# MATH310S26 - Day 10: Fourier Modes and Complex Form ## Course Information **Course**: MATH310 - Introduction to Mathematical Modeling **Date**: February 6, 2026 (Day 10) **Topics**: Fourier modes on strings, complex Fourier series, Euler's identity **Previous Day**: [[MATH310S26-Day9-Modeling Discussions]] ## Administrative Notes - Next work day: February 13 (Friday) - President's Day off will affect schedule - All MR01 and MR02 feedback has been posted - Career day was on Wednesday - Golden retriever count: 5 on the way to campus ## Main Topics ### 1. Fourier Series Review **Core Concept**: Creating shapes/waves/signals through constructive and destructive interference of waves (continuing from [[MATH310S26-Day7-Notes]] and [[MATH310S26-Day8-Notes]]) $f(x) = a_0 + \sum_{n=1}^{\infty} \left[ a_n \cos(\omega_n x) + b_n \sin(\omega_n x) \right]$ where $\omega_n = \frac{n\pi}{L}$ for a domain of length $L$   ### 2. Vibrating String Model **Physical Setup**: String of length $L$ with fixed endpoints (like a stringed instrument) - **[Boundary conditions](https://en.wikipedia.org/wiki/Boundary_value_problem)**: $f(0) = 0$ and $f(L) = 0$ - **Energy finite shapes**: Area under the square of the curve must be finite, i.e., $\displaystyle \left<f,f\right>=||f||=\int_{-L}^{L}f^2 dt<\infty.$ - **Scaled Fourier series**: Adapted to interval $[-L, L]$ instead of $[-\pi, \pi]$ **Key Insight**: The red shape on the string isn't periodic, but we can create an **[odd periodic extension](https://en.wikipedia.org/wiki/Even_and_odd_functions)** to apply Fourier analysis: - Mirror the shape into quadrant 2 - Mirror down into quadrant 3 - Repeat the pattern For [odd functions](https://en.wikipedia.org/wiki/Even_and_odd_functions#Odd_functions): $a_0 = 0$ and all $a_n = 0$, leaving only: $f(x) = \sum_{n=1}^{\infty} b_n \sin(\omega_n x)$ ### 3. Modes and Nodes **Terminology**: - **[Mode](https://en.wikipedia.org/wiki/Normal_mode)**: The $n$-th basis function (sinusoid/wave/basis vector) - **[Nodes](https://en.wikipedia.org/wiki/Node_(physics))**: Zeros of the mode function (also called roots) **Pattern**: - First mode ($n=1$): One half-wave, 2 nodes at endpoints - Second mode ($n=2$): Two half-waves, 3 nodes (including midpoint) - Higher modes: More oscillations, more nodes ### 4. Musical Acoustics Demonstrations #### Ukulele String Example Demonstrated plucking string and using finger placement to isolate modes: - Finger at $L/2$ suppresses [fundamental frequency](https://en.wikipedia.org/wiki/Fundamental_frequency), emphasizes second [harmonic](https://en.wikipedia.org/wiki/Harmonic) - Physical demonstration connects to mathematical modes #### YouTube Demonstrations 1. **String harmonics visualization**: Showed fundamental plus harmonics combining (embedding of shorts is not supported in Obsidian. So, here is the [link](https://youtube.com/shorts/fzo8gcmcds8?si=Um9bRQ4cxH3bSePy)) ) 2. **Flute [frequency spectrum](https://en.wikipedia.org/wiki/Frequency_spectrum)**: - Fundamental frequency ≈ 500 Hz defines the note - Harmonics create [timbre](https://en.wikipedia.org/wiki/Timbre)/tone color - Background noise appears as baseline in [frequency domain](https://en.wikipedia.org/wiki/Frequency_domain)  **Key Insight**: "Your ear is great" - we perceive the fundamental as the note, while harmonics provide richness and timbre ### 5. Complex Fourier Series #### [Euler's Identity](https://en.wikipedia.org/wiki/Euler%27s_identity) $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ **Special case**: $e^{i\pi} + 1 = 0$ ([Feynman](https://en.wikipedia.org/wiki/Richard_Feynman): "one of the most important formulas") #### Deriving Complex Form Using Euler's identity to rewrite [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions): $\cos(\theta) = \frac{e^{i\theta} + e^{-i\theta}}{2}$ $\sin(\theta) = \frac{e^{i\theta} - e^{-i\theta}}{2i}$ Substituting into Fourier series yields: $f(x) = a_0 + \sum_{n=1}^{\infty} \left[ c_n e^{i\omega_n x} + \overline{c_n} e^{-i\omega_n x} \right]$ where: - $c_n = \frac{a_n - ib_n}{2}$ - $\overline{c_n}$ is the [complex conjugate](https://en.wikipedia.org/wiki/Complex_conjugate) #### Final Complex Form $f(x) = \sum_{n=-\infty}^{\infty} c_n e^{i\omega_n x}$ where: $c_n = \frac{1}{2L} \int_{-L}^{L} f(x) e^{-i\omega_n x} dx$ **Key advantages**: - Single formula for all coefficients (including $n=0$) - Symmetric summation from $-\infty$ to $\infty$ - Natural connection to Fourier transforms ### 6. Looking Ahead: Fourier Transforms Current limitation: Only allowing discrete frequencies $\omega_n = \frac{n\pi}{L}$ **Goal**: Extend to all frequencies by taking $L \to \infty$, leading to: - Continuous frequency spectrum - [Fast Fourier Transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT) algorithms - Practical data analysis tools ## Check Your Understanding 1. **Prove Euler's identity** using [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) expansion - Show $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ - Assume ratio test works with complex numbers 2. **Mode analysis**: For a string of length $L$, how many nodes does the $n$-th sine mode have? Where are they located? 3. **Complex conjugates**: Show that if $f(x)$ is real-valued, then $c_{-n} = \overline{c_n}$ in the complex Fourier series ## Key Takeaways 1. **Physical-mathematical connection**: Vibrating strings naturally produce Fourier modes 2. **Nodes define modes**: Higher modes have more nodes and higher frequencies 3. **Complex form unifies**: The complex exponential form simplifies formulas and reveals deeper structure 4. **Frequency domain insight**: What we hear as "notes" are superpositions of many frequencies 5. **Mathematical elegance**: Euler's identity connects exponentials, trigonometry, and complex numbers ## Mathematical Connections - **Previous**: [[MATH310S26-Day3-WorkdayMaterials (Two-by-two matrices, normal equations, and ordinary least squares)|Ordinary least squares]] as projection (Days 1-3) → [[MATH310S26-Day5-Notes|PCA and eigenanalysis]] (Days 4-6) → Fourier as projection onto sinusoids - **Current**: Building from real to complex representations - **Next**: [Fourier transforms](https://en.wikipedia.org/wiki/Fourier_transform) for continuous frequency analysis ## References - [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula) - [Fourier series](https://en.wikipedia.org/wiki/Fourier_series) - [Harmonic series (music)](https://en.wikipedia.org/wiki/Harmonic_series_(music)) - [Timbre](https://en.wikipedia.org/wiki/Timbre) - [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon) - [Partial differential equations](https://en.wikipedia.org/wiki/Partial_differential_equation) - [Signal processing](https://en.wikipedia.org/wiki/Signal_processing)