[QMSE] W02 - Plane Wave and Interference

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

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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

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generalizing wave equation to 3D

• applying gradient operator to 1D wave equation gives
  • ${\nabla }^2\varphi -\frac{1}{c^2}\frac{{\partial }^2\varphi }{\partial t^2}=0$
  • 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
  • ${\nabla }^2=\nabla \cdot \nabla$
  • ${\nabla }^2=(\hat{x}\frac{\partial }{\partial x}+\hat{y}\frac{\partial }{\partial y}+\hat{z}\frac{\partial }{\partial z})\cdot (\hat{x}\frac{\partial }{\partial x}+\hat{y}\frac{\partial }{\partial y}+\hat{z}\frac{\partial }{\partial z})$

plane wave solutions

• a (monochromatic) plane wave is given by

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

  • where

    • $\text{r}=x\hat{\text{x}}+y\hat{\text{y}}+z\hat{\text{z}}$

    • $\text{k}=k_x\hat{\text{x}}+k_y\hat{\text{y}}+k_z\hat{\text{z}}$

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

time partials of plane wave

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

verify solution plane wave in 3D wave

• 3D wave:
  • ${\nabla }^2\varphi -\frac{1}{c^2}\frac{{\partial }^2\varphi }{\partial t^2}=0$

• plugging in space and time partials of plane wave:

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

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

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

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

conclusion

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

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wave interference

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

• consider a second plane ${\text{k}}_2$
  • which is a also a 3D wave equation solution

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