# Pulse

## Motivation

Our motivation here is to show that the equations we have chosen can, in certain circumstances, give a pulse which travels from one side of our object to the other. This is what happens in normal heart behaviour: there is a pulse that occurs periodically at one side of the heart, travels across the heart and dissipates, and then the process is repeated.

## The Model

### The surface

The surface we use for this simulation is the pulsating ellipsoid with deformation \( \alpha =0.1 \) and period \( \beta =0.1 \) (see the section on moving surfaces for the definitions).

### The Initial Conditions

We now change the initial conditions for both \(u \) and \( v \) to

\begin{align*} u_0 (x,y) &= 0 \\ v_0 ( x,y ) &=0. \end{align*}

### The New Equations

We add an extra term to the right hand side of our \( u \) equation:

\begin{equation*} (1-u) \mathbb I _{y >0.95} \mathbb I _ {u <0.99} \mathbb I _ { \mathbb Z < t < \mathbb Z +0.05} \end{equation*}

This term has the effect of creating an impulse in the value of \( u \). We do not want multiple waves on our surface so we only allow the impulse if \( u \) is decreasing (\( u<0.99 \)) . We also confine the impulse to a small circular region in the upper hemisphere (\(y>0.95\) ). The impulse occurs every 1 time unit and lasts for \( 0.05 \) time units

\begin{align*} \dot u + u \nabla_\Gamma \cdot \mathbf{v} - a \Delta_\Gamma u &= f(u,v) + (1-u) \mathbb I _{y >0.3} \mathbb I _ {u <0.99} \mathbb I _ { \mathbb Z < t < \mathbb Z +0.05} \\ \dot v + v \nabla _ \Gamma \cdot \mathbf{v} &= g(u,v) \end{align*}

## Results