Main Page Chapter 1 Chapter 2 Chapter 3 Chapter 4 List of Topics

CHAPTER 2

WAVE EQUATION

in three dimension ………………1(a)

two dimension ……………….1(b)

one dimension ………………1(c)

There are three main methods of solution of wave equation

(i) Separation of variables/standing wave solution

(ii) D'Alembert's solution

(iii) Numerical solution

SEPARATED SOLUTION OF 1(c) :

Let u = X(x) T(t) ………………(2)

be a solution of equation 1(c).

Substituting in equation 1(c) gives

or ………………….(3)

Three cases may occur :

CASE 1 l = 0 and then from (3)

i.e. ……………….(4)

where A, B, C, D are arbitrary constants.

\ The required solution is

CASE 2 suppose and hence from (3)

\ …………………(5)

A, B, C, D are arbitrary constants.

\ The required solution is

CASE 3 . Suppose and hence from (3)

\ ..……(6)

where A, B, C, D are arbitrary constants and the required solution is

VIBRATIONS OF A STRETCHED STRING :

Solve wave equation 1(c) subject to the boundary conditions

and initial conditions which hold at t = 0 :

(initial displacement)

and (zero initial velocity)

SOLUTION

We know separable solution

……………………(i)

since for all

…………………..(ii)

Hence from equation (4) ; B = 0 and Aa + B = 0

\ A = B = 0 and so u = 0

which is not acceptable and is .

Again from equation (5) ;

i.e. , B = 0

so, A = B = 0 and also u = 0.

For non-zero solution .

* For vibration problems solutions (4) and (5) are useless.

Now, from solution (6), we get

A = 0 and

or,

\

\ , where n is an integer i.e., n=1, 2, 3, 4,………

or,

\ and

and displacement becomes

which we can write as

……………..(iii)

n = 1, 2, 3,……….

Here is called the n^th eigen function or characteristic function on n^th normal mode of vibrating string and the values are called eigen values or characteristic values of the vibrating string. The set is called the spectrum.

i.e., u_n represents a harmonic motion having frequency cycles/sec with a period of sec. Since , the frequency is . For first normal mode n = 1 and , which is lowest frequency and called the fundamental frequency of the string whereas others are known as overtones.

Now using initial condition , from equation (iii), we get D_n = 0 and it becomes

\ ……………(iv)

[general solution]

Now, initial displacement

i.e. , …………………(v)

which is a half-range Fourier sine series, so using Euler formula

, (n = 1, 2, 3,………) ……………(vi)

Thus the displacement of the string at all times i.e., the solution of equation 1(c) is

EXAMPLE 1

Solve the equation for the vibrations of a stretched string with its ends fixed at x = 0 and x = a such that

(a)

(b)

(c)

SOLUTION

Let the separable solution be

……………..(1)

Hence for stretched string

and …………….(2)

where p is a constant.

Now using (a) for all , equation (2) becomes

and

or, [since A = 0]

or,

\

\

i.e., ……………..(3)

From equation (1),

\ using (c), we get

Now from (2),

\

So,

i.e., …………………(4)

\

Hence general solution

…………………(5)

Now using (b), ,

Using Euler's formula,