Answer:
Explanation:
A particular solution for the 1D wave equation has the form
[tex]\Psi(x,t) \ = \ A \ sin ( \ k x \ + \omega t \ + \phi \ )[/tex]
where A its the amplitude, k the wavenumber, ω the angular frequency and φ the phase angle.
Now, for any given position [tex]x_0[/tex], we can use:
[tex]\phi_0 \ = \ \phi \ + \ k x_0[/tex]
so, the equation its:
[tex]\Psi(x_0,t) \ = \ A \ sin ( \omega t \ + \ \phi_0 )[/tex].
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 [tex]\delta[/tex] distance from the original position, we got:
[tex]x_1 = x_0 + \delta[/tex]
and
[tex]\phi_1 = \phi \ + k x_1[/tex]
now, this its
[tex]\phi_1 = \phi \ + k ( x_0 + \delta)[/tex]
[tex]\phi_1 = \phi \ + k x_0 + k \delta[/tex]
[tex]\phi_1 = \phi_0 + k \delta[/tex]
So, there its a phase angle difference of [tex]k \delta[/tex]. We can model this simply by starting the simple harmonic oscillation with a different phase angle.
