# 莫列波紋

 莫列波紋, 由兩組平行線構成, 其中一組, 相對於另一組, 傾斜了 5° 在鸚鵡羽毛上出現的莫列波紋 兩個格狀柵欄圖形相疊產生的莫列波紋

## 計算

### 平行圖樣中的摩列

#### 幾何手法

the patterns are superimposed in the mid-width of the figure

$n = \frac{p}{2 \delta p}.$

$d = n \cdot p = \frac{p^2}{2 \delta p}$

$2d = \frac{p^2}{\delta p}$

當然，當 $\delta p = \frac{p}{2}$，無例外的可得一均勻灰階圖案。


#### 數學方程手法

於此節中我們將給出一個數學上的實例，與一種(出於許多種) 得出圖樣與摩列效應可以數學表示的方法。


$f = \frac{1 + \sin(k x)}{2}$

1的存在使得方程恆正，而除以2避免方程結果大於1。

$k$為強度週期/單位距離， 表示圖樣灰階強度的週期變動。因正弦方程對 $2 \pi$有循環，當 $k \Delta x = 2 \pi$，或 $\Delta x = \frac{2 \pi}{k}$時，可得每強度週期（波長）之距離增長。

$f_1 = \frac{1 + \sin(k_1 x)}{2}$
$f_2 = \frac{1 + \sin(k_2 x)}{2}$

$f_3 = \frac{f_1 + f_2}{2}$
$= \frac12 + \frac{\sin(k_1 x) + \sin(k_2 x)}{4}$
$= \frac{1 + \sin(A x) \cos(B x)}{2}$

$A = \frac{k_1 + k_2}{2}$

$B = \frac{k_1 - k_2}{2}.$

### 旋轉圖樣

"網"單位格; "ligne claire" ， "亮紋"

$(2 D)^2 = d^2 (1 + \cos \alpha)^2 + p^2$

id est

$(2D)^2 = \frac{p^2}{\sin^2 \alpha}(1+ \cos \alpha)^2 + p^2 = p^2 \cdot \left ( \frac{(1 + \cos \alpha)^2}{\sin^2 \alpha} + 1\right )$

$(2D)^2 = 2 p^2 \cdot \frac{1+\cos \alpha}{\sin^2 \alpha}$ or $D = \frac{p}{2} / \sin\frac{\alpha}{2}.$

$\alpha$ 極小 ($\alpha < \frac{\pi}{6}$)， 可做以下近似：

$\sin \alpha \approx \alpha$
$\cos \alpha \approx 1$

$D \approx \frac{p}{\alpha}.$

$\alpha \approx \frac{p}{D}$

## 意義與應用

### 列印全彩圖案

The product of two "beat tracks" of slightly different speeds overlaid, producing an audible moiré pattern; if the beats of one track correspond to where in space a black dot or line exists and the beats of the other track correspond to the points in space where a camera is sampling light, because the frequencies are not exactly the same and aligned perfectly together, beats (or samples) will align closely at some moments in time and far apart at other times. The closer together beats are, the darker it is at that spot; the farther apart, the lighter. The result is periodic in the same way as a graphic moiré pattern. See: phase (music).

## 參考

1. ^ Alexander Trabas. Beacons. Online-list-of-lights.info. [2012-10-30].