Tools

This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

Excel simulations

• Breakwater: We show a simulation of a breakwater protecting a harbor. It starts with a single “pulse” (tsunami?) outside the harbor, $u(20,3,0)=1$, and $u(x,y,0)=0$ for the rest of $(x,y)$, and $u(x,y,1)=0$ for all $(x,y)$. The breakwater is represented by a row of fixed values: $u(x,10,t)=0$ for all $x\ne 20$, with a single gap. The large waves outside produce very mild, circular waves inside. video spreadsheet background
• Drum: A drum is a circular membrane the edge of which is attached to a ring. In other words, the boundary condition is $u(x,y,t)=0$ when $(x,y)$ is on the circle. We choose the initial shape of the drum to be a rotated sinusoid and zero initial velocity. video spreadsheet background
• Heater in the middle of a room: For each t, our function $u(⋅,⋅,t)$ is a function of two variables and can be visualized by its graph. For example, a square heater in the middle of a room will produce the following dynamics of distribution of the temperature over the floor. video spreadsheet background
• Traveling wave: The simulation starts with an arbitrary shape given by the first initial condition $u(x,0)$. It is a small piece of a sinusoid. The second initial condition is just a shift, $u(x,1)=u(x−1,0)$, of the first. The simulation continues to produce this shift at every iteration; in fact, the magnitude of the initial shift is the speed of propagation of the shape. The rope behaves like a whip. This effect is known as a “traveling wave”. video spreadsheet background
• Vibrating string: The simulation starts with a sinusoid given by the first initial condition $u(x,0)=\sin(\pi x/n)$. The second initial condition is an exact copy, $u(x,1)=u(x,0)$, of the first. The initial velocity, then, is zero (there is of course acceleration). The simulation produces more sinusoids and a perfect vibration. video spreadsheet background