Tính đối xứng Fourier và siêuposition sóng
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| harmonics/hɑːˈmɒnɪks/ | Frequencies that are whole-number multiples of the fundamental. |
| Fourier series/ˈfɔːrɪə ˈsɪəriːz/ | A sum of sines and cosines that represents a periodic function. |
| even/ˈiːvən/ | Unchanged when x is replaced by minus x. |
Harmonics 谐波, but no sines
- The triangle wave $|x|$ contains infinitely many harmonics, yet not a single sine term. Symmetry tells you this before any integration.
- GRE Fourier and superposition questions test coefficients, symmetry and coherent addition.
Fourier coefficients
- A Fourier series 傅里叶级数 writes $f(x) = \dfrac{a_0}{2} + \sum\left[a_n\cos nx + b_n\sin nx\right]$ with $a_n = \frac1\pi\int_{-\pi}^{\pi} f\cos nx\,dx$, $b_n = \frac1\pi\int_{-\pi}^{\pi} f\sin nx\,dx$.
- The mean value is $a_0/2$, not $a_0$.
Symmetry kills terms
- Even 偶 $f$: all $b_n = 0$. Odd $f$: mean and all $a_n = 0$.
- Symmetry depends on the origin: shifting a signal can mix sines and cosines without changing its harmonic frequencies.
Which Fourier terms?
Predict Fourier content from symmetry about x = 0.
The periodic extension of $f(x) = x$ on $(-\pi, \pi)$ contains
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$f(x) = x$ is odd, and an odd function has only odd (sine) components in its Fourier series.
An odd periodic function has zero mean over a period.
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True. For an odd function the contributions from $x$ and $-x$ cancel, so the average over a symmetric period is zero. Even functions such as $|x|$ need not have zero mean.
The triangle wave
- $|x|$ on $[-\pi, \pi]$: mean $\pi/2$; $a_n = \dfrac{2[(-1)^n - 1]}{\pi n^2}$, so $a_n = -\dfrac{4}{\pi n^2}$ for odd $n$ and 0 for even $n$.
- $1/n^2$ decay: continuous with corners. A jump (square wave) gives slower $1/n$ decay and overshoot.
Mean value of the periodic extension of $|x|$ on $[-\pi, \pi]$ (three decimal places)?
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The mean of $|x|$ over $[-\pi, \pi]$ is $\tfrac{1}{2\pi}\int_{-\pi}^{\pi}|x|\,dx = \pi/2 = 1.571$.
The coefficient $a_1$ of $\cos x$ for $|x|$ (three decimal places)?
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$a_1 = \tfrac1\pi\int_{-\pi}^{\pi}|x|\cos x\,dx = \tfrac2\pi\int_0^\pi x\cos x\,dx = \tfrac2\pi(-2) = -4/\pi = -1.273$.
Coherent superposition
- Equal coherent waves with phase difference $\varphi$: $I = 2I_0(1 + \cos\varphi)$. At $\varphi = \pi/2$, $2I_0$; at $\pi$, zero.
- Incoherent light averages away the cross term: $2I_0$ always.
First approximation to $|x|$: $\tfrac\pi2 - \tfrac4\pi\cos x$. At $x = 0$ it gives $\tfrac\pi2 - \tfrac4\pi \approx 0.298$, approaching the true 0 as more odd terms are added.
Two equal coherent waves (each $I_0$) with phase difference $2\pi/3$. Intensity in units of $I_0$?
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$I = 2I_0(1 + \cos\varphi)$ with $\cos(2\pi/3) = -1/2$ gives $2I_0 \times 1/2 = I_0$.
Recap
- Mean $= a_0/2$.
- Even → cosines only; odd → sines only (about the chosen origin).
- Coherent: add amplitudes; incoherent: add intensities.
- harmonics: Frequencies that are whole-number multiples of the fundamental.
- Fourier series: A sum of sines and cosines that represents a periodic function.
- even: Unchanged when x is replaced by minus x.