Re: trivial pedagogy

[Herb Schulz <herbs@INTERACCESS.COM>]

At 01:12 PM -0500 on 2/16/99, Ludwik Kowalski wrote:
> I just discovered a nice way of introducing the wave equation,
> y=A*sin(kx-wt), to students in a non-calculus physics course.
> Let me share the methodology.
>
> 1) Start by saying that a simple wave is a "moving shape".
>    Draw a picture of y=f(x) with white chalk.
> 2) Review the harmonic oscillator (which they studied already)
>    and draw the picture of y=f(t). Show what T is on the graph.
> 3) How to describe y=f(t) mathematically? y=A*sin(2*PI*t/T).
> ...............................................................................
> ..........
>
> 4) Go back to the picture of y=f(x) Call it "a frozen shape".
>     Introduce the concept of wavelength, L. Compare periodicity
>     in space and periodicity in time (L and T).
> 5) By analogy with y=f(t) how would you describe "this frozen
>    shape" mathematically? In the discussion help them to "invent"
>    y=A*sin(2*PI*x/L).
> 6) But the shape representing a wave is not frozen; it moves from
>    left to right. With red chalk draw a slightly displaced shape.
>     Is this red shape also described by  y=A*sin(2*PI*x/L)? Not
>     really. It is the same shape (same A and L) but displaced.
> 7) How can this little displacement (shift) of the shape be
>    described mathematically?
>
> This kind of questioning naturally leads to the concept of
> phase (something in radians to be subtracted from 2*PI*x/L)
> and to the wave equation which includes the speed of the moving
> shape. I still had difficulties in convincing them that they must
> subtract when the wave (shape) is moving to the right and add
> when it is moving to the left. Why do they often say "plus to
> the right and minus to the left" (which is wrong)?
>
> Nothing profound but worth sharing, I hope.
> Ludwik Kowalski

Howdy,

This is just a special case of knowing that given any function f(x) then
f(x-d) shifts the function in the +x direction by displacement d without
changing its shape.  Let d=vt and you have an undistorted wave of any shape
travelling in the +x direction with velocity v.  A negative value for v
gives the same shape moving in the -x direction.

In terms of coordinate systems I make sure that students know that +/-
directions are only defined if you first define the coordinate system you
are using.  Until the coordinate system is defined you MUST use descriptive
terms to define direction.  I DEMAND that students define the coordinate
system (origin and +/- direction in 1-dimension) or they will lose points
on an exam.

Good Luck,

Herb Schulz
(herbs@interaccess.com)