From 6558dd596e839fa1dcfd991aec14386f4000146c Mon Sep 17 00:00:00 2001
From: Hadrien Montanelli <hadrien.montanelli@gmail.com>
Date: Fri, 2 Feb 2024 23:17:26 +0100
Subject: [PATCH] fix typo

---
 2024-02-04.html | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/2024-02-04.html b/2024-02-04.html
index 6856c19..9b60c24 100755
--- a/2024-02-04.html
+++ b/2024-02-04.html
@@ -129,7 +129,7 @@ <h2>Introduction</h2>
 
                     <h2>Modified Helmholtz&ndash;Kirchoff identity</h2>
 
-                    <p>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.</p>
+                    <p>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.</p>
 
                     <p>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
                     $$