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 σ.  We suppose that the amplitude of the vibration is so small that the tension T of the string can be regarded to be constant.

The position of the string may be represented as a function


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 σdx.  If the angles the vector T at the ends x and x+dx of the element forms with the direction of the x-axis are α and β, then the scalar force F of all on dm (the gravitation omitted) are


Since the angles α and β are very small, the ratio


having the expression-tanβ+α2, also is very small.  Therefore we can omit the horizontal component Fx and think that the vibration of all elements is strictly vertical.  Because of the smallness of the angles α and β, their sines in the expression of Fy may be replaced with their tangents, and accordingly


the last form due to the mean-value theorem.

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


Equating both expressions, dividing by Tdx and denoting  Tσ=c,  we obtain the partial differential equationMathworldPlanetmath

yxx′′=1c2ytt′′ (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 conditionsMathworldPlanetmath

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

y(x, 0)=f(x),yt(x, 0)=g(x). (3)

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

2u=1c2utt′′ (4)

where  u=u(x,y,z,t).  The equation (4) rules the spatial waves in .  The number c can be shown to be the velocity of propagation of the wave motion.


  • 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