Wave equation of a string fixed on both the ends

Question

I am trying to understand the wave equation of a string fixed on both the ends, which looks like this:

$$ \frac{\partial^2 y}{\partial x^2} - \frac1{c^2}\frac{\partial^2y}{\partial t^2} - \gamma\frac{\partial y}{\partial t}- l^2\frac{\partial^4y}{\partial x^4} = 0 $$

For the below equations, $\rho$ stands for linear density, $b$ for damping coefficient, $T$ for tension, and $c$ stands for the wave's speed.

I am comfortable in the case where I do not consider damping. I could bring this result:

$$ \frac{\partial^2 y}{\partial x^2} -\frac1{c^2}\frac{\partial^2y}{\partial t^2} = 0 $$

from

$$ F = T\cdot\sin(\theta + \Delta\theta) - T\cdot\sin (\theta) $$

and $$ F = \rho\Delta x \frac{\partial ^2y}{\partial t^2} $$

However, when I take damping into consideration and derive it this way:

$$ F = T\cdot\sin(\theta + \Delta\theta) - T\cdot\sin(\theta) - b\cdot \frac {\partial y}{\partial t} $$

$$ \rho\Delta x \cdot \frac{\partial^2 y}{\partial t^2} = T\cdot \sin (\theta +\Delta \theta) - T\cdot\sin(\theta) -b\cdot\frac{\partial y}{\partial t} $$

$$ \rho\Delta x \cdot \frac{\partial^2 y}{\partial t^2} = T\cdot \tan (\theta +\Delta \theta) - T\cdot\tan(\theta) -b\cdot\frac{\partial y}{\partial t} $$

$$ \frac{\partial^2 y}{\partial t^2} = \frac1{\rho\Delta x}\left(T\cdot\left(\frac{\partial y (x+\Delta x,t)}{\partial x} - \frac{\partial y(x,t)}{\partial x}\right) - b\cdot\frac{\partial y}{\partial t}\right) $$

$$ \frac{\partial ^2y}{\partial x^2}-\frac1{c^2}\frac{\partial^2y}{\partial t^2}-\frac\gamma{\Delta x}\frac{\partial y}{\partial t} = 0 $$

This is the problem. I have a $\Delta x$ term where I am not supposed to.

Answer 1

$$F=T\sin (\theta+\Delta\theta)-T\sin\theta-b \frac{\partial y}{\partial t}$$

This is the wrong starting point. $F$ is the force on an arbitrarily small length $\Delta x$ of string. As $\Delta x$ goes to 0, the force has to go to 0. (Otherwise the acceleration would go to infinity!)

That constraint is correctly respected by the first two terms, which cancel in the limit that $\Delta x\to 0$, because then $\Delta \theta\to 0$ also (assuming the string is smooth). But the last term fails to satisfy this constraint.