Question

See also the slides on Moire patterns. A plane sinusoidal wave travelling in the x- direction is $p_1(x, t) = Ae^{i(k_1x - \omega t)}$ A plane sinusoidal wave travelling in the $\hat{n}$ - direction is expressed: $p_2(x, t) = Ae^{i(\vec{k_2} \cdot \vec{r} - \omega t)}$ where $\vec{k_2} = \frac{2\pi}{\lambda} \hat{n}$ and $\vec{r} = (x, y, z)$ is the position. Two waves of the same frequency and thus the same magnitude wavenumber ($k = k_1 = |\vec{k_2}|$), travelling in different directions interfere. Let $\vec{k_2} = k(cos \theta, sin \theta)$ and let $\vec{r} = (x, y)$. The bright regions in the Moire pattern from the slides represent constructive interference. a. Find the equation for the lines of constructive interference. They should be of the form $y = ax + b$. The slope, $a$, should be expressed in terms of $\theta$. The y- intercept, $b$, should be in terms of $\theta$ and an integer. b. Show that as $\theta \to 0$ the slope $a$ approaches zero. What is the spacing along the y- axis of adjacent lines of constructive interference? Show that as $\theta \to 0$ the spacing between adjacent lines approaches $\infty$.

          See also the slides on Moire patterns. A plane sinusoidal wave travelling in the x-
direction is
$p_1(x, t) = Ae^{i(k_1x - \omega t)}$
A plane sinusoidal wave travelling in the $\hat{n}$ - direction is expressed:
$p_2(x, t) = Ae^{i(\vec{k_2} \cdot \vec{r} - \omega t)}$
where $\vec{k_2} = \frac{2\pi}{\lambda} \hat{n}$ and $\vec{r} = (x, y, z)$ is the position. Two waves of the same frequency
and thus the same magnitude wavenumber ($k = k_1 = |\vec{k_2}|$), travelling in different
directions interfere. Let $\vec{k_2} = k(cos \theta, sin \theta)$ and let $\vec{r} = (x, y)$. The bright regions
in the Moire pattern from the slides represent constructive interference.
a. Find the equation for the lines of constructive interference. They should be of the
form $y = ax + b$. The slope, $a$, should be expressed in terms of $\theta$. The y-
intercept, $b$, should be in terms of $\theta$ and an integer.
b. Show that as $\theta \to 0$ the slope $a$ approaches zero. What is the spacing along the y-
axis of adjacent lines of constructive interference? Show that as $\theta \to 0$ the
spacing between adjacent lines approaches $\infty$.

See also the slides on Moire patterns. A plane sinusoidal wave travelling in the x-
direction is
p1(x, t) = Ae^i(k1x - ω t)
A plane sinusoidal wave travelling in the n̂ - direction is expressed:
p2(x, t) = Ae^i(k⃗2⃗·r⃗ - ω t)
where k⃗2⃗ = (2π)/(λ)n̂ and r⃗ = (x, y, z) is the position. Two waves of the same frequency
and thus the same magnitude wavenumber (k = k1 = |k⃗2⃗|), travelling in different
directions interfere. Let k⃗2⃗ = k(cos θ, sin θ) and let r⃗ = (x, y). The bright regions
in the Moire pattern from the slides represent constructive interference.
a. Find the equation for the lines of constructive interference. They should be of the
form y = ax + b. The slope, a, should be expressed in terms of θ. The y-
intercept, b, should be in terms of θ and an integer.
b. Show that as θ→ 0 the slope a approaches zero. What is the spacing along the y-
axis of adjacent lines of constructive interference? Show that as θ→ 0 the
spacing between adjacent lines approaches ∞.

Added by Kayla G.

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