void standing_wave_exact ( double x, double t, double c, double &u, 
  double &ut, double &utt, double &ux, double &uxx );
void standing_wave_parameters ( double *c_in, double *xmin_in, double *xmax_in, 
  double *t0_in, double *tstop_in, double *c_out, double *xmin_out, 
  double *xmax_out, double *t0_out, double *tstop_out );
void standing_wave_residual ( double x, double t, double c, double &r );