18 Application of Fourier Series
1D Wave Equation¶
Assuming that the vibration only happens in one direction.
\[ \begin{aligned} a^2 &= \frac{T}{m} > 0 \\ \frac{\partial^2 y}{\partial t^2} &= a^2 \left( \frac{\partial^2 y}{\partial x^2} \right) \end{aligned} \]
\(a \ne\) acceleration
Conditions¶
| For | |||
|---|---|---|---|
| Initial Vertical Displacement at Left End | \(y(0, t)\) | \(0\) | \(\forall t\) |
| Initial Vertical Displacement at Right End | \(y(\pi, t)\) | \(0\) | \(\forall t\) |
| Vertical Velocity | \(\frac{\partial y}{\partial t}(x, 0)\) | \(0\) | \(0 \le x \le \pi\) |
| The function | \(y(x, 0)\) | \(f(x)\) | \(0 \le x \le \pi\) |
Solution¶
Solution of equation under the initial conditions
\[ \begin{aligned} y(x, t) &= \sum_{n = 1}^\infty b_n \sin(nx) \textcolor{hotpink}{\cos (nat)} \\ b_n &= \frac{2}{\pi} \int\limits_0^\pi f(x) \sin(nx) dx \end{aligned} \]
1D Heat Equation¶
Fourier Thermal Law¶
The amount of heat flowing through a heat-producing body \(H\)
- \(H \propto\) temperature gradient
- \(H \propto\) area of cross-section
- \(H \frac{1}{\propto}\) resistance
Time 0 is the time at which the external temperature is placed
Formula¶
\(\alpha^2\) is thermal diffusability.
\[ \begin{aligned} \alpha^2 &= \frac{k}{\rho c} > 0\\ \frac{\partial u}{\partial t} &= \alpha^2 \left( \frac{\partial^2 u}{\partial x^2} \right) \end{aligned} \]
Conditions¶
| For | |||
|---|---|---|---|
| Initial Heat at Left End | \(u(0, t)\) | 0 | \(\forall t\) |
| Initial Heat at Right End | \(u(\pi, t)\) | 0 | \(\forall t\) |
| \(u(x, 0)\) | \(f(x)\) | \(0 \le x \le \pi\) |
Solution¶
Solution of equation under the initial conditions
\[ \begin{aligned} u(x, t) &= \sum_{n = 1}^\infty b_n \sin (nx) \textcolor{hotpink}{e^{-n^2 \alpha^2 t}} \\ b_n &= \frac{2}{\pi} \int\limits_0^\pi f(x) \sin(nx) dx \end{aligned} \]