derivation of wave equation

Let a string of 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 of the string be the constant $\sigma$.  We suppose that the amplitude of the vibration is so small that the tension $\overrightarrow{T}$ 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 $\overrightarrow{T}$ 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 force $\overrightarrow{F}$ of all on $dm$ (the gravitation omitted) are

$$F_x=-T\cos \alpha +T\cos \beta ,F_y=-T\sin \alpha +T\sin \beta .$$

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

$$F_y=T\cdot (\tan \beta -\tan \alpha )=T[y_x^{\prime }(x+dx,t)-y_x^{\prime }(x,t)]=T{y}_{xx}^{\prime \prime }(x,t)dx,$$

the last form due to the mean-value theorem.

On the other hand, by Newton the force equals the mass times the acceleration:

$$F_y=\sigma dx{y}_{tt}^{\prime \prime }(x,t)$$

Equating both expressions, dividing by $Tdx$ and denoting  $\sqrt{{\displaystyle \frac{T}{\sigma }}}=c$,  we obtain the partial differential equation

$y_{xx}^{\prime \prime }={\displaystyle \frac{1}{c^2}}{y}_{tt}^{\prime \prime }$(1)

for the equation of the 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 to satisfy the boundary conditions

$y(0,t)=y(p,t)=\mathrm{\hspace{0.33em}0}$(2)

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

$y(x,\mathrm{\hspace{0.17em}0})=f(x),y_t^{\prime }(x,\mathrm{\hspace{0.17em}0})=g(x).$(3)

The equation (1) is a special case of the general wave equation

${\nabla }^{2}u={\displaystyle \frac{1}{c^2}}{u}_{tt}^{\prime \prime }$(4)

where  $u=u(x,y,z,t)$.  The equation (4) rules the spatial waves in $ℝ$.  The number $c$ can be shown to be the velocity of propagation of the wave motion.