2.3. Homogeneous $1$D Wave equation


2.3. Homogeneous $1$D Wave equation

Consider equation $$u_{tt}-c^2u_{xx}=0. \label{eq-2.3.1}$$

Physical examples

Remark 1. As we mentioned in Section 1.4 this equation describes a lot of things.

Example 1. Consider a string with the points deviating from the original position (along $x$) in the orthogonal direction ($y$); so the string is described by $y=u(x,t)$ at the moment $t$ (so $u$ is a displacement along $y$). In this case $c^2= T/\rho$ where $T$ is a tension of the string and $\rho$ is a linear density of it.

Example 2. This equation also describes compression-rarefication waves in elastic $1$-dimensional media. Then $u(x,t)$ is displacement along $x$.

Example 3. Consider a pipe filled by an ideal gas. Then $c^2= p(\rho)/\rho$ where $\rho$ is a density and $p(\rho)$ is a pressure (for an ideal gas at the given temperature such ratio is constant and due to Mendeleev-Clapeyron equation it is proportional to absolute temperature $T$ which is assumed to be a constant). Then f.e. $u$ may denote a density $\rho(x,t)$ at point $x$ at time $t$.

Remark 2. $c$ has a dimension of the speed. In the example above $c$ is a speed of sound.

General solution

Let us rewrite formally equation (\ref{eq-2.3.1}) as $$(\partial_t^2 -c^2 \partial_x^2)u= (\partial_t-c \partial_x)(\partial_t+c \partial_x)u=0. \label{eq-2.3.2}$$ Denoting $v=(\partial_t+c \partial_x)u= u_t+cu_x$ and $w=(\partial_t-c \partial_x)u= u_t-cu_x$ we have \begin{align} &v_t-cv_x=0,\label{eq-2.3.3}\\ &w_t+cw_x=0.\label{eq-2.3.4} \end{align} But from Section 2.1 we know how to solve these equations \begin{align} &v=2c\phi'(x+ct),\label{eq-2.3.5}\\ &w=-2c\psi'(x-ct) \label{eq-2.3.6} \end{align} where $\phi'$ and $\psi'$ are arbitrary functions. We find convenient to have factors $2c$ and $-2c$ and to denote by $\phi$ and $\psi$ their primitives (aka indefinite integrals). Recalling definitions of $v$ and $w$ we have \begin{align*} &u_t+cu_x=2c\phi'(x+ct),\\ &u_t-cu_x=-2c\psi'(x-ct). \end{align*} Then \begin{align*} c^{-1}&u_t=\phi'(x+ct)-\psi'(x-ct),\\ &u_x=\phi'(x+ct)+\psi'(x-ct). \end{align*} The second equation implies that $u=\phi(x+ct)+\psi(x-ct)+\Phi(t)$ and plugging to the first equation we get $\Phi'=0$, thus $\Phi=\const$.

So, $$u=\phi(x+ct)+\psi(x-ct) \label{eq-2.3.7}$$ is a general solution to (\ref{eq-2.3.1}). This solution is a superposition of two waves $u_1=\phi(x+ct)$ and $u_2=\psi(x-ct)$ running to the left and to the rightrespectively with the speed $c$. So $c$ is a propagation speed.

Remark 3. Adding constant $C$ to $\phi$ and $-C$ to $\psi$ we get the same solution $u$. However it is the only arbitrarness.

Cauchy problem

Let us consider IVP (initial–value problem, aka Cauchy problem) for (\ref{eq-2.3.1}): \begin{align} &u_{tt}-c^2u_{xx}=0,\label{eq-2.3.8}\\ &u|_{t=0}=g(x), &&u_t|_{t=0}=h(x). \label{eq-2.3.9} \end{align} Plugging (\ref{eq-2.3.7}) into initial conditions we have \begin{align} &\phi(x)+\psi(x)=g(x),\label{eq-2.3.10}\\ &c\phi'(x)-c\psi'(x)=h(x) \implies \phi(x)-\psi(x)=\frac{1}{c}\int^x h(y)\,dy.\label{eq-2.3.11} \end{align} Then \begin{align} &\phi(x)= \frac{1}{2} g(x)+\frac{1}{2c}\int^x h(y)\,dy,\label{eq-2.3.12}\\ &\psi(x)= \frac{1}{2} g(x)-\frac{1}{2c}\int^x h(y)\,dy.\label{eq-2.3.13} \end{align} Plugging into (\ref{eq-2.3.7}) and using property of an integral we get D'Alembert formula $$u(x,t)=\frac{1}{2}\bigl[g(x+ct)+g(x-ct)\bigr]+ \frac{1}{2c}\int_{x-ct}^{x+ct} h(y)\,dy. \label{eq-2.3.14}$$

Remark 4. Later we generalize it to the case of inhomogeneous equation (with the right-hand expression $f(x,t)$ in (\ref{eq-2.3.8}).