Intensity of interfering surface waves

Two point sources emit surface waves that interfere with each other. The amplitudes of the waves emitted by the two sources are given by,

$z_1 = A_1\cos (k|\vec{r}-\vec{r}_1|-\omega t +\phi_1)/\sqrt{|\vec{r}-\vec{r}_1|}$ cm

$z_2 = A_2\cos (k|\vec{r}-\vec{r}_2|-\omega t +\phi_2)/\sqrt{|\vec{r}-\vec{r}_2|}$ cm

Where $|\vec{r}-\vec{r}_1|$ is the distance from source 1 and $|\vec{r}-\vec{r}_2|$ is the distance from source 2. The wavenumber $k$ is related to the wavelength $\lambda$, $k=\frac{2\pi}{\lambda}$, and the angular frequency $\omega$ is related to the period $T$, $\omega=\frac{2\pi}{T}$.

If you focus on one point, it executes simple harmonic motion with a certain amplitude. Simple harmonic motion is the real part of circular motion in the complex plane. Written in complex form, the oscillations are,

$ \large \frac{A_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1|-\omega t +\phi_1)}+\frac{A_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| -\omega t+\phi_2)} = \left( \frac{A_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1|+\phi_1)}+\frac{A_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| +\phi_2)}\right)e^{-i\omega t}.$

The right side of this equation describes a phasor that moves in a circle in the complex plane with an amplitude,

$\large A= \frac{A_1}{\sqrt{|\vec{r}-\vec{r}_1|}} e^{ i(k|\vec{r}-\vec{r}_1|+\phi_1)}+\frac{A_2}{\sqrt{|\vec{r}-\vec{r}_2|}} e^{ i(k|\vec{r}-\vec{r}_2| +\phi_2)}.$

The intensity is the square of the amplitude,

$$ I = A^*A = \left(\frac{A_1}{\sqrt{|\vec{r}-\vec{r}_1|}} \cos(k|\vec{r}-\vec{r}_1|+\phi_1)+\frac{A_2}{\sqrt{|\vec{r}-\vec{r}_2|}} \cos(k|\vec{r}-\vec{r}_2| +\phi_2)\right)^2 \\ + \left(\frac{A_1}{\sqrt{|\vec{r}-\vec{r}_1|}} \sin(k|\vec{r}-\vec{r}_1|+\phi_1)+\frac{A_2}{\sqrt{|\vec{r}-\vec{r}_2|}} \sin(k|\vec{r}-\vec{r}_2| +\phi_2)\right)^2.$$

The form below plots the intensity of the waves as a function of the position.

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$|A|=$ [cm]

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$A_1=$

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$A_2=$

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$x_1=$

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$x_2=$

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$y_1=$

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$y_2=$

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$\phi_1=$

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$\phi_2=$

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$\lambda=$

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$P_x=$

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$P_y=$

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Helligkeit =

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.