PX154 - F3a - principle of superposition

to add waves together
- eg: diffraction experiment with a laser + diffraction grating
to add waves of different frequencies

phase differenece

the wave function can be written as:

\begin{array}{r} u_{1} (x, t) = A \cos (k x - ω t) . . . [1] \\ u_{2} (x, t) = A \sin (k x - ω t) . . . [2] \end{array}

- for $[2]$ , $\sin$ is used instead of $\cos$ , but they both behave similarly

the phase difference, $ϕ$ , can be introduced as:

u_{3} (x, t) = A \cos (k x - ω t + ϕ)

- typically, $(0 \leq ϕ \leq 2 π)$ , or ( $- 2 π \leq ϕ \leq 2 π)$

if $ϕ = π$ :

u_{3} (x, t) = A \cos (k x - ω t + π) \equiv - A \cos (k x - ω t)

$u_{1} + u_{3} = 0$ : destructive interference

waves of the same frequency

eg: two loudspeakers emitting middle C $(261.63 H z)$
$u_{1} (x, t) = A \cos (k x - ω t + ϕ_{1})$
$u_{2} (x, t) = A \cos (k x - ω t + ϕ_{2})$
- let both have the same amplitude, $A$
  - helpful identity: $$\cos A + \cos B = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2}) ;...[3]$$
- adding the waves:
$u_{1} + u_{2} = 2 A \cos (k x - ω t + \frac{ϕ_{1} + ϕ_{2}}{2}) \cos (\frac{ϕ_{1} - ϕ_{2}}{2})$ - first $\cos$ is equation of a wave
- second $\cos$ depends on the phase difference, and has no time or position dependence
- if waves are in phase: $Δ ϕ = ϕ_{1} - ϕ_{2} = 0$
  - then, $u_{t o t a l} = u_{1} + u_{2} = 2 u_{1} = 2 u_{2}$
- if waves are out of phase: $Δ ϕ = π$
  - then, $\cos (\frac{ϕ_{1} - ϕ_{2}}{2}) = \cos (\frac{π}{2}) = 0$
    $∴ u_{t o t a l} = 0$