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The wave equation is an essential part of understanding how waves propagate through different media. In the context of a uniform bar, the wave equation describes how mechanical waves traverse the length of the bar. Given the wave speed, denoted as \(a\), the wave equation is expressed as:\[\frac{\partial^2 u}{\partial t^2} = a^2 \frac{\partial^2 u}{\partial x^2},\]where \(u(x,t)\) represents the displacement along the bar. Here, \(a = \sqrt{\frac{E}{\rho}}\) is the wave speed dependent on the material properties such as Young's modulus \(E\) and the material density \(\rho\). This equation shows that the acceleration of a point on the bar is proportional to the curvature of the displacement along the x-axis.Understanding the wave equation helps us realize that the forces acting on the bar impact the displacement in a periodic manner. This equation is vital in predicting how externally applied forces on one end of the bar will cause oscillations throughout the structure.

When exploring a mechanical system subjected to periodic forces, we often seek a steady-state or steady periodic solution. This means that over time, the system reaches a pattern or regular state of oscillation that continues unless disturbed by an external force.For the given problem, a steady periodic solution is modeled as:\[ u(x, t) = X(x)T(t), \]where \(T(t) = \sin(\omega t)\) reflects the temporal periodicity resulting from the sinusoidal force applied. This assumption implies that the oscillations are regular and persist without change over time.By solving the accompanying spatial and temporal components of the wave equation separately, we combine them to find the full expression for the displacement: \[ u(x,t) = \frac{F_0 a}{A E \omega \cos \left(\frac{\omega L}{a} \right)} \sin\left( \frac{\omega}{a}x \right) \sin(\omega t). \]This steady solution shows how the displacement is distributed across the length of the bar as time evolves and reaches a regular cycle based on the applied force.

Boundary conditions are critical in solving boundary value problems as they define the constraints the solution must satisfy. In this exercise, the boundary conditions ensure that the solution for the wave equation remains physically viable. The problem specifies two key boundary conditions:

• The fixed end condition, \( u(0, t) = 0 \), implies that there is no displacement at one end of the bar, ensuring it is anchored tightly.
• At the free end \( x = L \), the force condition \( A E \frac{\partial u}{\partial x}(L, t) = F_0 \sin(\omega t) \) ties the displacement gradient to the applied periodic force, conveying how force alters the bar's displacement at that point.

Using these conditions, solutions for the spatial part of the displacement \(X(x)\) are adjusted to obey both criteria. This leads to simplification of constants in the solution and assures that the periodical application of force results in a matching steady-state oscillation of the system.