Fourier symmetry and wave superposition
| English | 中文 | Pinyin |
|---|---|---|
| orthogonality | 正交性 | zhèng jiāo xìng |
| Fourier coefficient | 傅里叶系数 | fù lǐ yè xì shù |
A decision before an answer
- A waveform can contain many harmonics while having no sine terms at all. Reflection symmetry tells you which coefficients vanish before integration.
- Your goal: Use orthogonality to identify Fourier coefficients.
Read the relationship
- For a sufficiently regular real 2π-periodic function, write f(x)=a0/2+Σ[a_n cos(nx)+b_n sin(nx)]. The coefficients are a_n=(1/π)∫from−πtoπ f(x)cos(nx)dx and b_n=(1/π)∫from−πtoπ f(x)sin(nx)dx, with a0 using n=0. Cosines and sines are orthogonal on a full period. The mean is a0/2, not a0; retain the constant term even when all nonzero-frequency coefficients of one family vanish.
- Exploit even and odd parity without discarding the constant term.
A real even 2π-periodic waveform about x=0 necessarily has:
Even times sine is odd, so each symmetric sine-coefficient integral is zero.
Use the defining rule
- If f is even about the chosen origin, f(x)sin(nx) is odd and its symmetric integral is zero, so all b_n vanish. If f is odd, its mean and all a_n vanish. Symmetry depends on the origin: shifting the same physical signal can mix sine and cosine coefficients while leaving its harmonic frequencies unchanged. A nonnegative triangular waveform symmetric about x=0 can therefore have cosine harmonics and a nonzero mean without any sine harmonics.
- Relate component amplitudes to interference and physical waveforms.
For the periodic extension of |x|, a2 is:
a_n=2[(-1)^n−1]/(πn²); even n gives zero.
Check the conditions
- For the 2π-periodic extension of f(x)=|x| on [−π,π], the mean is π/2. Integrating x cos(nx) by parts on [0,π] gives a_n=2[(-1)^n−1]/(πn²), so even-n cosine coefficients are zero and odd-n coefficients are −4/(πn²). All sine coefficients are zero. The 1/n² decay reflects a continuous function with a slope discontinuity. A jump discontinuity often gives slower coefficient decay and partial-sum overshoot; do not infer the same convergence behaviour for every waveform.
- Relate component amplitudes to interference and physical waveforms.
For f(x)=|x| periodically extended, the first cosine approximation is π/2−(4/π)cos x. It is even and has zero sine coefficients. At x=0 it gives π/2−4/π≈0.298, which approaches the exact 0 as more odd cosine terms are added. Two equal coherent waves at phase difference π/2 have intensity 2I0; their fields do not cancel completely.
If a0=6, the waveform mean is ____.
The constant term is a0/2.
Apply the task format
- A Fourier sum adds amplitudes with their phases. For equal coherent monochromatic amplitudes A meeting with phase difference φ, resultant squared amplitude is 2A²(1+cosφ), so intensity is 2I0(1+cosφ). It is 4I0 in phase and zero at φ=π. Incoherent averaging removes the cross term, yielding 2I0. Harmonics at different frequencies can construct a shape over time; their instantaneous sum is not the sum of their individual intensities. Identify whether the question concerns a waveform, time-average power or coherent interference.
- Relate component amplitudes to interference and physical waveforms.
Even symmetry removes sine coefficients, not the constant term or every cosine harmonic. Check the symmetry origin and whether fields are coherent before adding intensities.
Which answer fits this case?
Use orthogonality to identify Fourier coefficients
Shifting the origin can mix sine and cosine coefficients without changing the harmonic frequencies.
The angle-addition identities mix coefficients at the same frequency.
Keep the distinctions
- orthogonality 正交性 — A zero integral of the product of distinct basis functions over the specified interval.
- Fourier coefficient 傅里叶系数 — Weight of a sine, cosine or constant basis component in a periodic expansion.
- Use orthogonality to identify Fourier coefficients.
- Exploit even and odd parity without discarding the constant term.
- Relate component amplitudes to interference and physical waveforms.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.