Many founding fathers of science have underscored the importance of beauty in mathematical representations of natural phenomena and their connection with the beauty of the objects they represent. Paul Dirac, for example, believed that the beauty of a mathematical equation might be an indication that it describes a fundamental law of nature; that it is a feature of nature in that fundamental physical laws are typically described in terms of great beauty. However, despite the growing body of research on beauty in nature and in mathematical representations of natural phenomena, we are unaware of studies in physics and mathematics devoted to the objective beauty induced by motion, regardless of the aesthetic qualities of the moving body. We undertake this objective by focusing on the Doppler formula, which describes the shifts in wave frequencies caused by the motion of the wave's source relative to a human observer or receiver. We underscore the apparent beauty of the equation and uncover several fascinating golden and silver ratios of the base formula and its mathematical moments. Furthermore, we allude to existing applications of the Doppler Effect and golden ratio aesthetics in computer-generated music, and sonar image-detection technology. We also propose a similar usage in the rapidly developing applications of Wi-Fi and smartphones to sense human motion. We point to appearances of the Doppler formula and its moments in quantum physics and the relativity of information, and contemplate the possibility of a deeper level of physical reality.