## 3 Frequency Domain Spectral Finite Element Method (FDSFEM)  
The utilisation of the FDSFEM for wave propagation problem analysis is a quite new methodology. The crucial for this method is integral transform [55], in particular, Fourier transform [45].  
The continuous Fourier transform (CFT) has been applied to the solution of wave propagation problems as a standard procedure as stated by early work of [54]. However, to get the time signal, it is necessary to perform an inverse transform, which is rather difficult to do in an exact analytical manner. As a consequence, many approximate and asymptotic schemes have been developed and these give fine results for the far-field behaviour, e.g., for seismological studies [45]. For structural wave propagation, these schemes are not sufficient. Some important information is lost due to the reflections of the interacting stress waves at numerous boundaries. In addition, analytical transforms are achievable only for relatively simple transform functions. This is not the case for most practical problems [45]. The lack of a adequate inversion technique has stopped the growth of CFT based methods and directed the research into development of the discrete Fourier transform (DFT), which is an approximation of the integral involved in CFT. However, the application of the discrete Fourier series was still quite limited, with some early exceptions being in [56; 57; 58]. The reason of this has been the absence of suitable forward and inverse transform schemes, which have been required for processing all general inputs and outputs. Since the publication of a Fast Fourier Transform (FFT) algorithm [59], the numerical signal analysis has been revolutionised.  
An insight into DFT based analysis of wave propagation, where the DFT is performed by the FFT algorithm, is given in [79] by [PERSON]. In this book, in its simplest terms, the solution to a propagating wave problem is represented as:  
\[u(x,t)=\sum_{n}\hat{P_{n}}(\hat{G_{1}}(k_{1n}x)+\hat{G_{2}}(k_{2n}x)+...)e^{i \omega_{n}t}=\sum\hat{P_{n}}\hat{G}(k_{mn}x)e^{i\omega_{n}t}, \tag{24}\]where \(\hat{G}\) is the analytically known transfer function of the problem, typically a function of position \(x\) and has different numerical values at each frequency, \(i^{2}=-1\) and \(\omega_{n}\) equals [45]:  
\[\omega_{n}=n\Delta\omega=\frac{n\omega_{f}}{N}=\frac{n}{N\Delta t}=\frac{n}{T}, \tag{25}\]  
where \(\Delta t\) is the rate of time sampling, and \(\omega_{f}\) is the highest frequency captured by \(\Delta t\). The \(\hat{P_{n}}\) is the amplitude spectrum, which is known from the input conditions or from some measurement. The \(\hat{P_{n}}\hat{G}\) is known as the Fourier transform of the solution. It is different at each position, but, while evaluated at a particular position, its inverse immediately gives the time history of the solution at that point. A simple diagram is an illustration of an algorithm to calculate the propagating wave (Figure 1, the abbreviations used have been explained at the end of the paper).  
The time input \(P(t)\) is converted into its spectrum \(\hat{P_{n}}\) through the use of the forward FFT. The transformed solution is then obtained by evaluating the product  
\[\hat{t_{n}}=\hat{P_{n}}\hat{G}(k_{mn}x) \tag{26}\]  
at each frequency and some position. This is finally reconstructed in the time domain by use of the inverse FFT. The reconstructed time history will be real only when \(\hat{P_{n}}\hat{G}\) will be evaluated only up to Nyquist frequency as a complex conjugate of the initial part. The signal reconstruction procedure is similar for 1D elements, but for 2D the formulation of FDSFEs for structural wave propagation is much more complex. A brief explanation of this has been presented in Section 3.3.