Y [vibrations] The displacement, u(x, t), of a rod is a...

Y [vibrations] The displacement, $u(x, t)$, of a rod is a function of position $x$ and time $t$ : $$ \begin{aligned} u(x, t)=&\left(A \sin \left(\frac{\omega}{\alpha} x\right)+B \cos \left(\frac{\omega}{\alpha} x\right)\right) \\ & \times[\operatorname{Csin}(\omega t)+D \cos (\omega t)] \end{aligned} $$ where $\alpha=\sqrt{\frac{E}{\rho}},(E$ is modulus of elasticity, $\rho$ is density of rod and $\omega$ is natural frequency of vibration). Show that $u(x, t)$ satisfies the wave equation: $$ \alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}} $$

Y [vibrations] The displacement, $u(x, t)$, of a rod is a function of position $x$ and time $t$ :
$$
\begin{aligned}
u(x, t)=&\left(A \sin \left(\frac{\omega}{\alpha} x\right)+B \cos \left(\frac{\omega}{\alpha} x\right)\right) \\
& \times[\operatorname{Csin}(\omega t)+D \cos (\omega t)]
\end{aligned}
$$
where $\alpha=\sqrt{\frac{E}{\rho}},(E$ is modulus of elasticity, $\rho$ is density of rod and $\omega$ is natural frequency of vibration). Show that $u(x, t)$ satisfies the wave equation:
$$
\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial^{2} u}{\partial t^{2}}
$$

Key Concepts

Harmonic Oscillations

Harmonic oscillations describe periodic motions that can be represented using sine and cosine functions. These functions are fundamental in modeling systems that exhibit regular oscillatory behavior, such as vibrations in elastic media, and are often integral components of solutions to the wave equation.

Chain Rule in Differentiation

The chain rule is an essential concept in calculus for taking derivatives of composite functions. In the context of partial differential equations, it allows the correct computation of higher-order derivatives when the function involves nested functions of the independent variables.

Separation of Variables

Separation of variables is a mathematical technique used to solve partial differential equations by expressing the solution as a product of functions, each depending on a single independent variable. This method reduces a complex PDE into simpler ordinary differential equations, facilitating finding solutions that satisfy the original equation.

Wave Equation

The wave equation is a fundamental partial differential equation that models how waves travel through a medium. It captures the relationship between spatial variation and temporal variation of a disturbance, stating that the second spatial derivative (curvature) is proportional to the second time derivative (acceleration) of the field variable.

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