7 7.28, §7 Comparison

Separation of variables solution found as:

$$u = \sum_{n=1}^\infty \left[ f_n \cos\left(\frac{(2n-1)\... ...l}\right) \right] \cos\left(\frac{(2n-1)\pi x}{2\ell}\right)$$

• Shows the natural frequencies (tones) to be $\pi a/2\ell$, $3 \pi a/2\ell$,...
• Shows the energy in each harmonic.
• Not restricted to the 1D wave equation.

D’Alembert:

$$u(x,t) = \frac{\bar f(x-at) + \bar f(x+at)}{2} + \frac{1}{2a} \int_{x-at}^{x+at} \bar g(\xi) \d\xi$$

• I can evaluate the pressure at any point without doing big sums.
• Shows how wave fronts propagate.
• Shows regions of influence and dependence.