ABSTRACT
CH A P T E R 4 Introduction to Integral Transforms
Wave propagation analysis involves two important steps, namely, deriving the governing equation of motion and performing spectral analysis. The governing differential equation of motion for structures made from different material systems can be derived using the concepts outlined in Chapter 2 and Chapter 3. The details of spectral analysis will be discussed in Chapter 5. In spectral analysis, the governing differential equation needs to be transformed to the frequency domain for determination of wave parameters such as wavenumbers, phase speeds, group speeds, cut-off frequencies and band gaps. In addition, wave propagation analysis deals with many different and complex experimentally obtained time signals, which may require fine tuning and manipulation, such as removing white noise, filtering unwanted signals, etc. Some of these operations can be effectively handled in the frequency domain. Also, if one needs to understand the physics of wave propagation, one has to perform spectral analysis in the frequency domain. The governing equations can be transformed into the frequency domain using any of the available integral transforms. There are a number of integral transforms available to transform a time domain variable into the frequency domain. Among those, are four important integral transforms, namely, the Fourier Transforms, Short-Term Fourier Transforms, Wavelet Transforms, and Laplace Transforms, which will be discussed in this chapter.
