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Nauzad,

You've asked many good questions! Before I talk specifically about them, I will give some general information about waves. You can read quickly through the parts you already understand, just to learn how I'm using the notation.

In its simplest form, a **progressive wave
** is just a repeating, measurable pattern that
*moves* through space. An example is when you quickly
move one end of a rope up and down. A pulse moves down the
rope, as in this figure, showing the rope at three different
times:

It's important to clarify what moves in what direction.
Individual points on the rope ('particles'--like the red dot
in the figure) move up and down, and we can measure their
height. The *pattern* of high-low-high-low moves
horizontally, so we can watch, for example, the "high spots"
(circled) move along the rope. Of course, the movement of
the pattern is related to the movement of the particles,
since the pattern we see is made up of different particle
heights.

Now for the math. An equation for a travelling wave must
relate (at least) *three* variables:

*y*- the signal or property being measured, such as air pressure in a sound wave, or rope height in our example,
*x*- the distance travelled by the pattern,
*t*- time.

* y(x,t)=a sin (kx+wt). *

(This isn't quite the equation in your question--we'll
get to that.) Notice that the height of the rope depends on
*where* on the rope you are (*x*) and also on
*when* you look at it (*t*) . In addtion to
the variables, there are three parameters in the equation
which determine the character of the wave:

*a*- the
**amplitude**. This is the maximum value of the signal; in our case, the maximum vertical displacement of the string. *k*- the
**wavenumber**is 2*pi divided by the**wavelength**, which is the spatial length of the pattern. For the rope, the wavelength is the distance between high spots. Writing it out,*kx*= 2*pi**x*/wavelength -- in other words distance is now measured by wavelengths, instead of by meters or centimeters or whatever. The 2*pi (pi=3.14125...) is thrown in because the sine function completes a cycle every 2*pi units, not every unit. -
*w* - (often written as the greek letter
**omega**) is defined as -2*pi divided by the**period**, which is the amount of time taken for one cycle of the pattern to occur or to pass a particular spot. This is exactly the same idea as was true for*kx*: time is now being measured in periods, instead of, say, seconds. Making omega negative is for later convenience; we could work through the problem with it positive if we wanted to.As you remarked,

*w*is sometimes called the*angular velocity*, which you noticed was strange since there is no circular motion in the wave we are studying. In some applications, it really does mean angular velocity! For instance, the height*h*of an object as it travels in a vertical circle is given by*h = r sin wt*, where*r*is the radius, and*w*is the actual angular velocity of the object. (If*w*is negative, is the object moving clockwise or counter-clockwise?) But the term can make sense in other uses, too, if you consider that anything moving in a circle repeats itself regularly, and represents a cycle. So some people use "angular velocity" for any cyclical process, in an analogy to circular motion. Other people avoid this usage, because as you know, it can cause confusion.

Let's examine the wave equation from both points of view:
studying the cycle of displacement at one spot, and looking
at the motion of the pattern. We'll use the figure above,
which was plotted with *a* = *k* = 1 and
*w* = -1.

*y=a sin(wt)*.

If we were to put the bead somewhere else, say
*x=3*, the equation would be *y=a
sin(k*3+wt)*. (In that case *k*3* would take the
place of *d* in the equation from your question. That
is a hint about what *d* is -- more below.)

The rope "particles" don't move in *x*, but we
definitely see waves moving in *x*. What's going on?

What we see moving is the pattern. You can see the "high points" on the rope moving along as the wave progresses. This is because one particle is highest one moment, then another particle, in another location, is highest the next. So the location of the highest particle has changed. Then the next particle is highest, and so on. The high point moves smoothly down the rope. Same for the low point, or points at the central height, or any other fixed height.

Any specific point in the cycle is called the
**phase. If we ask, "what the phase?", we
mean, at what point in the high-low-high cycle are we?
Mathematically (and more specfically), the phase is the
argument of the sine function: kx+wt. If at some
place and time, the phase is equal to pi/2, then the rope is
at its maximum height, and there's a high spot at that place
and that time (leftmost open circle in the figure). A short
time later, t is larger. The phase (kx+wt)
of this particular high point always equals pi/2. If
t is larger then wt is more negative
(remember w is negative!). For the phase to equal
pi/2, x must be larger. So the high point must have
moved to larger x (center open circle in the
figure). A short time later, t and x have
increased again, and the high point where kx+wt =
pi/2 has moved again (rightmost open circle). (Notice that
this wave is moving from left to right. What would you have
to change in the equation to make it move from right to
left?)
**

**How fast does the wave move? We can make this question
specific by asking, how fast does a particular phase, like a
high spot, move? We can compare its postition at time
t_{1} to its position at a later time
t_{2}. For the circled high spot in the
figure, kx+wt always equals pi/2. That means
kx_{1}+wt_{1}= pi/2 =
kx_{2}+wt_{2}, which leads to
**

*(x _{2}-
x_{1})/(t_{2}- t_{1}) = -w/k.*

**This has the units of a speed (length/time), and it is
the speed at which the pattern moves down the rope. It is
called the phase speed.
**

If I understand your question, you want to know how far a
particle moves in *y* when the pulse, or pattern,
moves a certain distance in *x*. We can now answer
this. Here's how:

- First we use the phase speed to calculate how long
it takes for the pulse to move the distance in
*x*. - Then we use the equation describing the wave to
determine how far a particle moved in
*y*during that time.

Say the wave pattern moves a distance *dx*. We
know that the time it takes to do that, *dt*, is
*dx* divided by the phase speed, so *dt =
-k*dx/w.* (Remember *w* is negative, so
*dt* is positive.)

How far does a particle move in the interval *dt*?
It depends which particle it is! Say the particle we are
interested in is at *x=x _{0}*. At time

*y _{1}=a
sin(kx_{0}+wt_{0}).*

After *dt*, the particle's height is

*y _{2}=a
sin[kx_{0}+w(t_{0}+dt)] = a
sin[kx_{0}+w(t_{0}-k*dx/w)] = a
sin[k(x_{0}-dx)+wt_{0}].*

In other words, the height at (*x _{0}*,

The displacement *dy =
y _{2}-y_{1}* is the vertical
displacement of the particle. Using trigonometric
identities, you can show that this reduces to

*dy = -2a cos[
k(x _{0}-dx/2)+wt_{0} ]sin[ k*dx/2 ]. *

I hope I understood your question correctly! If I didn't, then I hope the information I've given helps you find the answer. Otherwise, please feel free either to rephrase and resubmit your question, or to e-mail me directly.

You asked what *d* was in the equation *y=a
sin(wt+d)*. A constant term in the argument of the sine
function is called the **phase lag**. It
serves to reset the value of the function at *t = 0*.
This shifts the function so you can model things which vary
sinusoidally but whose initial positions are not at *y =
0*. Note that for a simple progressive wave with a phase
lag, the equation is

*y = a sin(kx + wt + d)*.

Looking at the expression, you can interpret *d*
as a shift in time, or space, or both! When we calculated
the displacements, we saw that you could calculate the
change in particle height by looking at the same place at a
different time, or at the same time in a different place!
One of the beautiful things about waves is that time and
space are, in a sense, interchangeable: you could let
*x* stand for time, and *t* stand for
distance, and (except for redefining *k* and
*w*) it wouldn't make any difference.

The connections between mathematical models and nature are complex, compelling, baffling, and fun--I hope you continue to explore them!

Dan Goldner,
goldner@mit.edu

MIT /
Woods Hole
Joint Program in Oceanography

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