Transducers used in acoustic logging while drilling (ALWD) must be mounted on a drill collar, and their radiation performance is dependent on the employed mounting method. Herein, the complex transmitting voltage response of a while-drilling (WD) monopole acoustic source was calculated through finite-element harmonic-response analysis. Subsequently, the acoustic pressure waveform radiated by the source driven by a half-sine excitation voltage signal was calculated using the complex transmitting voltage response. The calculation results were compared with those obtained using finite-element transient analysis to verify the accuracy of the calculation method. The influence of transducer-mounting methods on the radiation performance of the monopole acoustic source was examined by modifying the material and structural dimensions of the coupling medium between the transducer and drill collar as well as the material and thickness of the protective cover. Numerical simulations were performed, and a transducer-mounting method suitable for ALWD was proposed based on the simulation results. Results showed that soft rubber (as the coupling material; thickness = 2 mm) enabled the WD monopole acoustic source to radiate robust acoustic energy in an infinite fluid. Increasing the height of the coupling material enhanced the radiated acoustic energy and reduced axial vibrations on the drill collar. The radiated acoustic pressure signal was unaffected by a steel protective cover (thickness = 0.5 mm). Conversely, increasing the cover thickness reduced the energy of the radiated acoustic signal. With increasing pulse width of the half-sine excitation voltage signal, the amplitude of the radiated acoustic pressure of the transducer initially increased and then declined, reaching a maximum at a pulse width that was 0.6 times the resonant period. Overall, the findings help in designing acoustic-source structures and excitation signals for ALWD tools.