Fig. 1. Sound focusing in homogeneous waveguides.

Fig. 2. Configuration of rigid waveguides with varying cross-sections.

Fig. 3. Acoustic focusing field in the waveguide as calculated by the present method. The foci are located at (a) (3.2, 0.9) in transmission region and (b) in scattering region, respectively. The blue solid lines in (c) and (d) are corresponding to (a) and (b), respectively, and the black dotted lines are generated by (plane wave). The insets plot the modulus of the corresponding incident waves.

Fig. 4. (a) Sound two-point focusing field in the waveguide with varying cross-section, the foci are located at and ; (b) modulus of the optimal incident pressure; (c) the blue solid line represents in (a); the red dot-dashed line shows when the wave focus only at , which is same as the blue solid line in Fig. 3(c); and the black dashed line shows when the wave focus only at . The frequency and geometries of the waveguide are same as Fig. 3.

Fig. 5. Configuration of rigid waveguides involving a scatterer.

Fig. 6. (a) Sound focusing field in the waveguide with a scatterer. The focus is located at ; (b) sound field generated by a plane wave ; (c) modulus of the pressure of optimal incident wave in (a) and that of the plane incident wave in (b); (d) in (a) (blue solid line) and (b) (black dotted line).

Fig. 7. (a) Sound focusing field in the waveguide with negative sound-speed gradient and a scatterer. The focus is located at , where is the depth; (b)the sound speed profile; (c) (blue solid line) compared with that generated by . The insetplotsthe modulus of the optimal incident pressure (blue solid line) and (black dashed line).

Fig. 8. Sound focusing fields when the optimal incident wave is discretized: (a) Half-wavelength spacing; (b) single-wavelength spacing; (c) the moduli of the two spaced incident waves; (d) the red dashed line and the black dot-dashed line are the corresponding generated by the incident waves in (c). The blue solid line is the theoretical result which is same as that in Fig. 3(c).

Fig. 9. Sound focusing fields when (a) the moduli and (b) the arguments of the optimal incident wave are perturbed; (c) the red dashed line is the incident wave with perturbed moduli, and the black dot-dashed line is that with perturbed arguments; (d) the red dashed line and the black dot-dashed line are the corresponding generated by the incident waves in (c). The blue solid line is the result without perturbation which is same as that in Fig. 3(c).