The Wave Equation

The 1D Wave Equation: In its simplest form, the one-dimensional wave equation is expressed as $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$. Here, $u(x,t)$ represents the displacement of the wave at position $x$ and time $t$.

Wave Speed ($c$): The constant $c$ represents the propagation speed of the wave through the medium. It is a physical property determined by the characteristics of the medium, such as tension and linear density in a string, or bulk modulus and density in a fluid.

Second-Order Nature: The equation is second-order in both time and space. This implies that the wave's behavior depends on its local acceleration (second time derivative) and its concavity or 'curviness' (second spatial derivative).

Separation of Variables: This technique assumes a solution of the form $u(x,t) = X(x)T(t)$. By substituting this into the PDE, the equation is split into two ordinary differential equations (ODEs), which are then solved using boundary conditions to find eigenvalues and eigenfunctions.

d'Alembert's Solution: For an infinite domain, the general solution is $u(x,t) = f(x - ct) + g(x + ct)$. This represents two arbitrary wave shapes, $f$ and $g$, traveling in opposite directions at speed $c$ without changing form.

Fourier Series Integration: When dealing with finite domains (like a vibrating string fixed at both ends), the solution is expressed as an infinite sum of sine and cosine terms. The coefficients are determined by the initial displacement and initial velocity of the medium.

Homogeneous vs. Non-homogeneous: A homogeneous wave equation has no external forcing terms, while a non-homogeneous equation includes a source term $Q(x,t)$ on the right side, representing external driving forces.

Boundary Conditions: Dirichlet conditions specify the displacement at the boundaries (e.g., fixed ends), while Neumann conditions specify the slope or derivative (e.g., free ends).

Identify the Wave Speed: Always look for the coefficient of the spatial derivative. If the equation is $u_{tt} = A u_{xx}$, remember that $c^2 = A$, so the wave speed is $c = \sqrt{A}$.

Check the Dimensions: Ensure that the units of $c^2$ match $[Length^2 / Time^2]$. This is a quick way to verify if a derived equation or a given constant is physically plausible.

Verify Boundary Conditions: In problems involving finite strings, check if the ends are fixed ($u=0$) or free ($u_x=0$). This determines whether your solution should use sine terms, cosine terms, or both.

Initial Velocity vs. Displacement: If the initial velocity is zero, the solution will only contain terms that are stationary at $t=0$ (usually $\cos(ct)$ terms). If the initial displacement is zero, look for $\sin(ct)$ terms.

Confusing $c$ and $c^2$: A very common error is using $c$ as the coefficient in the PDE instead of $c^2$. This leads to incorrect calculations of wave speed and frequency.

Sign Errors in d'Alembert: Students often mix up the signs in $f(x - ct)$ and $g(x + ct)$. Remember that $x - ct$ represents a wave moving in the positive $x$ direction.

Ignoring Initial Velocity: When solving for the full motion of a string, both the initial position $u(x,0)$ and the initial velocity $u_t(x,0)$ must be accounted for. Forgetting the velocity term results in an incomplete solution.