# 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 $\vec{T}$ 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 force $\vec{F}$ of all 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

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

the last form due to the mean-value theorem.

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

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

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

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

 $\displaystyle y(0,\,t)\;=\;y(p,\,t)\;=\;0$ (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

 $\displaystyle y(x,\,0)\;=\;f(x),\quad y^{\prime}_{t}(x,\,0)\;=\;g(x).$ (3)

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

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

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.

## References

• 1 K. Väisälä: Matematiikka IV.  Handout Nr. 141. Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).
Title derivation of wave equation DerivationOfWaveEquation 2013-03-22 18:46:36 2013-03-22 18:46:36 pahio (2872) pahio (2872) 10 pahio (2872) Derivation msc 35L05 Slope MeanValueTheorem DerivationOfHeatEquation