Cortical bone thickness is an important biomarker of bone fragility that reveals the risk of fractures. However, ultrasound bone assessment is challenging due to the complex nature of bone, such as varying porosity and microarchitecture, and the large difference between bone and soft tissue acoustic impedances. The research objective of this study is to develop a method to estimate cortical bone thickness by using spectral analysis, while avoiding traditional speed of sound measurements' figures due to the porous structure of bone tissue. In this study, multi-frequency ultrasound acquisitions have been used to cover a wide range of bone thickness and porosity values. Frequency modulated chirp waveforms are used as a transmit signal to increase the measurement SNR and the continuous wavelet transformation (CWT) is employed for the spectral analysis. The feasibility of the proposed methodology is demonstrated on simulated datasets and via experiments using ex vivo bone tissue. The preliminary experimental results showed a potential for cortical thickness classification using the received RF data.