ECE3300 Lecture 3-2 waves (sinusoidal)

Uploaded by cfurse on 26.08.2009

Now let's talk about a wave and how it
propagates. If we had a direction X and we had a
sinusoidal wave, it would propagate something like this
as a function of time. We could describe this wave,
which we're going to call Y, as a function of X and T as
amplitude times the cosine of omega T minus beta X
plus Fe, and that would be our wave. In this case, Y
would be the amplitude of that wave as a function of
distance and time, A, would be the amplitude, the
maximum value that that could be reached. So this is
the amplitude and it's also the max of that wave.
Omega is the angular frequency. Omega is two
pi times the frequency given in hertz, and it's also
equal to two pi divided by the period. The period of the
wave, if we drew it in time -- here's a time picture of
this wave so right now what I'm going to do is choose
one point, let's choose that point right there. We're
just going to draw that wave as a function of time. It
has its amplitude and this right here is the period of
that wave, and that period would be given in seconds or
some other measurement of time. So omega is given in
radians per second, the frequency is given in hertz,
which is just one per second so omegas radians per
second F is hertz or inverse seconds and T, the period,
is given in seconds.
Beta is our propagation constant. Beta is given
in radians per meter and that is equal to two pi divided
by the wavelength. The wavelength is given in meters.
Right here, on our distance plot, this is meters; this
represents the wavelength of our wave. So beta is
radians per meter, wavelength here is meters, and beta
is our propagation constant.
The last variable that we have here is Phi. Phi is
given in radians, and that tells us something about the
shift of the wave. We'll be talking about waves that
lead and lag in a minute.
So take a look at what this generally means.
Right here, we have a wave that's propagating as a
function of time and as a function of distance. The
negative sign right there, that negative sign, tells me
that it's going in the positive X direction. If it were a
positive sign that would mean that the wave was going
in the negative X direction. So we have a wave with an
amplitude of A, propagating with the frequency of
omega in the X direction with a phase of Fe.
Now, here's a picture of this wave both as a
function of distance and as a function of time. Right
here, is this a cosine wave or a sine wave? This is our
cosine wave. It's starting right there at its maximum
amplitude and it's moving in space. What that means is
that at some instant of time, this value is zero, this
value is maximum, this value is zero. Right here, we can
see that the wave repeats every wavelength so here's
the repeat point. This point is the same as this point,
A, so the wave repeats every wavelength. A wavelength
is two pi radians or 360 degrees. Over here, on the
time sample, we're going to look at this point, right
here, where X is equal to zero and we can see that it
starts out its maximum value and it decreases and
increases and decreases and increases and decreases.
It repeats every period and a period is also 360
degrees or two pi radians.