1. Wave & Wave Equation
Longitudinal Waves

2. 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.

3. 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.

4. 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.

5. 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.

6. Wave Equation

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

8. 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.

9. Elastic Wave A L
Rod made of elastic substance

10. Disturbance in the rod

11. Young’s Modulus

12. Elasticity : Spring constant

13. i
i-1
i+1

14. Displacement of ith mass satisfies differential equation

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

16. In the Continuum limit

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

22. Taylor series expansion

23. and

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

26. For disturbance propagating in all directions
(Laplacian operator)

27. 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  PAI/P