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

${\begin{matrix}\nabla ^{2}u\;=\;{\frac {1}{c^{2}}}u''_{tt}\end{matrix}}$

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.