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| ===== Example 16.2 (Imagining the modes) ===== | ===== Example 16.2 (Imagining the modes) ===== | ||
| - | I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$. | + | I want to help you imagine what the modes, $u_n$, look like. For simplicity, take the length $L = \pi$. Also, we may take $c = 1$. The individual modes we want to examine are: |
| + | $$ | ||
| + | u_n(x, t) = \sin\left(nx\right) \left[ A_n \cos(nct) + B_n \sin(nct)\right] | ||
| + | $$ | ||