# Math Help - exponential decay

1. ## exponential decay

Hi!

I don't understand the following:

F = 1/tf * V

F... vector quantity, represents the friction
tf ... friction timescale
V ... velocity of an air parcel V= (u,v,w), also called wind velocity

In the Article "Ray tracing volume densities" Kajiya and Von Herzen (1984) writes:
"Frictional effects are approximated by a simple relation yielding an exponential decay of wind velocities with time"
(the wind velocities are described by a simplified equation of motion and also
change at every time step)

Why is this a exponential decay ?!

hope you can help me
ldenk

2. Originally Posted by ldenk
Hi!

I don't understand the following:

F = 1/tf * V

F... vector quantity, represents the friction
tf ... friction timescale
V ... velocity of an air parcel V= (u,v,w), also called wind velocity

In the Article "Ray tracing volume densities" Kajiya and Von Herzen (1984) writes:
"Frictional effects are approximated by a simple relation yielding an exponential decay of wind velocities with time"
(the wind velocities are described by a simplified equation of motion and also
change at every time step)

Why is this a exponential decay ?!

hope you can help me
ldenk
The friction on the parcel should be in the opposite direction to the
velocity so you should have:

$F = -V/t_f$
Then Newton tells us that:

$\frac{dV}{dt}\ =\ m.F$
which is:

$\frac{dV}{dt}\ =\ -m.V/t_f$
and as both $m$ and $t_f$ are positive the solution
of the differential equation has the form of exponential
decay.

(I would also assume that at some point things have been normalised
so that F is the frictional force per unit mass, and so the value of
$m$ that appears here should be $1$.)

RonL