Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.1 IMPORTANT PARTIAL DIFFERENTIAL EQUATIONS

u

x x

T

T

∆s

x+∆x

θ 1

θ 2

Figure 20.1 The forces acting on an element of a string under uniform

tensionT.

Dividing both sides by ∆xwe obtain, for the vibrations of the string, the one-dimensional
wave equation

∂^2 u

∂x^2

=

1

c^2

∂^2 u

∂t^2

,

wherec^2 =T/ρ.

The longitudinal vibrations of an elastic rod obey a very similar equation to

that derived in the above example, namely

∂^2 u

∂x^2

=

ρ

E

∂^2 u

∂t^2

;

hereρis the mass per unit volume andEis Young’s modulus.

The wave equation can be generalised slightly. For example, in the case of the

vibrating string, there could also be an external upward vertical forcef(x, t)per

unit length acting on the string at timet. The transverse vibrations would then

satisfy the equation

T

∂^2 u

∂x^2

+f(x, t)=ρ

∂^2 u

∂t^2

,

which is clearly of the form ‘upward force per unit length = mass per unit length

×upward acceleration’.

Similar examples, but involving two or three spatial dimensions rather than one,

are provided by the equation governing the transverse vibrations of a stretched

membrane subject to an external vertical force densityf(x, y, t),

T

(

∂^2 u

∂x^2

+

∂^2 u

∂y^2

)

+f(x, y, t)=ρ(x, y)

∂^2 u

∂t^2

,

whereρis the mass per unit area of the membrane andTis the tension.