wave equation rod made of elastic substance a elastic wave ©sb/spk
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Wave Equation
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Rod made of elastic substance
A
Elastic Wave
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Disturbance in the rod
)(xx
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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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©SB/SPK
ii+1i-1
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Displacement of ith mass satisfies differential equationi
)(k)(kdt
dm iiii
i112
2
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Let
a: separation between the masses
a where 0x x
)t,x(i is a function of two continuous variable x and t
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In the Continuum limit
)t,xx(
)t,xx(
)t,x(
i
i
i
1
1
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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(
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2
22
1, ( ) ( ) ......
2x x t x x x
x x
Taylor series expansion
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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
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22
2
2
2
)x(x
kt
m
22
2
2
2
)x(xm
k
t
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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
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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
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•Wave equation
2 2
2 2 2
1 x c t
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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