# 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 $\overrightarrow{T}$ 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 $\overrightarrow{T}$ at the ends $x$ and $x+dx$ of the element forms with the direction of the $x$-axis are $\alpha $ and $\beta $, then the scalar force $\overrightarrow{F}$ of all on $dm$ (the gravitation omitted) are

$${F}_{x}=-T\mathrm{cos}\alpha +T\mathrm{cos}\beta ,{F}_{y}=-T\mathrm{sin}\alpha +T\mathrm{sin}\beta .$$ |

Since the angles $\alpha $ and $\beta $ are very small, the ratio

$$\frac{{F}_{x}}{{F}_{y}}=\frac{\mathrm{cos}\beta -\mathrm{cos}\alpha}{\mathrm{sin}\beta -\mathrm{sin}\alpha}=\frac{-2\mathrm{sin}\frac{\beta -\alpha}{2}\mathrm{sin}\frac{\beta +\alpha}{2}}{2\mathrm{sin}\frac{\beta -\alpha}{2}\mathrm{cos}\frac{\beta +\alpha}{2}},$$ |

having the expression $-\mathrm{tan}\frac{\beta +\alpha}{2}$, 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

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

the last form due to the mean-value theorem.

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

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

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

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

$y(0,t)=y(p,t)=\mathrm{\hspace{0.33em}0}$ | (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

$y(x,\mathrm{\hspace{0.17em}0})=f(x),{y}_{t}^{\prime}(x,\mathrm{\hspace{0.17em}0})=g(x).$ | (3) |

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

${\nabla}^{2}u={\displaystyle \frac{1}{{c}^{2}}}{u}_{tt}^{\prime \prime}$ | (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 |
---|---|

Canonical name | DerivationOfWaveEquation |

Date of creation | 2013-03-22 18:46:36 |

Last modified on | 2013-03-22 18:46:36 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 10 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 35L05 |

Related topic | Slope |

Related topic | MeanValueTheorem |

Related topic | DerivationOfHeatEquation |