4  Bernoulli’s solution

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We will now solve the wave equation again in a totally different way from d’Alembert, following instead Bernoulli. This is a method you have seen before, namely the method of separation of variables.

4.1 Separation of variables

We make the Ansatz that the solution factorises into one function of only \(x\) and one function of only \(t\): \[ y(x,t) = X(x)T(t). \tag{4.1}\] Substituting this into the wave equation \[ \partial_x^2y(x,t)=\frac{1}{c^2}\partial_t^2y(x,t) \tag{4.2}\] we get \[ X''(x)T(t)=\frac{1}{c^2}X(x)T''(t) . \tag{4.3}\] We divide both sides by \(X(x)T(t)\), which will separate the variables: \[ \frac{X''(x)}{X(x)}=\frac{1}{c^2}\frac{T''(t)}{T(t)}. \tag{4.4}\] It the last equation, we have a function of \(x\) only on the left hand side and a function of \(t\) only on the right hand side. The equation must hold for all \(x\) and \(t\). This is only possible if both functions are equal to a constant. It will turn out to be convenient to write this constant as \(-k^2\) for some new constant \(k\). Thus we set \[ \frac{X''(x)}{X(x)}=\frac{1}{c^2}\frac{T''(t)}{T(t)} =- k^2. \tag{4.5}\] This means that \(X(x)\) and \(T(t)\) must be solutions of the ODEs \[ X''=-k^2 X \quad \text{and}\quad T''=-k^2 c^2 T. \tag{4.6}\] The general solutions of these ODEs are \[\begin{split} X(x)&= A \, \sin(kx) + B \, \cos(kx), \\ T(t)&= F \, \sin(kct) + G \, \cos(kct), \end{split} \tag{4.7}\] where \(A, B, F, G\) and \(k\) are arbitrary constants.

4.2 Finite string with fixed ends

Bernoulli was most interested in the finite string. Let us put the ends of the string at \(x=0\) and \(x=\pi\). Now let the ends of the string be fixed. This means that we need to impose homogeneous Dirichlet boundary conditions \[ y(0,t)=0 \quad \text{and} \quad y(\pi,t)=0 \quad \text{for all} \ \ t . \tag{4.8}\] These conditions will be satisfied if \(X(0)=0\) and \(X(\pi)=0\). Imposing the condition \(X(0)=0\) on the general solution for \(X(x)\), we find that we need \(B=0\). We then have from \(X(\pi)=0\) that \[ A\sin(k\pi) = 0. \tag{4.9}\] The last equation implies that either \(A=0\) or \(\sin(k\pi)=0\). We reject the first option because it results in a zero solution. The second option yields \[ \sin(k\pi) = 0 \quad \Rightarrow k\in\mathbb{Z}. \tag{4.10}\]

Thus, for each integer \(k\) we have a solution of the wave equation of the form \[ y_k(x,t) = \sin\left(k x\right)\left(F_k \, \sin(k c t) + G_k \, \cos(k c t)\right) \tag{4.11}\]

4.3 Standing waves and superposition

The solutions we obtained in the previous section are standing waves. They don’t change their shape and don’t move, only their amplitude oscillates with time. In the lecture videos I make various sketches and animations to illustrate this.

So we now have a stark contrast between what d’Alembert found and what Bernoulli found. According to d’Alembert, the solutions of the wave equation are travelling waves of arbitrary shape. According to Bernoulli, the solutions are standing waves that have sine shape. It is always wonderful when one has two very different ways of looking at the same phenomenon. That is where deep understanding comes from.

To resolve the apparent conflict between d’Alembert’s solution and Bernoulli’s solution we have to use the superposition principle: For any set of linear homogeneous equations, any sum of solutions is also a solution.

Since the wave equation is linear and also the boundary conditions we imposed were linear, a linear combination of any number of harmonic standing waves is also a solution. So, we can construct a general solution of the wave equation by summing up all 1 the harmonic standing waves from Eq. 4.11: \[ y(x,t) = \sum_{k=1}^{\infty}\sin\left(k x\right)\left(F_k \, \sin(k c t) + G_k \, \cos(k c t)\right). \tag{4.12}\] Here the \(G_k\) and the \(F_k\) are undetermined constants, to be fixed from the initial conditions.

  • 1 We do not need negative \(k\) because they will produce the same solutions, and we do not need \(k=0\) because it yields zero solution. Also, we have absorbed the constant \(A\) into the constants \(F_k\) and \(G_k\).

    • The resolution of the apparent paradox is that a superposition of standing waves can give a travelling wave, and that a superposition of sine waves can give any shape.

    4.4 Initial value problem

    In Section 2.4 we imposed initial conditions on d’Alembert’s general solution and obtained d’Alembert’s formula in Eq. 2.23. We now similarly impose initial conditions on Bernoulli’s solution. Suppose now that we are given the initial conditions \[ y(x,0) = y_0(x), \quad \partial_t y(x,0)=v_0(x) \quad \text{for } x\in[0,\pi]. \tag{4.13}\] We want to use these initial conditions to determine the unknown coefficients \(F_k\) and \(G_k\) for all \(k\in\mathbb{N}\). As always, the procedure is to substitute the general solution into the initial conditions. When we substitute Eq. 4.12 into Eq. 4.13 we get \[ y_0(x) = \sum_{k=1}^{\infty}G_k \, \sin\left(k x\right), \quad v_0(x) = \sum_{k=1}^{\infty}F_k\, k\, c \, \sin\left(k x\right). \tag{4.14}\] These formulae look like Fourier series for the functions \(y_0\) and \(v_0\). So, to find the coefficients \(F_k\) and \(G_k\) we have to perform the inverse Fourier transform. This uses the identity \[ \int\limits_{0}^{\pi}\sin\left(k x\right)\sin(l x)dx = \frac{\pi}{2}\delta_{kl}, \] where \(\delta_{kl}\) is the Kronecker delta: \[ \delta_{kl} =\begin{cases} 1 &\text{if \ } k=l \\ 0 &\text{if \ } k\neq l \end{cases} \] We multiply each of Eq. 4.14 by \(2/\pi \,\sin(l x)\) and integrate over the interval \([0,\pi]\): \[\begin{split} \frac{2}{\pi}\int\limits_{0}^{\pi}y_0(x)\sin(l x)dx &=\frac{2}{\pi} \sum_{k=1}^{\infty}G_k \, \int\limits_{0}^{\pi}\sin\left(k x\right)\sin(l x)dx\\ &=\sum_{k=1}^{\infty}G_k \delta_{kl}=G_l \end{split} \tag{4.15}\] \[\begin{split} \frac{2}{\pi}\int\limits_{0}^{\pi}v_0(x)\sin(l x)dx &= \frac{2}{\pi}\sum_{k=1}^{\infty}F_k\, k c \, \int\limits_{0}^{\pi}\sin\left(k x\right)\sin(l x)dx\\ &=\sum_{k=1}^{\infty}F_k\, k c\, \delta_{kl}= F_l\, l c \end{split} \tag{4.16}\] Thus we know how to determine the constants \(F_k\) and \(G_k\) in the general solution Eq. 4.12 so as to satisfy given initial conditions.

    The method of separation of variables that we have used in this lecture to solve the wave equation is a powerful technique that can be used to solve a wide range of partial differential equations beyond the wave equation. The idea behind this method is to assume that the solution of a partial differential equation can be expressed as a product of two functions, one depending only on the spatial variables and the other only on the temporal variables. This reduces the partial differential equation to a set of ordinary differential equations, which can be solved independently. You will encounter this method repeatedly, in fluid mechanics, quantum mechanics, mathematical ecology and epidemiology and many other fields.