# 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*}