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

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

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).



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