In integrated sensing and communications (ISAC), a distinguishing feature of 6G wireless networks, the main challenge lies in integrating the two distinct functions of sensing and communication within the same waveform. In this paper, the ISAC waveform synthesis is studied in the wideband regime, since a large bandwidth can simplify the analysis and is justified by the employment of millimeter wave or higher frequency band. Standard orthogonal frequency division multiplexing (OFDM) signaling is assumed, and the wideband analysis of sensing is a counterpart of the existing studies on wideband communications. It is proposed that the phase over such OFDM subcarriers is for modulating communication messages while the power spectral density (PSD) is shaped for the sensing performance. Beyond OFDM, we further reveal a duality between the proposed PSD-shaping rule and the orthogonal time frequency space (OTFS) waveform. Flattening the OTFS delay-axis PSD produces the same integrated sidelobe level (ISL) reduction effect in the delay-Doppler domain as PSD control achieves for OFDM in the frequency domain. To balance communication and sensing performance over frequency-selective channels, we propose a low-complexity, water-filling-like allocator with an explicit PSD-flatness (variance) constraint. The performance of the proposed wideband ISAC scheme is demonstrated using both numerical simulations and hardware experiments.