simple harmonic oscillation

\( \frac{d^2 y}{d t^2} = - \omega^2 y \)

  • where:
    • \( \omega = \sqrt{\frac{K}{M}} = 2\pi f\): angular frequency
    • \( y \): displacement
    • \( M \): mass
    • \( K \): spring constant
    • \( t \): time variable

wave equation

\( \frac{\partial^2 y}{d z^2} - \frac{1}{v^2} \frac{\partial^2 y}{d t^2} = 0 \)

  • where
    • \( v = \sqrt{\frac{T}{\rho}} \) wave velocity
    • \(T\): tension in string,
      • influenced by mass and acceleration due to field potential
    • \(\rho = \frac{M}{L}\): density of string
    • \(z\): spatial variable
    • \(t\): time variable
  • this is for a wave in 1D over time

solution

  • the solutions to this equations are of the form
    • \(f(z-vt)\): wave moving to right
    • \(g(z+vt)\): wave moving left

generalizing wave equation to 3D

  • applying gradient operator to 1D wave equation gives
    • \( \nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = 0 \)
    • where:
      • \( \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \)
  • 3D wave equation with unit vectors \( \hat{x}, \hat{y}, \hat{z} \) becomes
    • \( \nabla^2 = \nabla \cdot \nabla \)
    • \( \nabla^2 = \left( \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z} \right) \cdot \left( \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y} + \hat{z} \frac{\partial}{\partial z} \right) \)

plane wave solutions

  • a (monochromatic) plane wave is given by

    • \( \exp{\left[ i(\textbf{k}\cdot\textbf{r} - \omega t) \right]} \)

    • where

      • \( \textbf{r} = x \hat{ \textbf{x} } + y \hat{ \textbf{y} } + z \hat{ \textbf{z} } \)

      • \( \textbf{k} = k_x \hat{ \textbf{x} } + k_y \hat{ \textbf{y} } + k_z \hat{ \textbf{z} } \)

        • this is the wave vector
  • a plane wave is a solution to the 3D wave equation when
    • \( k = \frac{\omega}{t} \)

gradient operator for space partials of plane wave

  • first partial:
    • \( \nabla \exp{ \left[ i(\textbf{k} \cdot \textbf{r} - \omega t ) \right] } = i \textbf{k}\exp{\left[ i (\textbf{k} \cdot\textbf{r} - \omega t) \right]} \)
  • second partial:
    • \( \nabla^2 \exp{ \left[ i(\textbf{k} \cdot \textbf{r} - \omega t ) \right] } = -k^2 \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t) \right] } \)
    • where: \( k^2 = k_x^2 + k_y^2 + k_z^2 \)

time partials of plane wave

  • \( \frac{\partial^2}{\partial t^2} \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t) \right]} = -\omega^2 \exp{\left[ i (\textbf{k} \cdot \textbf{t}) - \omega t \right]} \)

verify solution plane wave in 3D wave

  • 3D wave:
    • \( \nabla^2 \phi - \frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2} = 0 \)

  • plugging in space and time partials of plane wave:

    • \( \nabla^2 \exp{\left[ i (\textbf{k}\cdot\textbf{r} - \omega t) \right]} - \frac{1}{c^2} \frac{\partial^2 \exp{\left[ i (\textbf{k}\cdot\textbf{r} - \omega t) \right]}}{\partial t^2} = 0 \)

    • \( (-k^2 + \frac{\omega^2}{c^2}) \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t) \right] } = 0 \)

    • \( ( -k^2 + k^2) \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t) \right] } = 0 \)

    • \( 0 \cdot \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t) \right] } = 0 \)

conclusion

  • \( \exp{ \left[ i (\textbf{k} \cdot \textbf{r} - \omega t ) \right] } \) is a solution for any vector direction \( \textbf{k} \) provided \( k = \frac{\omega}{c} \)
    • where: \( k^2 = k_x^2 + k_y^2 + k_z^2 \)

wave interference

  • consider a plane wave with wave-vector \(\textbf{k}_1 \)
    • which is a solution to the 3D wave equation

K-1

  • consider a second plane \(\textbf{k}_2 \)
    • which is a also a 3D wave equation solution

K-2

  • now, the linear combination of these two waves is a solution
    • the two waves \(\textbf{k}_1 \) and \(\textbf{k}_2 \) show interference

K1 + K2