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A traveling wave is a disturbance that propagates through space over time, defined to be of the (rough) form,

y(x,t) = f(x - vt) + g(x + vt)

where f is the right-moving part of the wave, g is the left-moving part of the wave, and v is the speed of the wave. Traveling waves are solutions of the harmonic equation,

yxx(x,t) = v2 ytt(x,t)

(Subscripts here denote partial differentiation -- i.e., fx means the partial derivative of f with respect to x.)

This is the pure mathematical description of a traveling wave. For a more intuitive understanding, imagine that what you see below is a stretched slinky:

------------------------------------------------------
Now let's give the left end a quick shake and see what happens:
\
 -----------------------------------------------------
\
 -----------------------------------------------------
\
 \
  ----------------------------------------------------
_
 \
  \
   ---------------------------------------------------
__
  \
   \
    --------------------------------------------------
 __
/  \
    \
     -------------------------------------------------
  __
 /  \
/    \
      -------------------------------------------------
   __
  /  \
 /    \
/      ------------------------------------------------
    __
   /  \
  /    \
_/      ------------------------------------------------
     __
    /  \
   /    \
__/      -----------------------------------------------
      __
     /  \
    /    \
___/      ----------------------------------------------
       __
      /  \
     /    \
____/      ---------------------------------------------
        __
       /  \
      /    \
_____/      --------------------------------------------
         __
        /  \           --->
       /    \
______/      -------------------------------------------

As you can see, we have created a pulse which is traveling rightwards down the slinky. This is one example of a traveling wave, but there are many others: ocean waves, sound waves, and light waves, to name a few. Anywhere that you have a disturbance propagating through a medium, whether be water, air, or slinky, you have a form of traveling wave.

The mathematics behind a traveling wave are very elegant. Returning to our slinky example for a moment, suppose that you have a function, y(x,t) which gives you how far away the slinky at the point x units from the left is away from it's natural place at time t -- i.e., when the slinky is perfectly straight, y=0 everywhere along the slinky. Here's a table of y for several values of x and t in our example:

y(x,t) | x=0 | x=1 | x=2 | x=3 | x=4 | x=5 | x=6
-------+-----+-----+-----+-----+-----+-----+-----
   t=0 |  0  |  0  |  0  |  0  |  0  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=1 |  1  |  0  |  0  |  0  |  0  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=2 |  2  |  1  |  0  |  0  |  0  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=3 |  3  |  2  |  1  |  0  |  0  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=4 |  3  |  3  |  2  |  1  |  0  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=5 |  2  |  3  |  3  |  2  |  1  |  0  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=6 |  1  |  2  |  3  |  3  |  2  |  1  |  0  
-------+-----+-----+-----+-----+-----+-----+-----
   t=7 |  0  |  1  |  2  |  3  |  3  |  2  |  1  
-------+-----+-----+-----+-----+-----+-----+-----
   t=8 |  0  |  0  |  1  |  2  |  3  |  3  |  2  
-------+-----+-----+-----+-----+-----+-----+-----
   t=9 |  0  |  0  |  0  |  1  |  2  |  3  |  3  

Pick any place on the table, and you will see that if you move right (towards greater values of x) you see exactly the same thing that you do if you move up (towards lesser values of t). Similarly, what you see by moving left is the same as what you see by moving up. What we have just observed is that holding time still and moving in space is exactly the same as staying in the same spot and moving in time. This suggests that there is a symmetry between the x and t variables, so instead of looking at y as being a function of two variables, we can instead think of it as being equal to some f(ξ) such that

y(x,t) = f(x - t)

where f has the following values:

 f(-7) | f(-6) | f(-5) | f(-4) | f(-3) | f(-2) | f(-1) | f(0)
-------+-------+-------+-------+-------+-------+-------+-------
   0   |   1   |   2   |   3   |   3   |   2   |   1   |   0 

...i.e, f(ξ) gives us the shape of the wave, independent of where or when it is.

By doing this, we have just taken a function of two variables and rewritten it as a function of only one variable! This is a very powerful feature of traveling waves, as it often allows us to simplify what would ordinarily be very difficult equations. For example, assuming a traveling wave form allows us to turn systems of partial differential equations into systems of ordinary differential equations, so that

axx(x,t) = btt(x,t)

becomes

c''(ξ) = d''(ξ).

(If you aren't familiar with multi-variable calculus, you'll just have to trust me when I say that the second is often easier. ;-) )

To get back to the definition I mentioned at the beginning, we shall make two more enhancements to our equation. First, we shall allow our wave to move at different speeds by introducing the wave speed, v, so that now

y(x,t) = f(x - vt)

In the case of our slinky, v happens to equal to one, but in general it could be set anything we want. A beautiful interpretation of v is that it is simply a conversion factor that lets us change units of time into units of space and vice versa. (i.e. seconds are to meters as meters are to feet -- they all measure length, just using different "rulers")

Finally, our waves will not always be traveling strictly to the right; sometimes we will have a disturbance which is propagating to the left. To do this, we need to add one last part to our equation: a function of the form g(x + vt), so that

y(x,t) = f(x - vt) + g(x + vt).

Note that all we had to do was flip the sign of the speed, v, indicating that the g part of the wave is traveling in the opposite direction from the f part.

At last, we have ended where we started -- hopefully with a better grasp of the ideas behind traveling waves.

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