ODE: The Wave Equation

The wave equation is one of the most fundamental partial differential equations (PDEs) in engineering and physics, serving as the cornerstone for modeling phenomena from vibrating violin strings and drumheads to seismic waves and electromagnetic signals. Mastering its solutions provides you with the analytical tools to predict how energy propagates through a medium, design structures to avoid resonant failure, and understand the very fabric of wave-based communication.

Derivation: The Vibrating String Model

We begin by deriving the one-dimensional wave equation from first principles, considering a perfectly flexible, elastic string under uniform tension $T$. We analyze a tiny segment of the string with length $\Delta x$, assuming its displacement $u(x,t)$ from equilibrium is small, leading to small angles $\theta$.

The net vertical force on the segment is the difference in the vertical components of tension at each end. For small angles, $\sin \theta \approx \tan \theta =\frac{\partial u}{\partial x}$. Therefore, the vertical force from tension is: $$F_{vert}=T{\frac{\partial u}{\partial x}|}_{x+\Delta x}-T{\frac{\partial u}{\partial x}|}_x$$

Applying Newton's second law ($F=ma$) to the segment, where its mass is density $\rho$ times length $\Delta x$ and its vertical acceleration is $\frac{{\partial }^{2}u}{\partial t^2}$, gives: $$T\left({\frac{\partial u}{\partial x}|}_{x+\Delta x}-{\frac{\partial u}{\partial x}|}_x\right)=\rho \Delta x\frac{{\partial }^{2}u}{\partial t^2}$$

Dividing by $\Delta x$ and taking the limit as $\Delta x\to 0$ turns the left side into a derivative with respect to $x$: $$T\frac{{\partial }^{2}u}{\partial x^2}=\rho \frac{{\partial }^{2}u}{\partial t^2}$$

This is usually written in its standard form: $$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$$ where $c=\sqrt{T/\rho }$ is the wave speed, a property of the medium. This equation states that the acceleration of a point on the string is proportional to the local curvature.

d'Alembert's Solution: The Traveling Wave Form

A powerful general solution to the wave equation on an infinite string was devised by Jean le Rond d'Alembert. It states that any solution can be written as the sum of two arbitrary wave shapes traveling in opposite directions: $$u(x,t)=F(x-ct)+G(x+ct)$$

Here, $F$ represents a wave profile traveling to the right with speed $c$, while $G$ represents a profile traveling to the left with the same speed. The beauty of this solution lies in how it directly satisfies the initial displacement and velocity conditions. Given initial conditions $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$, d'Alembert's formula provides the explicit solution: $$u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds$$

This formula elegantly shows how the initial displacement splits into two half-amplitude waves moving apart, while the initial velocity contributes to the solution via an integral over the interval of dependence $[x-ct,x+ct]$.

Separation of Variables and Standing Waves

For a finite string of length $L$ fixed at both ends (boundary conditions $u(0,t)=u(L,t)=0$), the method of separation of variables is essential. We assume a product solution of the form $u(x,t)=X(x)T(t)$. Substituting into the wave equation and separating variables leads to two ordinary differential equations: $$X^{\prime \prime }(x)+\lambda X(x)=0\text{and}{T}^{\prime \prime }(t)+c^2\lambda T(t)=0$$

The boundary conditions force $X(0)=X(L)=0$, which is a Sturm-Liouville eigenvalue problem. The only non-trivial solutions occur for eigenvalues ${\lambda }_n=(n\pi /L{)}^2$, giving spatial eigenfunctions or modes: $$X_n(x)=\sin \left(\frac{n\pi x}{L}\right),n=1,2,3,...$$

Natural Frequencies and Standing Wave Solutions

The corresponding temporal equation becomes $T_n^{\prime \prime }+(cn\pi /L{)}^2{T}_n=0$, which is a simple harmonic oscillator equation. Its solutions are sines and cosines with natural frequencies (or eigenfrequencies): $${\omega }_n=c\frac{n\pi }{L}=\frac{n\pi }{L}\sqrt{\frac{T}{\rho }}$$

The full solution for each mode is a standing wave solution: $$u_n(x,t)=\sin \left(\frac{n\pi x}{L}\right)\left[A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)\right]$$

The general solution is a superposition (infinite series) of all these modes: $$u(x,t)=\sum _{n=1}^{\infty }\sin \left(\frac{n\pi x}{L}\right)\left[A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)\right]$$

The coefficients $A_n$ and $B_n$ are determined by projecting the initial displacement $f(x)$ and initial velocity $g(x)$ onto the eigenfunctions, using Fourier sine series: $$A_n=\frac{2}{L}{\int }_0^{L}f(x)\sin \left(\frac{n\pi x}{L}\right)dx,B_n=\frac{2}{{\omega }_{n}L}{\int }_0^{L}g(x)\sin \left(\frac{n\pi x}{L}\right)dx$$

Energy in Wave Systems

A vibrating string possesses both kinetic and potential energy. For a string described by $u(x,t)$, the total mechanical energy is a conserved quantity in the absence of damping. The kinetic energy of a small segment is $\frac{1}{2}(\rho dx)(\frac{\partial u}{\partial t}{)}^2$. The potential energy is the work done by tension in stretching the string from its equilibrium length, which for small displacements is $\frac{1}{2}T(\frac{\partial u}{\partial x}{)}^{2}dx$.

Integrating over the string gives the total energy: $$E(t)=\frac{1}{2}{\int }_0^L\left[\rho {\left(\frac{\partial u}{\partial t}\right)}^2+T{\left(\frac{\partial u}{\partial x}\right)}^2\right]dx$$

For a solution constructed from separated variables, you can show that the total energy is the sum of the energies in each individual mode, and that it remains constant over time. This principle is crucial for analyzing energy transfer, damping, and resonance in complex systems.

Common Pitfalls

1. Misapplying d'Alembert's Solution to Bounded Domains: d'Alembert's formula is derived for an infinite domain $(-\infty ,\infty )$. Applying it directly to a finite string (e.g., $0\le x\le L$) without using the method of reflections to account for boundaries will yield an incorrect answer. Remember: for finite domains with fixed ends, separation of variables is the standard approach.
2. Confusing Wave Speed with Particle Velocity: The constant $c$ is the phase speed of the wave disturbance propagating along the string. The velocity of a specific point on the string, however, is the partial derivative $\frac{\partial u}{\partial t}$. These are completely different physical quantities. A point on the string moves transversely, while the wave moves horizontally.
3. Incorrectly Handling Non-Zero Initial Velocity: When using separation of variables, the initial velocity condition determines the $B_n$ coefficients. A common error is to forget the factor of $1/{\omega }_n$ in the formula for $B_n$, or to treat a non-zero $g(x)$ as if it were an initial displacement. Always clearly separate your conditions: $u(x,0)=f(x)$ sets the $A_n$ coefficients, and $u_t(x,0)=g(x)$ sets the $B_n$ coefficients.
4. Neglecting Boundary Conditions When Finding Modes: The spatial solutions $X(x)$ are entirely dictated by the boundary conditions. For a string fixed at both ends, we used $X(0)=X(L)=0$ to find $\sin$ eigenfunctions. If one end is free (a Neumann condition $\frac{\partial u}{\partial x}=0$), the eigenfunctions change to cosines. Always start a separation of variables problem by solving the spatial ODE with its boundary conditions to find the correct eigenvalues and eigenfunctions.

Summary

• The one-dimensional wave equation $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$ models small transverse vibrations of a taut string, where the wave speed $c=\sqrt{T/\rho }$ is determined by tension and linear density.
• d'Alembert's solution $u(x,t)=F(x-ct)+G(x+ct)$ describes traveling waves and provides an explicit formula for problems on an infinite domain using initial conditions for displacement and velocity.
• For a finite string with fixed ends, the method of separation of variables leads to a superposition of standing wave solutions $u_n(x,t)=\sin (n\pi x/L)[A_n\cos ({\omega }_{n}t)+B_n\sin ({\omega }_{n}t)]$, each oscillating at a distinct natural frequency ${\omega }_n=n\pi c/L$.
• The initial conditions are satisfied by expanding them in a Fourier sine series to determine the coefficients $A_n$ and $B_n$ for each vibrational mode.
• The total energy of the vibrating string is conserved and is the sum of kinetic energy (depending on $u_t$) and potential energy (depending on $u_x$), integrated over the length of the string.