Synthetic Aperture Sonar (SAS) is an advanced sonar imaging technique that uses multiple pulses or high duty cycle waveforms from a moving surface vehicle to create a synthetic aperture array for imaging. One of the key challenges of SAS systems is the design of continuous waveforms that enable high-quality imaging under practical constraints (e.g., unimodular constraint). This thesis addresses the design of such waveforms and observes that waveform design for a SAS system over a short time window can be approximated as a wideband ambiguity function shaping problem. To tackle this, we formulate the wideband ambiguity function shaping problem as a non-convex optimization problem and propose four methods to solve it. Among the proposed methods, the wideband gradient descent method is proven to be the most efficient and effective in minimizing the average sidelobe energy in the region of interest of the wideband ambiguity function. Simulation and field trial results show that, although waveform design for SAS systems is not strictly equivalent to wideband ambiguity function shaping, the waveform obtained through this approach still yields good SAS imaging performance compared to conventional sonar waveforms, such as the random Binary Phase-Shift Keying (BPSK) waveform. These findings provide a direction for the waveform design of future SAS systems. Beyond SAS, the proposed wideband ambiguity function shaping methods also show potential for other wideband applications, such as waveform design for underwater target tracking.