Question

A perfect elastic string of length l which is under tension T and its fixed at both ends, has a linear mass density. The string is given initial deflection and initial velocity at its various points and is released at time t=0.The string execute small transverse vibrations. The initial deflection and initial velocity of the spring at any point x are denoted by h(x) and v(x) respectively. Find the different normal modes of vibrations and deflection of the string at any point x at t=0

Answer 1

Answer:

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Answer 2

Given:

String length = l which is fixed at both ends.

To Find:

The different normal modes of vibrations and deflection of the string at any point x at t=0

Solution:

We know that both the ends of the string are fixed.

• The displacement of the string should be zero at both the ends.

• Any transverse wave propagating along the string in any direction will be reflected back at  the other end..

• Both the  waves super impose to produce the total deflection in the string.

• Y( x, t ) = ym sin(kx -ωt) (Forward direction)+ ym sin(kx +ωt)(Backward direction)

• Y(x,t) = 2ym sin(kx) cos(ωt)

A string fixed at one end will produce nodes at :

• We know if A = wavelength,

For first mode ,

• l  = A/2 = > A = 2l

• fA = V , V = Speed of wave

• f = V/2l

For nth mode,

• f = nV/2l

• Therefore prorogation constant k = ω/V = 2πf/V = nπ/l

Therefore, displacement/ deflection of string at point x :

• Y(x,t ) = 2ym sin (nπx/l ) cos (2πft )

We can consider t = 0 ,

• Y(x , 0 ) = 2ym sin ( nπx/l)

The different normal modes of vibrations will be at x = l/n where n = 1,2,3, .. and

deflection of the string at any point x at t = 0 Y(x , 0 ) = 2ym sin ( nπx/l).