# Questions on energy conservation of the wave equation

I'm reading this book. In Ch. 3.4, it studies the wave equation $u_{tt}=c^2u_{xx}$ with BCs $u_x(0,t)=0,\,u_x(L,t)=0$, and ICs $u(x,0)=f(x),\,u_t(x,0)=g(x)$.

The total energy of a string is the summation of the kinetic energy and the potential energy: $E=E_k+E_p=\frac{1}{2}\int_0^Lu_t^2dx+\frac{c^2}{2}\int_0^Lu_x^2dx$. I know that $E_k=mv^2/2$ and $E_p=kx^2/2$.

However, in the integral, $E_k$ term does not contain mass and the $E_p$ term has $u_x^2$ in stead of the square of the displacement. How can I figure out the unit and the dimension?

• It is a bit hard to answer without having the book. Anyway, in this kind of mathematical works it is very common to normalize constants to $1$, thus losing the possibility of doing dimensional analysis but simplifying formulas a bit. Commented Feb 27, 2016 at 16:18
• Anyway, this might be useful. Note that the term with $u_x^2$ accounts for the deformation of an element of the string, not its displacement. The string gains potential energy by deforming itself. Commented Feb 27, 2016 at 16:19

I have wondered similar things myself many times in the past (here an example). Mathematical books usually simplify physical constants to $1$, and moreover, they occasionally abuse of language. (Example. The word "energy" is one of the most common in mathematics, but it does not always match its physical meaning).
$$\mu \frac{\partial^2 u}{\partial t^2} - T\frac{\partial^2 u}{\partial x^2}=0,$$
with the notations of the linked post (i.e. $\mu=$ mass density, $T=$ tension). This is dimensionally consistent: $u$ is a displacement, so $$[u]=L,$$ ("dimension of $u$ is Length"), $\mu$ is a mass density, so $$[\mu]=ML^{-1}$$ ("mass times length$^{-1}$"), and $T$ is tension, which is a force, so $$[T]=MLt^{-2}$$ ("mass times length times time$^{-2}$"). Therefore $$\left[ \mu \frac{\partial^2 u}{\partial t^2} \right] = Mt^{-2} = \left[ T\frac{\partial^2 u}{\partial x^2} \right].$$
The equation you wrote, that is $$u_{tt} - c^2 u_{xx}=0$$ corresponds to this one with $$c=\sqrt{\frac{T}{\mu}}.$$ You can check that this $c$ has the dimensions of speed: $[c]=Lt^{-1}$.