Partial Differential Equation (PDE)

An ordinary differential equation is a differential equation that has only one independent variable. For example, the angular position of a swinging pendulum as a function of time: =(t). However, most physical systems cannot be modeled by an ordinary differential equation because they usually depends on more than one variables.

A differential equation involving more than one independent variables is called a partial differential equation. For example, the equations governing tidal waves should deal with the description of wave propagation varying both in time and space. Wfront=Wfront(x,y,z,t).

The Wave Equation

Mechanical vibrations of a guitar string, or in the membrane of a drum, or a cantilever beam are governed by a partial differential equation, called wave equation, since they deal with variations taking place both in time and space taking a form of wave propagation. To derive the wave equation we consider an elastic string vibrating in a plane, as the string on a guitar. Assume u(x,t) is the displacement of the string away from its equilibrium position u=0. We can derive a partial differential equation governing the behavior of u(x,t) by applying the Newton’s second law and several simple assumptions (see chapter 11.2 in textbook)

u(x,t)

x

u

T(x+x,t)

T(x,t)

x x+x

2

2

From Newton's 2nd law: net force = mass acceleration

Along the vertical direction, T(x+ x,t)sin -T(x,t)sin = x (1)

where ,defined as the mass of string per unit length, is a constant.

On the ot

u

t

her hand, T(x+ x,t)cos =T(x,t)cos =T - (2) is a constant

since we assume there is only vertical motion with no horizontal accel.

2

2

T TTherefore, from equ(2) T(x+ x,t)= ,T(x,t)=

cos cos

sin sinfrom equ (1): tan tan x (3)

cos cos

tan is nothing but the local slope of the string at x+ x

u utan = and tan =

x x

Subst

x x x

uT T T

t

2

2

2 2 22

2 2 2

2 22

2 2

1 u uitute into equ(3)

x x x

As we take the limit x 0,

This is the famious one-dimensional wave equation:

x x x

u

T t

u u uc

x T t t

u uc

x t

In order to model the motion of the string, we need not only the wave equation

but also the boundary and initial conditions.

Since the equation is second order in both time and space, we will need two

boundary plus two initial cnditions.

Boundary condition example:

string is fixed at both ends ( 0, ) 0, ( , ) 0

Initial condition example:

given initial string displacement: ( , 0) ( )

and initia

u x t u x L t

u x t f x

l string velocity: ( , 0) ( )

Once the equation is solved a general solution with unknows will be

obtained. These unknown constants have to be determined by

using the boundary and initial conditio

ux t g x

t

ns specified.