2  d’Alembert’s solution

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In this lecture, we consider an infinitely long string (this is physically justified if we consider waves propagating far away from any boundaries). Mathematically, this means that we are looking for solutions of the wave equation \[ \partial_t^2 y - c^2\, \partial_{x}^2 y=0 \tag{2.1}\] on the whole real axis \(-\infty<x<+\infty.\) Note that I have switched to the convenient notation using subscripts on derivatives to specify the variable with respect to which we are differentiating.

2.1 Characteristic coordinates

To solve the wave equation we use the method of characteristics, which involves a change of variables that makes the equation much simpler. We change from the variables \(x\) and \(t\) to the characteristic coordinates \[ \xi=x+ct,\quad \eta=x-ct. \tag{2.2}\] By this we mean that for any function \(y\) that depends on the variables \(x\) and \(t\) we can introduce a function \(\tilde{y}\) that depends on the variables \(\xi\) and \(\eta\) in such a way that it has the same values as \(y\): \[ y(x,t)=\tilde{y}(\xi(x,t), \eta(x,t)) \text{ for all }x,t. \tag{2.3}\] It is a conventional abuse of notation to drop the tilde and denote both functions by \(y\). We will follow this abuse of notation.

We need to express the derivatives with respect to \(t\) and \(x\) via the derivatives with respect to \(\xi\) and \(\eta\). This is done using the chain rule: \[\begin{split} \partial_t \, y &=\frac{\partial {y}}{\partial {\xi}} \, \frac{\partial \xi}{\partial t} + \frac{\partial {y}}{\partial {\eta}} \, \frac{\partial \eta}{\partial t}\\ &= c \left( \partial_{\xi} - \partial_{\eta}\right) {y} \end{split} \tag{2.4}\] and \[\begin{split} \partial_x \, y &=\frac{\partial y}{\partial {\xi}} \, \frac{\partial \xi}{\partial x} + \frac{\partial y}{\partial {\eta}} \, \frac{\partial \eta}{\partial x}\\ &= \left( \partial_{\xi} + \partial_{\eta}\right) y. \end{split} \tag{2.5}\] Hence \[ \partial_t = c \left( \partial_{\xi} - \partial_{\eta}\right),~~~ \partial_x = \partial_{\xi} + \partial_{\eta}. \tag{2.6}\] Substituting these into the wave equation, we find that \[ c^2\left(\partial_{\xi} - \partial_{\eta}\right)^2 y - c^2\left(\partial_{\xi} + \partial_{\eta}\right)^2 y = 0. \tag{2.7}\] Expanding the squares and cancelling terms gives \[ -4c^2 \partial_{\xi} \partial_{\eta} y = 0. \tag{2.8}\] We can divide both sides by the nonzero constant \(-4c^2\). Thus the wave equation simplifies to \[ \partial_{\xi} \partial_{\eta} y = 0 . \tag{2.9}\]

While here we have seen that the method of characteristics simplifies the wave equation, its applications extend far beyond this specific example. The method of characteristics can be used to solve a wide range of partial differential equations that arise in various areas of applied mathematics, including fluid dynamics. It can also be used to study the behaviour of complex systems, such as traffic flow, chemical reactions, and population dynamics. The key idea behind the method of characteristics is to transform a partial differential equation into a system of ordinary differential equations along characteristic curves, which can then be solved using standard techniques.

2.2 General solution of wave equation

The wave equation in characteristic coordinates is really easy to solve. First, we integrate Eq. 2.9 in the variable \(\xi\): \[\begin{split} \int \partial_{\xi} \partial_{\eta} y(\xi,\eta) \, d\xi &= 0\\ \Leftrightarrow \quad \partial_{\eta} y(\xi,\eta) &= F(\eta) \end{split} \tag{2.10}\] where \(F\) is an arbitrary function of one variable 1.

  • 1 You can verify that this is true by direct differentiation of Eq. 2.10 with respect to \(\xi\).

    • Note

    When we integrate a function of two variables in one of the two variable, we need to add to the result an arbitrary function of the other variable. This is similar to adding a constant of integration when we integrate a function of one variable.

    Now we can integrate Eq. 2.10 in the variable \(\eta\): \[\begin{split} y(\xi,\eta) &= \int \partial_{\eta} y(\xi,\eta)\,d\eta\\ &= \int F(\eta) d\eta +g(\xi)\\ &= f(\eta) + g(\xi), \end{split} \tag{2.11}\] where \(g(\xi)\) is an arbitrary function of one variable and \(f'(\eta)=F(\eta)\). Note that since \(F\) is arbitrary, so is \(f\).

    Returning to variables \(x\) and \(t\), we can write the general solution of the wave equation as \[ y(x,t) = f(x-ct) + g(x+ct) \tag{2.12}\] where \(f\) and \(g\) are arbitrary functions of one variable.

    2.3 Travelling waves

    We will now gain an initial understanding of this solution by visualising the two special cases where either \(f\) or \(g\) are zero.

    If \(g = 0\), then \(y(x,t)=f(x-ct)\). At \(t=0\), the string has the shape described by the graph \(y=f(x)\). At time \(t>0\), it will have the same shape relative to the variable \(\eta=x-ct\): \(y=f(\eta)\). Since \(x=\eta + ct\), this means that the graph of \(y\) as a function of \(x\) for a fixed \(t>0\) is the graph of \(f(x)\) shifted to the right (in the direction of positive \(x\)) by distance \(ct\).

    If \(f = 0\), then \(y(x,t)=g(x+ct)\). At \(t=0\), the string has the shape described by the graph \(y=g(x)\). At time \(t>0\), it will have the same shape relative to the variable \(\xi=x+ct\): \(y=g(\xi)\). Since \(x=\xi - ct\), this means that the graph of \(y\) as a function of \(x\) for a fixed \(t>0\) is the graph of \(g(x)\) shifted to the left (in the direction of negative \(x\)) by distance \(ct\).

    Thus, \(f(x-ct)\) and \(g(x+ct)\) describe waves that propagate (without changing shape) to the right and to the left, respectively, and the general solution Eq. 2.12 represent the sum of such waves.

    2.4 Initial value problem and d’Alembert’s formula

    The initial-value problem is to solve the wave equation \[ \partial_t^2 y - c^2 \partial_x^2 y =0 \tag{2.13}\] for \(-\infty <x<+\infty\) and \(0<t<+\infty\) with the initial conditions \[ y(x,0)=y_0(x),\quad \partial_t y(x,0)=v_0(x) \tag{2.14}\] for \(-\infty <x<+\infty\), where \(y_0\) and \(v_0\) are given functions of \(x\). The first of the two initial conditions prescribes the initial displacement of the string, the second the initial velocity.

    To solve an initial value one has to substitute the general solution into the initial conditions. We substitute the solution from Eq. 2.12 into the initial conditions in Eq. 2.14 and obtain \[y_0(x)=f(x)+g(x), \tag{2.15}\] \[v_0(x)=-cf'(x) + cg'(x). \tag{2.16}\] So we have two equations for the two unknown functions \(f\) and \(g\). To solve them, we first integrate Eq. 2.16: \[ -c f(x)+c g(x) = \int_0^x v_0(s)ds + a = V(x), \tag{2.17}\] where \(a\) is an integration constant and \(V(x)\) is just introduced to save writing below.

    Next, we add and subtract Eq. 2.15 and Eq. 2.17 divided by \(c.\) This results in \[\begin{split} y_0(x)-\frac{1}{c} \, V(x)&=2 \, f(x),\\ y_0(x)+\frac{1}{c} \, V(x)&=2 \, g(x), \end{split} \tag{2.18}\] which implies that \[\begin{split} f(x)&=\frac{1}{2} \, y_0(x) - \frac{1}{2c} \, V(x), \\ g(x)&=\frac{1}{2} \, y_0(x) + \frac{1}{2c} \, V(x). \end{split} \tag{2.19}\] Substituting these into the formula for the general solution, we get \[\begin{split} y(x,t)=&\frac{1}{2} \, y_0(x-ct) - \frac{1}{2c} \, V(x-ct) \\ &+\frac{1}{2} \, y_0(x+ct) + \frac{1}{2c} \, V(x+ct) \end{split} \tag{2.20}\] or \[\begin{split} y(x,t)=&\frac{1}{2} \left[ y_0(x-ct) + y_0(x+ct)\right]\\ &+ \frac{1}{2c} \left[ V(x+ct) - V(x-ct)\right]. \end{split} \tag{2.21}\] Note that only the difference \(V(x+ct) - V(x-ct)\) appears, so the integration constant cancels and we can combine the two integrals into one because \[\begin{split} V(x+ct) - V(x-ct)&=\int\limits_0^{x+ct}v_0(s)\, ds - \int\limits_0^{x-ct}v_0(s)\, ds\\ &=\int\limits_{x-ct}^{x+ct}v_0(s)\, ds . \end{split} \tag{2.22}\] Using this, we have \[ y(x,t)=\frac12 [y_0(x+ct)+y_0(x-ct)]+ \frac{1}{2c} \int\limits_{x-ct}^{x+ct}v_0(s)\,ds. \tag{2.23}\] This is the solution formula for the initial-value problem (Eq. 2.13, Eq. 2.14) and it is called the d’Alembert formula.

    Remark. Once we have the d’Alembert formula, we can consider solutions of the initial-value problem (Eq. 2.13, Eq. 2.14) corresponding to piecewise smooth (or even piecewise continuous) initial functions \(y_0(x)\) and \(v_0(x)\). This will result in generalised solutions of the wave equation which are defined everywhere in the upper half of the \((x,t)\) plane except for a finite number of lines where values of \(y(x,t)\) and/or its first derivatives are discontinuous.