Question

Is it possible to create a visual model (stop motion) of a wave motion using the differential equation for simple harmonic oscillation and how so?

Answer 1

Answer:

• Yes.

Explanation:

A particular solution for the 1D wave equation has the form

$\Psi(x,t) \ = \ A \ sin ( \ k x \ + \omega t \ + \phi \ )$

where A its the amplitude, k the wavenumber, ω the angular frequency and φ the phase angle.

Now, for any given position $x_0$, we can use:

$\phi_0 \ = \ \phi \ + \ k x_0$

so, the equation its:

$\Psi(x_0,t) \ = \ A \ sin ( \omega t \ + \ \phi_0 )$.

This is the equation for a simple harmonic oscillation!

So, for any given point, we can use a simple harmonic oscillation as visual model. Now, when we move a $\delta$ distance from the original position, we got:

$x_1 = x_0 + \delta$

and

$\phi_1 = \phi \ + k x_1$

now, this its

$\phi_1 = \phi \ + k ( x_0 + \delta)$

$\phi_1 = \phi \ + k x_0 + k \delta$

$\phi_1 = \phi_0 + k \delta$

So, there its a phase angle difference of $k \delta$. We can model this simply by starting the simple harmonic oscillation with a different phase angle.