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Homederivation of wave equation

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# 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

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

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 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

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

## Mathematics Subject Classification

35L05*no label found*

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