\mathbf{y}) = 0, \]where \(\eta\) is the so-called coupling parameter. In matrix form the above can be written as
\[ \left[\mathbf{H} + \eta\mathbf{E}\right]\mathbf{p} +\left[\mathbf{G} + \eta\mathbf{F}\right]\mathbf{v}_n = \mathbf{0}. \]It has been shown[1] that the above system of matrices will have a minimum condition number when \(\eta = \frac{\mathrm{i}}{k}\).
| [1] | Marburg, Steffen. “The Burton and Miller Method: Unlocking Another Mystery of Its Coupling Parameter.” Journal of Computational Acoustics, vol. 24, no. 1, World Scientific Publishing Co. Pte Ltd, 2016, p. 1550016, doi:10.1142/S0218396X15500162. |