Wave & Wave Equation Longitudinal Waves.

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Presentation transcript:

Wave & Wave Equation Longitudinal Waves

Longitudinal Wave In Longitudinal waves, the particles in a medium oscillate back and forth about their equilibrium positions but it is the disturbance which travels, not the individual particles in the medium. http://www.kettering.edu/~drussell/Demos/waves-intro/waves-intro.html

Transverse wave In a transverse wave the particle displacement is perpendicular to the direction of wave propagation. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by. http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Water Wave Water waves are an example of waves that involve a combination of both longitudinal and transverse motions. As a wave travels through the water, the particles travel in circles. http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Rayleigh surface wave Both longitudinal and transverse motion may be found in solids as Rayleigh surface waves. The particles in a solid, through which a Rayleigh surface wave passes, move in elliptical paths, with the major axis of the ellipse perpendicular to the surface of the solid. http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html

Wave Equation

ξ1 ξ2 k k k0 Consider a case of two identical spring-mass system coupled by a third spring in the middle

Equation of motion of the two masses can be written as These two equations are a set of two 2nd order linear homogeneous coupled differential equations with constant coefficients.

Elastic Wave A L Rod made of elastic substance

Disturbance in the rod

Young’s Modulus

Elasticity : Spring constant

i i-1 i+1

Displacement of ith mass satisfies differential equation

In the Continuum limit Let a: separation between the masses a  where 0 is a function of two continuous variable x and t

In the Continuum limit

Notation of partial derivatives variation of with t while x is kept constant variation of with x while t is kept constant

Taylor series expansion

and

Longitudinal wave in elastic rod Y: Young’s modulus A: Cross sectional area r=mass density Wave equation cs: wave velocity

For disturbance propagating in all directions (Laplacian operator)

Reference 1. LECTURE NOTES FOR PHYSICS I SASTRY AND SARASWAT 2. THE PHYSICS OF VIBRATIONS AND WAVES AUTHOR: H.J. PAIN IIT KGP Central Library Class no. 530.124 PAI/P