$ \newcommand{\RR}{\mathbb{R}} \newcommand{\NN}{\mathbb{N}} \newcommand{\OO}{\mathcal{O}} \newcommand{\mathcow}{\OO} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\CC}{\mathbb{C}} \newcommand{\KK}{\mathbb{K}} \newcommand{\PP}{\mathcal{P}} \newcommand{\TT}{\mathcal{T}} \newcommand{\BB}{\mathcal{B}} \newcommand{\LL}{\mathcal{L}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\veca}{\vec{a}} \newcommand{\vecb}{\vec{b}} \newcommand{\vecd}{\vec{d}} \newcommand{\vece}{\vec{e}} \newcommand{\vecf}{\vec{f}} \newcommand{\vecn}{\vec{n}} \newcommand{\vecp}{\vec{p}} \newcommand{\vecr}{\vec{r}} \newcommand{\vecu}{\vec{u}} \newcommand{\vecv}{\vec{v}} \newcommand{\vecx}{\vec{x}} \newcommand{\vecy}{\vec{y}} \newcommand{\vecz}{\vec{z}} \renewcommand{\vec}[1]{\mathbf{#1}} $ Shaw Research Notes

July 7th, 2021

Derived the wave-response approximation for a spatially localized delta pulse stimulus, and compared to simulations. Prepared for the July 8th presentation to the research group.

To-Do List Snapshot

Spatially Localized Pulse

Here we explore the wave response to a spatially localized pulse stimulus $$ I(x,t) = I_0 H(\tfrac{\Delta x}{2} - |x - x_p|) \delta(t-t_0) $$ where $t_0$ denotes the time of the pulse, $x_p$ denotes the location of the center of the pulse (relative to the front of the pulse at $x=0$), and $\Delta x$ denotes the width of the pulse.

For $t_0 = 0$ the adjoint method gives the first order asymptotic approximation to the wave response function as $$ \eta(t) \approx \frac{\mu c + 1}{\mu \theta} I_0 \begin{cases} 0, & x_p \le - \tfrac{\Delta x}{2} \\ 1 - \exp \left( \frac{-(x_p + \tfrac{\Delta x}{2})}{c\mu} \right), & - \tfrac{\Delta x}{2} \le x_p < \tfrac{\Delta x}{2} \\ \exp \left( \frac{-(x_p - \tfrac{\Delta x}{2})}{c\mu} \right) - \exp \left( \frac{-(x_p + \tfrac{\Delta x}{2})}{c\mu} \right), & \tfrac{\Delta x}{2} \le x_p. \end{cases} $$

Figure 1 below shows our predicted wave response compared to simulation.

Fig 1. Wave response to the spatially localized pulse stimulus $I(x,t) = I_0 H(\tfrac{\Delta x}{2} - |x - x_p|) \delta(t-t_0)$. Here, $x_p$ denotes the center of the stimulated region and $\Delta x = 5$ denotes the width of the stimulated region.