# Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation

My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on either side of the element have to be equal since it is assumed it moves only in the vertical plane but why does it have to be constant? Don't they just have to be equal?

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Let $T_x(t,x)$ denote the horizontal component of the tension as a function of time $t$ and position $x$ along the string. Let a small element of length $\Delta x$ of the string with left endpoint at position $x$ be given. As you point out, if the string element is to never accelerate horizontally, then the net tension in the $x$ direction must vanish at every instant in time $t$. By Newton's second law, this means that $$T_x(t,x+\Delta x) - T_x(t,x) = 0$$ dividing both sides by $\Delta x$ and then taking the limit $\Delta x \to 0$ shows that the derivative of the $x$-component of the tension with respect to $x$ must vanish; $$\frac{\partial T_x}{\partial x}(x) = 0$$ This implies that $T_x$ is constant in $x$. In particular, this means that there is some function $f$ for which $$T_x(t, x) = f(t)$$ for all times $t$. The tension does not need to be constant in time, but it must be constant in position along the length of the string.