Currently, I am learning about mechanical waves in school, including sound and what-not. My teacher provided us with this formula for the energy per length of any given wave on a string: $ E/L = 2(π^2)(f^2)(A^2)μ $ where $E/L$ is the kinetic energy per unit length, $f$ is frequency, $A$ is amplitude, and $μ$ is the linear density of the string. Being a basic introductory physics class, the teacher didn't get into the derivation of this formula, but more so just threw it at us. However, I noticed that $E/L$ has the same units as force $(kg \cdot m/s^2)$ so I was wondering if kinetic energy per unit length is another way of representing the force of a wave. Which way would this wave force point? Is it of any significance?
It is indeed a force, the tension in a taut string. Tension takes units of energy per unit length for a string. An analogous situation is surface tension of the surface of a liquid, which has units of energy per unit area. Both are properties that affect wave propagation on the string, or the surface of a liquid respectively. For example, wave velocity would depend on the elastic properties of the medium, which in this case is the tension. At each point, it tries to minimize the length of the string, or the area of the surface. You could think of it as springs holding a lattice of balls together to get a sense of the direction of the force.