PlanetPhysics/Derivation of Wave Equation

Let a string of homogeneous matter be tightened between the points \,$$x = 0$$\, and\, $$x = p$$\, of the $$x$$-axis and let the string be made vibrate in the $$xy$$-plane.\, Let the line density of mass of the string be the constant $$\sigma$$.\, We suppose that the amplitude of the vibration is so small that the tension $$\vec{T}$$ of the string can be regarded to be constant.

The position of the string may be represented as a function $$y \;=\; y(x,\,t)$$ where $$t$$ is the time.\, We consider an element $$dm$$ of the string situated on a tiny interval \, $$[x,\,x\!+\!dx]$$;\, thus its mass is $$\sigma\,dx$$.\, If the angles the vector $$\vec{T}$$ at the ends $$x$$ and $$x\!+\!dx$$ of the element forms with the direction of the $$x$$-axis are $$\alpha$$ and $$\beta$$, then the scalar components of the resultant force $$\vec{F}$$ of all forces on $$dm$$ (the gravitation omitted) are $$F_x \;=\; -T\cos\alpha+T\cos\beta, \quad F_y \;=\; -T\sin\alpha+T\sin\beta.$$ Since the angles $$\alpha$$ and $$\beta$$ are very small, the ratio $$\frac{F_x}{F_y} \;=\; \frac{\cos\beta-\cos\alpha}{\sin\beta-\sin\alpha} \;=\; \frac{-2\sin\frac{\beta-\alpha}{2}\sin\frac{\beta+\alpha}{2}}{2\sin\frac{\beta-\alpha}{2}\cos\frac{\beta+\alpha}{2}},$$ having the expression \,$$-\tan\frac{\beta+\alpha}{2}$$, also is very small.\, Therefore we can omit the horizontal component $$F_x$$ and think that the vibration of all elements is strictly vertical.\, Because of the smallness of the angles $$\alpha$$ and $$\beta$$, their sines in the expression of $$F_y$$ may be replaced with their tangents, and accordingly $$F_y \;=\; T\cdot(\tan\beta-\tan\alpha) \;=\; T\,[y'_x(x\!+\!dx,\,t)-y'_x(x,\,t)] \;=\; T\,y''_{xx}(x,\,t)\,dx,$$ the last form due to the mean-value theorem.

On the other hand, by Newton the force equals the mass times the acceleration: $$F_y \;=\; \sigma\,dx\,y''_{tt}(x,\,t)$$ Equating both expressions, dividing by $$T\,dx$$ and denoting\, $$\sqrt{\frac{T}{\sigma}} = c$$,\, we obtain the partial differential equation $$\begin{matrix} y_{xx} \;=\; \frac{1}{c^2}y_{tt} \end{matrix}$$ for the equation of the transversely vibrating string.\\

But the equation (1) don't suffice to entirely determine the vibration.\, Since the end of the string are immovable,the function\, $$y(x,\,t)$$\, has in addition to satisfy the boundary conditions $$\begin{matrix} y(0,\,t) \;=\; y(p,\,t) \;=\; 0 \end{matrix}$$ The vibration becomes completely determined when we know still e.g. at the beginning\, $$t = 0$$\, the position $$f(x)$$ of the string and the initial velocity $$g(x)$$ of the points of the string; so there should be the initial conditions $$\begin{matrix} y(x,\,0) \;=\; f(x), \quad y'_t(x,\,0) \;=\; g(x). \end{matrix}$$

The equation (1) is a special case of the general wave equation $$\begin{matrix} \nabla^2u \;=\; \frac{1}{c^2}u''_{tt} \end{matrix}$$ where\, $$u =u(x,\,y,\,z,\,t)$$.\, The equation (4) rules the spatial waves in $$\mathbb{R}$$.\, The number $$c$$ can be shown to be the velocity of propagation of the wave motion.