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