PHY.K02UF Molecular and Solid State Physics

Plotting the intensity of the scattered waves

When plane waves scatter off two point scatterers in two dimensions, the amplitude of the scattered waves can be factored into a time-independent part $\mathcal{A}(\vec{r})$ and a time-dependent factor $e^{-i\omega t}$,

$$\left( \frac{F_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1| +kx_1)}+\frac{F_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| +kx_2)} \right) e^{-i\omega t} = \mathcal{A}(\vec{r})e^{-i\omega t}.$$

The time-independent part is a complex number and the magnitude of this complex number $|\mathcal{A}(\vec{r})|$ is the amplitude of the oscillations at point $\vec{r}$. The intensity $I$ is proportional to the amplitude squared,

$$I\propto \mathcal{A}^*\mathcal{A}.$$

The form below plots the intensity of the waves as a function of the position. The intensity pattern is time-independent.

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$|A|=$ [cm]

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 $F_1=$ [cm3/2] $F_2=$ [cm3/2] $x_1=$ [cm] $x_2=$ [cm] $y_1=$ [cm] $y_2=$ [cm] $\lambda=$ [cm] $P_x=$ [cm] $P_y=$ [cm] Brightness =

There is a red point on the intensity pattern. The coordinates of this point are $P_x$ and $P_y$. To the left of the intensity pattern is a representation in the complex plane of the harmonic oscillations at the red point. The blue and green phasors represent the harmonic motion caused by the waves traveling from the two sources. The center of one wave source is indicated by the green point and the other is indicated by a blue point. The red phasor is the sum of the blue and green phasors. The real part of the red phasor is the motion observed at the red point.