In Partial Differential Equations by Walter Strauss, Ch 1.3 Example 2, in deriving the wave equation the author states that Newton's law $F=ma$ in its longitudinal (x) and transverse (u) components is
$$\frac{T}{\sqrt{1+u_x^2}} \Bigg | _{x_0}^{x_1} = 0 \quad \text{(longitudinal)} $$
$$\frac{T u_x}{\sqrt{1+u_x^2}} \Bigg | _{x_0}^{x_1} = \int_{x_0}^{x_1} \rho u_{tt} \,dx \quad \text{(transverse)} $$
Where do these equations come from? I understand longitudinal and transverse waves. Is it something along the lines of finding the force in the longitudinal (horizontal) direction and the transverse (vertical direction)? The right hand side of the transverse equation looks pretty much like mass times acceleration and since they are transverse waves the right hand side of the longitudinal equation is 0, but I'm not sure where the left hand sides are coming from.
EDIT: I just realised these equations are equivalent to:
$$Tcos(\theta) \Bigg | _{x_0}^{x_1} = 0$$
$$Tsin(\theta) \Bigg | _{x_0}^{x_1} = \int_{x_0}^{x_1} \rho u_{tt} dx$$
where $\theta$ is the left angle in the triangle in the figure above.
EDIT: Okay so the basic thing I didn't realise was that this is just the horizontal and vertical components of the force i.e.
$\| \vec{F_x} \| = \| \vec{F} \| cos(\theta)$
$\| \vec{F_y} \| = \| \vec{F} \| sin(\theta)$