## 1 Introduction  
Acoustic methods for localizing a source in an underwater acoustic waveguide, including direction finding and range and depth estimation, are a topic of great interest in the field of underwater acoustic signal processing. Popular methods for estimating the source range and depth are matched field processing (MFP) [1, 2] and matched mode processing (MMP) [3, 4], which require multiple hydrophones to form a vertical or horizontal array to receive the source radiated noise. Knowledge of the environment, such as the sound speed profile (SSP) and the geoacoustic parameters, is also needed to calculate the copy field vector for MFP or the modal coefficient vector for MMP using a propagation model to estimate source range and depth. The MFP and MMP methods are often plagued by environmental mismatch problems: if the assumed environment differs from the true environment, there may be large error in the final position estimate [5, 6]. A number of robust localization algorithms have been developed to solve the mismatch problem [7, 8, 9, 10], and will not be described here.  
In actual use, the hydrophone array is expensive, and it is cumbersome to deploy at sea. Therefore, the idea of using a single hydrophone to perform source range localization emerged. A single hydrophone can be deployed more quickly and flexibly, at a lower cost, and monitoring of a large area can be realized by mass deployment. The principle of the early single-hydrophone localization methods is similar to the MFP method [11; 12]. In general, a wideband acoustic signal or a pulsed acoustic signal is required, as well as accurate environmental information, so there is still the problem of environmental mismatch. Later, methods based on the inherent characteristics of the sound field were developed. [PERSON] and [PERSON] [13] developed a ranging method based on the waveguide invariant. This method calculates the slope of interference patterns in different regions on the range-frequency plane and uses the principle of waveguide invariance to achieve robust range measurement. [PERSON] and [PERSON] [14] pointed out that the magnitude of the square of the difference between the acoustic field at two different ranges contains information about the range interval, and the range rate can be ascertained. [PERSON] et al. [15] developed a single-hydrophone ranging method using the nonlinear signal processing method warping transform to filter the modes. Once modal arrivals have been separated, the source range can be estimated using the conventional modal dispersion technique. [PERSON] et al. [16] have also developed a single-hydrophone ranging method based on repeat Fourier transformation of the interference pattern formed during source motion.  
Here we derive a source range localization method based on single-hydrophone synthetic aperture beamforming. The horizontal wavenumber spectrum excited by the source can be obtained by the synthetic aperture beamforming of a single hydrophone. According to the relationship between the maximum value of the horizontal wavenumber spectrum, the average horizontal wavenumber, and the radial velocity of the source, range localization can be achieved.  
The method is derived and illustrated with simulated results in Sections 2 and 3, respectively. Then, the method is validated with data from a shallow water experiment in Section 4. Section 5 concludes the work.