
Periodic Functions
You may have noticed the common mathematical thread which runs
through all bloodflow related phenomena: they all repeat
over and over. The heart goes through its cycle of contraction
and relaxation (called a systole), performing
the same proccess 70 times a minute. Pressure
also goes up and down repeatedly, and so does blood flow 
when the heart contracts, pressure rises, when the heart relaxes
(for a split second) the pressure drops. This behavior is known
as periodic, and mathematicians have special functions that
describe this behavior  they are sines and cosines that we
learn in trigonometry (see figure).
In fact, blood flows through arteries, very much like a sine
wave, where the amplitude is the maximum blood flow, and
the frequency is the heart rate. There is a beatiful partial
differential equation known as the wave equation, which
describes an idelized wave, be it ocean waves, blood flowing
periodically, or ripples spreading in a pond from the place
where you dropped a pebble.
One can make a few comments about this equation,
without solving it just yet. What we mean by
solving this equation is finding a function W
of both x and t, which satisfies the equation. This
function, in our application, denotes the quantity
of blood passing through a point in the artery at a given moment.
We also see that equation involves the
second derivative in time (t) and the second derivative
in space (x), which should suggest that the solution
should look similar in time and in space. There is also
one constant c present in the equation, which happens
to be the speed with which the wave propagates. The
constant is determined by the environment: how
hard the heart is pumping, how thick your blood is,
etc.
We would like to find solutions of the wave
equation that we see in nature  the sine and cosine
waves. Below I will outline the steps that are
necessary to solve this old chestnut, and for more
details, just click on the links.
This function is a reasonably realistic model for blood flow in the
arteries (not the capillaries, though). Notice that the frequency
in time is ten times the frequency in space, which corresponds
to our chosen speed of propagation of 10 cm/sec. The solution is the
cosine wave in space that moves through time, like this:
M & B Exhibits  intro  pg 2  pg 3  pg 4  pg 5  pg 6  pg 7  pg 8
