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[parent] derivation of wave equation (Derivation)

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

$\displaystyle y \;=\; y(x,\,t)$
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

$\displaystyle F_x \;=\; -T\cos\alpha+T\cos\beta, \quad F_y \;=\; -T\sin\alpha+T\sin\beta.$
Since the angles $\alpha$ and $\beta$ are very small, the ratio

$\displaystyle \frac{F_x}{F_y} \;=\; \frac{\cos\beta-\cos\alpha}{\sin\beta-\sin\... ...\frac{\beta+\alpha}{2}}{2\sin\frac{\beta-\alpha}{2}\cos\frac{\beta+\alpha}{2}},$
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

$\displaystyle F_y \;=\; T\cdot(\tan\beta-\tan\alpha) \;=\; T\,[y'_x(x\!+\!dx,\,t)-y'_x(x,\,t)] \;=\; T\,y''_{xx}(x,\,t)\,dx,$
the last form due to the mean-value theorem.

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

$\displaystyle F_y \;=\; \sigma\,dx\,y''_{tt}(x,\,t)$
Equating both expressions, dividing by $T\,dx$ and denoting  $\displaystyle\sqrt{\frac{T}{\sigma}} = c$,  we obtain the partial differential equation
$\displaystyle y''_{xx} \;=\; \frac{1}{c^2}y''_{tt}$ (1)
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

$\displaystyle y(0,\,t) \;=\; y(p,\,t) \;=\; 0$ (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
$\displaystyle y(x,\,0) \;=\; f(x), \quad y'_t(x,\,0) \;=\; g(x).$ (3)

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

$\displaystyle \nabla^2u \;=\; \frac{1}{c^2}u''_{tt}$ (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.

Bibliography

1
K. V¨AISÄLÄ: Matematiikka IV.  Handout Nr. 141.    Teknillisen korkeakoulun ylioppilaskunta, Otaniemi, Finland (1967).



"derivation of wave equation" is owned by pahio.

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d'Alembert and D. Bernoulli solutions of wave equation (Topic) by pahio

Cross-references: motion, waves, wave equation, velocity, boundary, partial differential equation, acceleration, theorem, scalar, vector, function, position

This is version 2 of derivation of wave equation, born on 2009-01-26, modified 2009-01-27.
Object id is 434, canonical name is DerivationOfWaveEquation.
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Physics Classification: 03.75.-b (Matter waves )
 41.20.Jb (Electromagnetic wave propagation; radiowave propagation )
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