Uploaded by cfurse on 26.08.2009

Transcript:

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.

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.