What is a Wave?

1. What is a Wave? Wave is a disturbance or variation that transfers energy progressively from point to point in a medium

2. Transverse Waves 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.

3. Transverse Wave in string

4. Longitudinal Waves In a longitudinal wave the particle displacement is parallel to the direction of wave propagation. The particles do not move down the tube with the wave; they simply oscillate back and forth about their individual equilibrium positions.

5. Water Waves Water waves are an example of waves that involve a combination of both longitudinal and transverse motions. As a wave travels through the waver, the particles travel in clockwise circles. The radius of the circles decreases as the depth into the water increases.

6. Transverse and Longitudinal Wave pulse Transverse wave in string travels with a speed, The speed of mechanical wave depends upon the inertia and elasticity of the medium in which wave is travelling. Longitudinal wave in a medium travels with a speed,

7. Equation of Travelling wave

8. Equation of Travelling wave

9. Illustration

10. Solution

11. Solution

12. Solution

13. Sinusoidal Wave Motion in Time and Space • The equation of wave is a function of both space and time expressed by, Y = f(x-vt) Y = f(t-x/v) Y = A sin k(x - vt) Y = A sin w(t - x/v) where A is the amplitude of the wave, ω is the angular frequency of the wave and k is the wave number. The negative sign is used for a wave traveling in the positive x direction and the positive sign is used for a wave traveling in the negative x direction.

14. Illustration

15. Solution

16. Solution

17. x=10.25 cm Vibration of particle with time at x = 10.25 cm Snapshot at t = 27 s Plane Progressive wave

18. The Linear Wave Equation

19. The Linear Wave Equation

20. Illustration

21. Solution

22. The speed of transverse wave v

23. The speed of transverse wave

24. Illustration

25. Solution

26. Illustration

27. Solution

28. POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

29. POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

30. POWER TRANSMITTED ALONG THE STRING BY A SINE WAVE

31. INTENSITY TRANSMITTED ALONG THE STRING BY A SINE WAVE

32. Illustration

33. Solution

34. Superposition of Waves The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time. The waves pass through each other without being disturbed. The net displacement of the medium at any point in space or time, is simply the sum of the individual wave dispacements. This is true of waves which are finite in length (wave pulses) or which are continuous sine waves.

35. Superposition & Colliding Waves Principle of Superposition: When two or more waves are simultaneously at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave. Example: Two waves travel on a string in opposite directions. While they overlap, their amplitudes add. However, after the overlap is finished, they proceed unchanged from their initial shapes. Note: If the amplitude grows too large, superposition may fail. In that case, the medium is said to go “non-linear”.

37. More Waves in Collision Complete cancellation!

38. Illustration

39. Solution

40. Reflection of Wave 1. Reflection from a hard boundary The animation at left shows a wave pulse on a string moving from left to right towards the end which is rigidly clamped. As the wave pulse approaches the fixed end, the internal restoring forces which allow the wave to propagate exert an upward force on the end of the string. But, since the end is clamped, it cannot move. According to Newton's third law, the wall must be exerting an equal downward force on the end of the string. This new force creates a wave pulse that propagates from right to left, with the same speed and amplitude as the incident wave, but with opposite polarity (upside down). • at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change) Animation Equation

41. Reflection of Pulse from the Fixed Boundary

42. Reflection from a SOFT boundary • at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave • The animation at left shows a wave pulse on a string moving from left to right towards the end which is free to move vertically . The net vertical force at the free end must be zero. This boundary condition is mathematically equivalent to requiring that the slope of the string displacement be zero at the free end (look closely at the movie to verify that this is true). The reflected wave pulse propagates from right to left, with the same speed and amplitude as the incident wave, and with the same polarity (right-side up). Equation Animation

43. Reflection of Pulse from the Free Boundary

44. Reflection and transmission of waves

47. Refraction of Sound Waves The speed of a wave depends on the elastic and inertia properties of the medium through which it travels. When a wave encounters different medium where the wave speed is different, the wave will change directions. Most often refraction is encountered in a study of optics, with a ray of light incident upon a boundary between two media (air and glass, or air and water, or glass and water). Snell's law relates the directions of the wave before and after it crosses the boundary between the two media. Notice that as the wavefronts cross the boundary the wavelength changes, but the frequency remains constant.

48. Reflection from an impedance discontinuity • When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary. The exact behavior of reflection and transmission depends on the material properties on both sides of the boundary. One important property is the characteristic impedance of the material. The characteristic impedance of a material is the product of mass density and wave speed. • In the animations on next slide, two strings of different densities are connected so that they have the same tension. The density of the thick string is 4 times that of the thin string. • How do the wave speeds compare for the two strings?

49. From high speed to low speed (low density to high density) In this animation the incident wave is travelling from a low density (high wave speed) region towards a high density (low wave speed) region • How do the amplitudes of the reflected and transmitted waves compare to the amplitude of the incident wave? • How do the polarities of the reflected and transmitted waves compare to the polarity of the incident wave? • How do the widths of the reflected and transmitted waves compare to the width of the incident wave?

50. From low speed to high speed (high density to low density) In this animation the incident wave is travelling from a high density (low wave speed) region towards a low density (high wave speed) region. • How do the amplitudes of the reflected and transmitted waves compare to the amplitude of the incident wave? • How do the polarities of the reflected and transmitted waves compare to the polarity of the incident wave? • How do the widths of the reflected and transmitted waves compare to the width of the incident wave?