Wave Equation

Rod made of elastic substance

A

Elastic Wave

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Disturbance in the rod

)(xx

Stress /

Strain /

StressYoung's modulus(Y)

Strain

F A

L

FL

A

Young’s Modulus

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Elasticity : Spring constant

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ii+1i-1

Displacement of ith mass satisfies differential equationi

)(k)(kdt

dm iiii

i112

2

Let

a: separation between the masses

a where 0x x

)t,x(i is a function of two continuous variable x and t

In the Continuum limit

)t,xx(

)t,xx(

)t,x(

i

i

i

1

1

Notation of partial derivatives

: variation of with t while x is kept constant

: variation of with x while t is kept constant

t

)t,x(

)t,x(

x

)t,x(

)t,x(

2

22

1, ( ) ( ) ......

2x x t x x x

x x

Taylor series expansion

1

22

2

( , ) ( , ) ( , ) ( , )

1( )

2

i ix t x t x x t x t

x xx x

and

1

22

2

( , ) ( , ) ( , ) ( , )

1( ) ( )

2

i ix t x t x t x x t

x xx x

22

2

2

2

)x(x

kt

m

22

2

2

2

)x(xm

k

t

Longitudinal wave in elastic rod

x

YAk

Wave equation2 2

2 2

Y

t x

2 2

22 2

1

sx tc

m A x

cs: wave velocity

Y: Young’s modulusA: Cross sectional arear=mass density

For disturbance propagating in all directions

2

2

2

2

2

2

2

2

2

zyxx

(Laplacian operator)

2

2

22 1

tc)t,r(

s

•Wave equation

2 2

2 2 2

1 x c t

1. LECTURE NOTES FOR PHYSICS ISASTRY AND SARASWAT

2. THE PHYSICS OF VIBRATIONS AND WAVESAUTHOR: H.J. PAINIIT KGP Central LibraryClass no. 530.124 PAI/P