Energy Analysis

The frequency domain analysis above was applied to the case of the wave equation defined over an infinite domain ${\mathcal D}={\mathbb{R}}$, and thus boundary conditions have not been taken into account. Another way of examining the behaviour of the wave equation is similar to that which was discussed in Section 3.1.2, with regard to the oscillator, through the use of energetic techniques. Besides the fact that it may be applied to systems which are not linear and shift invariant, energy analysis also provides extremely useful insights regarding correct boundary termination, as well as important bounds on solution growth.

In the first instance, consider again the wave equation defined over the entire real line, i.e., for ${\mathcal D}={\mathbb{R}}$. Taking the inner product of (6.2) with $u_{t}$ over ${\mathcal D}$ gives

$$\langle u_{t}, u_{tt}\rangle_{{\mathcal D}} = \gamma^2\langle u_{t},u_{xx}\rangle_{\mathcal D}$$ (6.13)

and, employing integration by parts, one may arrive at

$$\langle u_{t}, u_{tt}\rangle_{{\mathcal D}} +\gamma^2\langle u_{tx},u_{x}\rangle_{\mathcal D} = 0$$ (6.14)

Both of the terms in the above equation may be written as total derivatives with respect to time, i.e.,

$$\frac{d}{dt}\left(\frac{1}{2}\Vert u_{t}^2\Vert^2_{{\mathcal D}} + \frac{\gamma^2}{2}\Vert u_{t}^2\Vert^2_{{\mathcal D}}\right) = 0$$ (6.15)

or, more simply,

$$\frac{d{\mathfrak{H}}}{dt}=0\qquad {\rm with}\qquad {\mathfrak{H}}={\mathfrak{T}}+{\mathfrak{V}}$$ (6.16)

and

$${\mathfrak{T}}=\frac{1}{2}\Vert u_{t}\Vert^2_{{\mathcal D}}\qquad{\mathfrak{V}} = \frac{\gamma^2}{2}\Vert u_{x}\Vert^2_{{\mathcal D}}$$ (6.17)

Equation (6.16), and the non-negativity of the terms ${\mathfrak{T}}$ and ${\mathfrak{V}}$ above imply that

$${\mathfrak{H}}(t)={\mathfrak{H}}(0)\geq 0$$ (6.18)

Clearly, the non-negativity of ${\mathfrak{T}}$ and ${\mathfrak{V}}$ also imply that

$$\Vert u_{t}\Vert _{{\mathcal D}} \leq \sqrt{2{\mathfrak{H}}(0)}\qquad\Vert u_{x}\Vert _{{\mathcal D}} \leq \frac{\sqrt{2{\mathfrak{H}}(0)}}{\gamma}$$ (6.19)