Geophysical time-series with non-stationary behaviour which contain transient excitations need specific tools to be analysed. Especially, it is difficult to detect and characterize transient excitations with attenuation by diffusion when their intensity is low compared to noise. This is a typical question in the search for the elusive translational motion of the inner core, the so-called Slichter triplet. This detection remains a non-trivial problem since the amplitude of the Slichter triplet is expected to be weak. Moreover, the characteristics (period, time of excitation and quality factor) of these low-intensity damped transient waves are poorly constrained. To improve the detection and characterization of such causal damped transient waves, we introduce a new continuous wavelet-based method using correlations with causal damped sinusoids that serve as the wavelet basis. This causal damped wavelet (CDW) method presents the advantage to give a direct estimate of the quality factor of the transient wave. After some synthetic tests, we present an application of the method to the time-varying gravity data recorded by superconducting gravimeters in the search for the Slichter triplet. No evidence of the Slichter triplet has been found in the 1-yr gravity data considered. However, we show that, thanks to the 3-D representation of our CDW method, it is possible to have a complete overview of the data set content and to constrain the detected events in terms of amplitude, quality factor, frequency and time.