Wave Wave Equation Longitudinal Waves Longitudinal Wave In

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Wave & Wave Equation Longitudinal Waves

Wave & Wave Equation Longitudinal Waves

Longitudinal Wave In Longitudinal waves, the particles in a medium oscillate back and forth

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. html

Transverse wave In a transverse wave the particle displacement is perpendicular to the direction

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/wavemotion. html

Water Wave Water waves are an example of waves that involve a combination of

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/wavemotion. html

Rayleigh surface wave Both longitudinal and transverse motion may be found in solids as

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/wavemotion. html

Wave Equation

Wave Equation

ξ 1 k ξ 2 k 0 k Consider a case of two identical

ξ 1 k ξ 2 k 0 k 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

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

Elastic Wave A L Rod made of elastic substance

Elastic Wave A L Rod made of elastic substance

Disturbance in the rod

Disturbance in the rod

Young’s Modulus

Young’s Modulus

Elasticity : Spring constant

Elasticity : Spring constant

i i-1 i+1

i i-1 i+1

Displacement of ith mass satisfies differential equation

Displacement of ith mass satisfies differential equation

In the Continuum limit Let a: separation between the masses a where 0 is

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

In the Continuum limit

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

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

Taylor series expansion

and

and

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

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)

For disturbance propagating in all directions (Laplacian operator)

1. LECTURE NOTES FOR PHYSICS I SASTRY AND SARASWAT 2. THE PHYSICS OF VIBRATIONS

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