Consider the incident field \(\phi(\boldsymbol{x},\boldsymbol{z}_\epsilon)\) generated by a point source located at \(\boldsymbol{z}_\epsilon=\lambda\epsilon^{-q}e^{i\theta_z}\) for some scalars \(\epsilon>0\), \(q>0\), and \(\theta_z\in[0,2\pi]\). Here, \(k>0\) is the wavenumber and \(\lambda=2\pi/k\) is the wavelength. Let \(D\) be an obstacle of size proportional to \(\lambda\) and independent on \(\epsilon\). Without loss of generality, we assume that \(D\) is centered at the origin. We also consider a small disk \(D_\epsilon=D_\epsilon(\boldsymbol{y}_\epsilon)\) of radius \(\rho(D_\epsilon)=\lambda\epsilon\) centered at \(\boldsymbol{y}_\epsilon = \lambda\epsilon^{-p}e^{i\theta_y}\) for some scalars \(0 < p < q\) and \(\theta_y\in[0,2\pi]\). Measurements will be taken inside the volume \(B\), which is consistent with our previous work.
+Consider the incident field \(\phi(\boldsymbol{x},\boldsymbol{z}_\epsilon)\) generated by a point source located at \(\boldsymbol{z}_\epsilon=\lambda\epsilon^{-q}e^{i\theta_z}\) for some scalars \(\epsilon>0\), \(q>0\), and \(\theta_z\in[0,2\pi]\). Here, \(k>0\) is the wavenumber and \(\lambda=2\pi/k\) is the wavelength. Let \(D\) be an obstacle of size proportional to \(\lambda\) and independent on \(\epsilon\). Without loss of generality, we assume that \(D\) is centered at the origin. We also consider a small disk \(D_\epsilon=D_\epsilon(\boldsymbol{y}_\epsilon)\) of radius \(\rho(D_\epsilon)=\lambda\epsilon\) centered at \(\boldsymbol{y}_\epsilon = \lambda\epsilon^{-p}e^{i\theta_y}\) for some scalars \(0 < p < q\) and \(\theta_y\in[0,2\pi]\). Finally, let \(B\subset\mathbb{R}^2\setminus\overline{D\cup D_\epsilon}\) be a compact set whose size and distance to \(D\) are proportional to \(\lambda\) and independent on \(\epsilon\). Measurements will be taken inside the volume \(B\), which is consistent with my previous work.
We examine the scattering of the incident field \(\phi(\cdot,\boldsymbol{z}_\epsilon)\) by \(D\) and \(D_\epsilon\), which generates the scattered field \(w^s_\epsilon\). More precisely, let \(w^s_\epsilon(\cdot,\boldsymbol{y}_\epsilon,\boldsymbol{z}_\epsilon)\) be the solution to the sound-soft scattering problem $$