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| This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval. | This lecture will do apply separation of variables and Fourier series in order to solve for the wave equation on a finite interval. | ||
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| + | < | ||
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| ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs ===== | ===== Definition 16.1 (1D wave equation with homogeneous Dirichlet BCs ===== | ||
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| + | ==== Implementation of the Fourier series ==== | ||
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| + | Now we need to look to solve for the coefficients of our series by applying the boundary conditions. We have that | ||
| + | $$ | ||
| + | u(x, t) = \sum_{n=0}^\infty \sin\left(\frac{n\pi x}{L}\right) \left[ A_n \cos\left(\frac{n\pi ct}{L}\right) + B_n \sin\left(\frac{n\pi ct}{L}\right)\right] | ||
| + | $$ | ||
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| + | Imposing the initial displacement, | ||
| + | $$ | ||
| + | u_0(x) = \sum_{n=0}^\infty A_n \sin\left(\frac{n\pi x}{L}\right), | ||
| + | $$ | ||
| + | |||
| + | which we recognise as the sine series for the odd 2L-periodic extension of the function u0(x) originally defined on [0,L]. So the coefficients are (see theorem 12.7) | ||
| + | $$ | ||
| + | A_n = \frac{2}{L} \int_0^L u_0(x) \sin\left(\frac{n\pi x}{L}\right). | ||
| + | $$ | ||
| + | |||
| + | Imposing the initial velocity, we have | ||
| + | $$ | ||
| + | v_0(x) = \sum_{n=1}^\infty \left(\frac{n\pi c}{L}\right)B_n \sin\left(\frac{n\pi x}{L}\right) | ||
| + | $$ | ||
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| + | Again we recognise this as the sine series, so we now need to equate | ||
| + | $$ | ||
| + | \left(\frac{n\pi c}{L}\right)B_n = \frac{2}{L} \int_0^L v_0(x) \sin\left(\frac{n\pi x}{L}\right). | ||
| + | $$ | ||
| + | |||
| + | //(The video gets very close to the end of this; we managed to get the An coefficients and need to address the Bn coefficients in lecture 26)// | ||