How do we describe wave propagation mathematically? Typically
the wave consists of some disturbance in the medium which depends
on space and time. We describe this disturbance as a function
If this function is to describe a traveling wave then the
x dependence at time 
must be a shifted version of the x
dependence at time 
. This means that
the x,t dependence must
take the form :
f takes on a specific value for positions and times satisfying
Therefore the displacement per time unit, in other words the
velocity of the wave front is
An interesting result of Eq. 8
this is that if I look at the spatial dependence of the disturbance at
fixed time and the temporal dependence of the disturbance at fixed
position I see qualitatively the same functional form though switched
around so what comes first with increasing x comes last with
increasing t and vise versa. We'll try to apply this to the
wave in an audience of people. Imagine we take a photograph
of the wave and then examine individuals as a function of increasing
x. I'll first see people sitting then people about to sit down then people standing tall then people about to stand up and then people that have not stood up yet. You'll notice that it is like watching
a movie of a single individual as the wave passes by him backwards.
These considerations also allow us to predict the length of the
region where people are standing. In time it takes 
to rise
then sit which means that in space it takes a distance
Waves can of course also travel in the opposite direction and then the
functional form for the disturbance is :
since this form describes a shift of a disturbance in the negative
x direction with increasing time.