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 $\overrightarrow{T}$ of the string can be regarded to be constant.

The [[../Position/|position]] of the string may be represented as a [[../Bijective/|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 [[../Vectors/|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 [[../Vectors/|scalar]] components of the resultant force $\overrightarrow{F}$ of all forces 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 $F_y=T\cdot (\tan \beta -\tan \alpha )=T[y{\text{'}}_x(x+dx,t)-y{\text{'}}_x(x,t)]=T{y}^{\prime }{\text{'}}_{xx}(x,t)dx,$ the last form due to the mean-value [[../Formula/|theorem]].

On the other hand, by Newton the force equals the mass times the [[../Acceleration/|acceleration]]: $F_y=\sigma dx{y}^{\prime }{\text{'}}_{tt}(x,t)$

Equating both expressions, dividing by

$Tdx$

and denoting\,

$\sqrt{\frac{T}{\sigma }}=c$

,\, we obtain the [[../DifferentialEquations/|partial differential equation]]

$\begin{array}{c}{y}^{\prime }{\text{'}}_{xx}=\frac{1}{c^2}{y}^{\prime }{\text{'}}_{tt}\end{array}$

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 [[../GenericityInOpenSystems/|boundary]] conditions

$\begin{array}{c}y(0,t)=y(p,t)=0\end{array}$

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/|velocity]] $g(x)$ of the points of the string; so there should be the initial conditions

$\begin{array}{c}y(x,0)=f(x),y{\text{'}}_t(x,0)=g(x).\end{array}$

The equation (1) is a special case of the general [[../TransversalWave/|wave equation]]

$\begin{array}{c}{\mathrm{\nabla }}^{2}u=\frac{1}{c^2}{u}^{\prime }{\text{'}}_{tt}\end{array}$

where\, $u=u(x,y,z,t)$.\, The equation (4) rules the spatial [[../CosmologicalConstant2/|waves]] in $ℝ$.\, The number $c$ can be shown to be the velocity of propagation of the wave [[../CosmologicalConstant/|motion]].