## 1 Introduction  
The Hopkinson pressure bar has been the primary means of recording expected high magnitude, short duration, initial loading for over one hundred years, since 1914 [1]. The technology is still widely used in the research fields of explosion [2, 3] and shock wave structural interaction [4]. As a sensitive element, the slender cylindrical bar is mainly optimized in material and size, while the measuring element is developed from detachable end-pieces [1], capacitance [5], and a strain gauge [6] attached to photonic doppler velocimetry (PDV) [7].  
The main problem that has puzzled the measurement scientists is that the axial elastic stress wave is dispersed due to the lateral inertia effect, which leads to inconsistency between the signal obtained by the measuring element and the actual load on the front end of the pressure bar. The theoretical basis of most dispersion analysis and correction methods is the analytical solution of axisymmetric load in an infinite isotropic elastic circular bar, independently proposed by [PERSON] [8] and [PERSON] [9], which is generally called the P-C frequency equation. In 1941, [PERSON] [10] used a numerical calculation method to solve the P-C frequency equation and obtained the first-order phase velocity curve and displacement along the radius of the section of the axial elastic stress wave in the cylindrical bar under different Poisson's ratios. In 1948, [PERSON] [5] experimentally verified the solution of the P-C equation and found that the group velocity of the high-frequency component of the stress pulse in the basic propagation mode is lower than the low-frequency component, so the high-frequency oscillation follows the main pulse. In 1978, [PERSON] [11] used the fast Fourier transform method to analyze the strain signals at two measuring points on the pressure bar, and the phase velocity of each frequency component of the axial longitudinal wave obtained was consistent with the theoretical value of the first-order propagation mode of the P-C equation. These studies prove that the finite length pressure bar can still be studied based on P-C theory in practical applications.  
[PERSON] [12] and [PERSON] [13] independently calculated phase shifts from the first mode of the Pochhammer-Chree dispersion relation and the sensor's distance from the pressure end. The incident signal is reconstructed from the frequency components, undispersed by the phase shifts. This frequency domain phase-shift method is already commonly employed in practical pressure bar work (e.g., [PERSON] [14], [PERSON] [15], [PERSON] [16]). In 1995, [PERSON] [17] constructed a group velocity curve from the Gaussian-windowed Fourier transform of the measured signal. The frequency components of the signal are shifted according to the mode with the dominant group velocity. [PERSON] [18; 19; 20] improved the dispersion correction method by the application of additional correction factors to the amplitude of the Fourier components, derived from the Pochhammer-Chree results for axial strain variation over the bar radius and stress-strain relationship. [PERSON] [21] presents a detailed review of the current status of Hopkinson pressure bar data analysis.  
For the free surface velocity measured in our system, we developed a local phase-amplitude joint correction method considering factors such as propagation mode of the elastic wave, frequency components' time of arrival, velocity variation over the bar axis, and the stress-velocity relation. This method has the capability of identifying the propagation mode and correcting multi-mode dispersion except for the non-axial elastic wave.