Fourier symmetry and wave superposition
GRE Subject Test · GRE Physics Topic 18 5:36 English narration · English + 中文 subtitles burned in
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A waveform can contain many harmonics and still have no sine terms at all: reflection symmetry tells you which coefficients vanish before you integrate.
一个波形可以包含许多谐波，却完全没有正弦项：反射对称性在积分之前就告诉你哪些系数为零。
This topic uses orthogonality to find Fourier coefficients, uses even and odd parity without dropping the constant term, and relates component amplitudes to interference and real waveforms.
本主题用正交性求傅里叶系数，利用奇偶性而不丢掉常数项，并把分量振幅与干涉和实际波形联系起来。
For a sufficiently regular real two pi-periodic function, write f of x equals a sub zero divided by two, plus the sum from n equals one to infinity of a sub n times cosine of n x, plus b sub n times sine of n x.
对于一个足够规则的二派周期实函数，将f(x)写为a₀/2加上从n等于1到无穷大的求和，即aₙ乘以cos(nx)加上bₙ乘以sin(nx)。
The coefficients are a sub n equals one over pi times the integral from negative pi to positive pi of f of x times cosine of n x, with respect to x.
系数aₙ等于一除以派乘以从负派到正派对f(x)乘以cos(nx)关于x的积分。
And b sub n equals one over pi times the integral from negative pi to positive pi of f of x times sine of n x, with respect to x.
bₙ等于一除以派乘以从负派到正派对f(x)乘以sin(nx)关于x的积分。
Note that a sub zero uses n equals zero.
注意a₀使用n等于零。
Cosines and sines are orthogonal on a full period.
余弦函数和正弦函数在一个完整周期内是正交的。
The mean is a sub zero divided by two, not a sub zero.
平均值是a₀除以二，而不是a₀。
Retain the constant term even when all nonzero-frequency coefficients of one family vanish.
即使某一族的所有非零频率系数都消失，也要保留常数项。
If f is even about the chosen origin, f of x times sine of n x is odd and its symmetric integral is zero, so all b sub n vanish.
如果f关于所选原点是偶函数，则f(x)乘以sin(nx)是奇函数，其对称积分为零，因此所有b_n均为零。
If f is odd, its mean and all a sub n vanish.
如果f是奇函数，则其均值和所有a_n均为零。
Symmetry depends on the origin: shifting the same physical signal can mix sine and cosine coefficients while leaving its harmonic frequencies unchanged.
对称性取决于原点：平移同一物理信号可能会混合正弦与余弦系数，但其谐波频率保持不变。
A nonnegative triangular waveform symmetric about x equals zero can therefore have cosine harmonics and a nonzero mean without any sine harmonics.
因此，一个关于x等于0对称的非负三角波形可以具有余弦谐波和非零均值，而不含任何正弦谐波。
For the 2 pi-periodic extension of f of x equals absolute value of x on negative pi to positive pi, the mean is pi over 2.
对于f(x) = |x|在区间 上的2π周期延拓，其均值为π/2。
Integrating x cos of n x by parts on 0 to pi gives a sub n equals 2 times open bracket negative 1 to the power of n minus 1 close bracket all over pi n squared. So even-n cosine coefficients are zero and odd-n coefficients are negative 4 over pi n squared.
在 上对x cos(nx)进行分部积分可得a_n = 2 /(πn²)，因此偶数项余弦系数为零，奇数项系数为−4/(πn²)。
All sine coefficients are zero.
所有正弦系数均为零。
The 1 over n squared decay reflects a continuous function with a slope discontinuity.
1/n²的衰减反映了该函数连续但存在斜率不连续点。
A jump discontinuity often gives slower coefficient decay and partial-sum overshoot. Do not infer the same convergence behaviour for every waveform.
跳跃间断点通常会导致更慢的系数衰减及部分和过冲现象；切勿对所有波形都推断出相同的收敛行为。
A Fourier sum adds amplitudes with their phases.
傅里叶求和是把振幅连同相位一起相加。
For equal coherent monochromatic amplitudes A meeting with phase difference phi, the resultant squared amplitude is 2 A squared times 1 plus cosine phi, so the intensity is 2 I nought times 1 plus cosine phi.
两列振幅同为 A 的相干单色波以相位差 φ 相遇时，合振幅的平方为 2 A 平方乘以 1 加 cos φ，因此强度为 2 I0 乘以 1 加 cos φ。
It is 4 I nought in phase and zero at phi equals pi.
同相时强度为 4 I0，φ 等于 π 时为零。
Incoherent averaging removes the cross term, yielding 2 I nought.
非相干平均消去交叉项，得到 2 I0。
Harmonics at different frequencies can construct a shape over time; their instantaneous sum is not the sum of their individual intensities.
不同频率的谐波可以随时间合成一个波形；它们的瞬时和并不等于各自强度之和。
Identify whether the question concerns a waveform, time-average power or coherent interference.
请判断问题涉及的是波形、时间平均功率还是相干干涉。
For f of x equals the absolute value of x, even symmetry removes every sine term.
对于 f(x) 等于 x 的绝对值，偶对称消去了所有正弦项。
Integrating by parts, a n is 2 times minus 1 to the n, minus 1, all over pi n squared.
分部积分得到 a_n 等于 2 乘以 (-1) 的 n 次方减 1，再除以 π n 的平方。
The mean is pi over 2, and only the odd cosine harmonics survive.
平均值是二分之 π，只有奇次余弦谐波保留下来。
Shift this waveform by pi over two. Let g of x equal f of x minus pi over two.
将此波形平移π/2：g(x) = f(x − π/2)。
Write its first nonconstant harmonic and its mean.
写出其第一个非恒定谐波及其均值。
Explain why its sine coefficient need not vanish, although the unshifted waveform is even.
解释为何尽管未平移的波形是偶函数，其正弦系数却未必为零。
Substitute the shift in f approximately equals pi over 2 minus 4 over pi times cosine of x, to get g approximately equals pi over 2 minus 4 over pi times sine of x.
将f约等于pi除以2减去4除以pi乘以cos x中的位移代入，得到g约等于pi除以2减去4除以pi乘以sin x。
Its mean remains pi over 2.
其均值仍为pi除以2。
Evenness depends on the chosen origin; g is not even about zero.
偶性取决于所选原点；g关于零点不是偶函数。
Frequencies stay unchanged.
频率保持不变。
Award substitution, harmonic or sign change, mean and origin or frequency interpretation.
对代换、谐波或符号变化、均值以及原点或频率的解释给予评分。
Even symmetry removes sine coefficients, not the constant term or every cosine harmonic.
偶对称会消除正弦系数，但不会消除常数项或每一个余弦谐波。
Check the symmetry origin and whether fields are coherent before adding intensities.
在叠加强度之前，请检查对称性的原点以及场是否相干。
orthogonality: A zero integral of the product of distinct basis functions over the specified interval.
正交性：指定区间内不同基函数乘积的积分为零。
Fourier coefficient: Weight of a sine, cosine or constant basis component in a periodic expansion.
傅里叶系数：周期展开中正弦、余弦或常数基分量的权重。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后更改原始传递中的一个参数并检验极限情况。