10 - Fourier Cosine and Sine Transforms

In civil engineering, many problems we encounter involve how structures behave under various conditions, like temperature changes or vibrations. To analyze these problems, engineers often need to convert functions that describe these situations into a different numerical space—the frequency domain. This is where Fourier transforms come into play. The standard Fourier Transform can handle functions defined over all real numbers; however, when we're only dealing with situations defined for x ≥ 0 (like one end of a material being fixed), we use the Fourier Cosine and Sine Transforms. These help us break down complex functions into simpler components—sine and cosine waves—that can match specific conditions of our problem, making it easier to find solutions.

The Fourier Cosine Transform (FCT) takes a function f(x) defined for all non-negative values and transforms it into the frequency domain. The formula shows that we take the integral of the function multiplied by
cos(sx) across the interval from 0 to infinity. For this transform to work properly, the function must meet certain conditions: it needs to be continuous over finite intervals and must not grow too quickly (absolutely integrable). The variable s represents different frequencies, helping us see how the function behaves in terms of its frequency components.

The Inverse Fourier Cosine Transform allows us to retrieve the original function from its transformed frequency representation. By taking an integral that involves multiplying the cosine of sx by F(s) across the frequency range, we can reconstruct the original function f(x). This process is essential because it allows engineers to switch between understanding a problem in the frequency domain and applying it back to the physical domain where the problem takes place.

The properties of the Fourier Cosine Transform help in simplifying computations and understanding the behavior of the transforms better. For example, linearity means that if you can express a function as a sum of two other functions, their transforms can be found separately and combined. The scaling property says if you stretch or compress the input function, the transform will change accordingly. Differentiation is useful when we want to analyze how a function changes; this property helps relate the transform of a function to the transform of its derivative. Finally, Parseval’s identity connects the total energy of a function in the spatial domain to its representation in the frequency domain, capturing the idea of energy conservation in these transformations.

In this example, we calculate the Fourier Cosine Transform of the function e^(-ax), which is commonly used in engineering because it represents a decaying function over time. Applying the formula, we integrate the product of e^(-ax) and cos(sx) from 0 to infinity, which requires careful handling of convergence due to the exponential term. The result shows how we can express the behavior of this exponential function in terms of its frequency components, which is crucial in real-world applications like transient heat conduction.

Similar to the Fourier Cosine Transform, the Fourier Sine Transform (FST) focuses on functions defined for non-negative values, but it multiplies the function by sin(sx) instead of cos(sx). This is particularly useful for functions that vanish at boundaries, such as those in problems with specific conditions at one end. The Inverse Fourier Sine Transform takes us back to the original function, showing the symmetry and properties associated with sine functions.

Just like the properties of the Fourier Cosine Transform, the properties of the Fourier Sine Transform are crucial for simplifying and solving problems using transforms. The linearity and scaling properties mean we can manage combinations and transformations of inputs similarly for sine functions. The differentiation property makes it possible to analyze how changes in the function affect its frequency representation, and Parseval’s identity shows how energy is conserved between the spatial and frequency domains.

This example shows how to apply the Fourier Sine Transform to the function e^(-ax). Just like the cosine case, we integrate the product of the function with sin(sx) over the interval from 0 to infinity. The resulting expression helps in understanding how this function can be represented in terms of its frequency components, which is particularly useful in cases where the function disappears at certain boundaries.

Fourier transforms play a crucial role in addressing real-world problems in civil engineering. For instance, in heat conduction problems involving semi-infinite materials like slabs, engineers use sine and cosine transforms to account for conditions that change at the material's boundaries. Similarly, for beams that are fixed at one end, the Fourier Cosine Transform helps resolve how they bend under load by turning complex differential equations into simpler algebraic forms. In scenarios with rods or strings, where one end is fixed, the Fourier Sine Transform helps analyze wave propagation, enabling engineers to predict how these structures will behave over time.

The full Fourier Transform handles functions that extend infinitely in both directions. However, when we restrict our analysis to functions defined only for positive values, we can classify them as even (symmetric) or odd (antisymmetric). Even functions transform nicely using the Cosine Transform, while odd functions are suited for the Sine Transform. This relationship isn't just about computational ease; it reveals significant geometric and physical insights about how these functions behave based on their symmetry properties.

This section lists standard pairs used in Fourier transforms, showcasing common functions with their respective cosine and sine transforms. These pairs are essential in practical applications, as they provide engineers and mathematicians with quick references to find transforms of frequently used functions. Understanding these pairs allows for efficient analysis and solutions in areas like structural dynamics, signal processing, and heat transfer.

In engineering, many situations can be modeled using partial differential equations (PDEs), which describe how physical quantities change across space and time. To solve these equations under specific boundary conditions, Fourier transforms become powerful tools. They simplify complicated equations into more manageable forms by decomposing the functions involved into their frequency components. This transformation makes it easier to apply boundary conditions and find solutions, particularly for problems defined in semi-infinite domains, like heat conduction in long materials or vibrations in structural elements.

In this application, we consider a semi-infinite rod that starts at uniform zero temperature, and we hold one end at a constant temperature T. The governing equation describes how heat flows through the rod. To find the temperature distribution over time, we apply the Fourier Cosine Transform to convert the problem into a more manageable form. We set up a transformed function U(s, t) and relate it to the original temperature function u(x, t). By applying the relevant boundary conditions and properties of the transforms, we ultimately derive a solution involving the complementary error function, which captures the heat distribution across the rod over time.

In this example, we study a cantilever beam that is fixed at one end and experiences a load along its length. The behavior of the beam under load is described by the Euler-Bernoulli equation. By applying the Fourier Cosine Transform to the equation and the associated boundary conditions, we reformulate the problem in the frequency domain. This leads us to a manageable form that allows us to express the deflection of the beam in terms of transforms of the applied load q(x). Finally, we can inverse transform the result to get back the physical deflection of the beam, providing practical insights into how engineers can predict structural behavior.

For many engineering problems, particularly where the solution is known to vanish at a specific boundary (like x = 0), the Fourier Sine Transform becomes the ideal tool to use. It allows us to effectively manage and solve partial differential equations (PDEs) under such conditions. Because the sine function naturally fits situations that meet these criteria, it helps streamline the solution process.

In this scenario, we analyze how a wave propagates in a one-dimensional system with fixed boundary conditions. Starting with the wave equation that describes wave propagation in time and space, we use the Fourier Sine Transform to convert this into the frequency domain. The resulting expression leads us to a second-order ordinary differential equation (ODE) which can be solved using initial conditions. The final solution reveals the displacement of the vibrating line (or rod) at any given time and position, showcasing the oscillatory nature of the system.

Fourier transforms not only help analyze functions but also serve as powerful tools for evaluating improper integrals. In this example, the integral of the product of x and sin(ax) over a specific range is computed using the Fourier Sine Transform. By transforming the function and using integral tables or advanced techniques like residue calculus, we can derive a solution that simplifies the evaluation process. This utility in simplifying complex integrals showcases another dimension of Fourier analysis in engineering.

Understanding how to deal with derivatives within Fourier Transforms is essential for solving boundary value problems that involve differential equations. The formulas presented show how the cosine and sine transforms relate to the first derivative of a function. In the case of the cosine transform, there's a negative sign which indicates how the transform captures the rate of change of the function. For the sine transform, the relationship accounts for both the derivative and the initial value at the boundary. These properties allow engineers to simplify complex equations into solvable forms, enhancing their ability to model systems accurately.