This study introduces new families of wavelets. The first is directly derived from the response of Second Order Underdamped Linear-Time-Invariant (SOULTI) systems, while the second is a generalization of the first to the complex domain and is similar to the Laplace transform kernel function. The first takes the acronym of SOULTI wavelet, while the second is named the Laplace wavelet. The most important criteria for a function or signal to be a wavelet is the ability to recover the original signal back from its continuous wavelet transform. It is shown that it is possible to recover back the original signal once the SOULTI or the Laplace wavelet transform is applied to decompose the signal. It is found that both wavelet transforms satisfy linear differential equations called the reconstructing differential equations, which are closely related to the differential equations that produce the wavelets. The new wavelets can have well defined Time-Frequency resolutions, and they have useful properties; a direct relation between the scale and the frequency, unique transform formulas that can be easily obtained for most elementary signals such as unit step, sinusoids, polynomials, and decaying harmonic signals, and linear relations between the wavelet transform of signals and the wavelet transform of their derivatives and integrals. The defined wavelets are applied to system analysis applications. The new wavelets showed accurate instantaneous frequency identification and modal decomposition of LTI Multi-Degree of Freedom (MDOF) systems and it showed better results than the Short-time Fourier Transform (STFT) and the other harmonic wavelets used in time-frequency analysis. The modal decomposition is applied for modal parameters identification, and the properties of the Laplace and the SOULTI wavelet transforms allows analytical and accurate identification methods. / Ph. D. / This study introduces new families of wavelets (small wave-like functions) derived from the response of Second Order Underdamped (oscillating) Linear-Time-Invariant systems. The first is named the SOULTI wavelets, while the second is named Laplace Wavelets. These functions can be used in a wavelet transform which transfers signals from the time domain to the time-frequency domain. It is shown that it is possible to recover back the original signal once the transform is applied. The new wavelets can have well defined Time-Frequency resolutions. The time-frequency resolution is the multiplication of the time resolution and the frequency resolution. A resolution is the smallest time range or frequency range that carries a feature of the signal. The new wavelets have useful properties; a direct relation between the scale and the frequency, unique transform formulas that can be easily obtained for most elementary signals such as unit step, sinusoids, polynomials, and decaying oscillating signals, and linear relations between the wavelet transform of signals and the wavelet transform of their derivatives and integrals. The defined wavelets are applied to system analysis applications. The new wavelets showed accurate instantaneous frequency identification, and decomposing signals into the basic oscillation frequencies, called the modes of vibration. In addition, the new wavelets are applied to infer the parameters of dynamic systems, and they show better results than the Short-time Fourier Transform (STFT) and the other wavelets used in time-frequency analysis.