4.5 Boundaries and reflection

On the half-line \(x > 0\) with a fixed end, \(u(0, t) = 0\), extend \(f\) and \(g\) to odd functions on \(\mathbb R\). The d’Alembert solution of the extended problem is odd in \(x\), so it vanishes at \(x = 0\), and by uniqueness it is the solution. For \(x \geq ct\) the extension plays no part, and for \(0 \leq x < ct\), \[ u(x, t) = \frac 12\bigl (f(x + ct) - f(ct - x)\bigr ) + \frac 1{2c}\int _{ct - x}^{x + ct}g(y)\,dy . \] A wave arriving at a fixed end is reflected upside down. For a free end, \(u_x(0, t) = 0\), the even extension gives a reflection the right way up. On a finite string \(0 \leq x \leq L\) with both ends fixed, the odd \(2L\)-periodic extension of the data gives the solution, which is therefore periodic in time with period \(2L/c\): the fundamental frequency of the string is \(c/2L = \sqrt {T/\rho }/2L\), Mersenne’s law. The decomposition into the overtones \(\sin (k\pi x/L)\), of frequencies \(kc/2L\), is the subject of the section on Fourier series.

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