1. H. SAIBI October 28 th, 2015

2. ©2008 by W.H. Freeman and Company Fig. Deep-Ocean Assessment and Reporting of Tsunamis (NOAA) in North Pacific

3. Waves are everywhere in nature  Sound waves,  visible light waves,  radio waves,  microwaves,  water waves,  sine waves,  telephone chord waves,  stadium waves,  earthquake waves,  waves on a string,  slinky waves 3

4. What is a wave? a wave is a disturbance that travels through a medium from one location to another. a wave is the motion of a disturbance 4

5. Simple Wave Motion Transverse Waves Transverse wave: The motion of the medium (the string) is perpendicular to the direction of the propagation of the disturbance. ©2008 by W.H. Freeman and Company Fig.1. Transverse wave pulse on a spring. The motion of the propagating medium is perpendicular to the direction of motion disturbance. Fig. 2.Three successive drawings of a transverse wave on a string traveling to the right.

6. Longitudinal Waves Longitudinal waves: The motion of the medium is parallel to the direction of propagation of the disturbance. Example: Sound waves ©2008 by W.H. Freeman and Company Fig.3. Longitudinal wave pulse on a spring.

7. Differences between Longitudinal and Transverse waves  The differences between the two can be seen 7

8. Anatomy of a Wave Now we can begin to describe the anatomy of our waves. We will use a transverse wave to describe this since it is easier to see the pieces. 8

9. Anatomy of a Wave In our wave here the dashed line represents the equilibrium position. Once the medium is disturbed, it moves away from this position and then returns to it 9

10. Anatomy of a Wave The points A and F are called the CRESTS of the wave. This is the point where the wave exhibits the maximum amount of positive or upwards displacement 10 crest

11. Anatomy of a Wave The points D and I are called the TROUGHS of the wave. These are the points where the wave exhibits its maximum negative or downward displacement. 11 trough

12. Anatomy of a Wave The distance between the dashed line and point A is called the Amplitude of the wave. This is the maximum displacement that the wave moves away from its equilibrium. 12 Amplitude

13. Anatomy of a Wave The distance between two consecutive similar points (in this case two crests) is called the wavelength. This is the length of the wave pulse. Between what other points is can a wavelength be measured? 13 wavelength

14. Anatomy of a Wave What else can we determine? We know that things that repeat have a frequency and a period. How could we find a frequency and a period of a wave? 14

15. Wave frequency We know that frequency measure how often something happens over a certain amount of time. We can measure how many times a pulse passes a fixed point over a given amount of time, and this will give us the frequency. 15

16. Wave frequency Suppose I wiggle a slinky back and forth, and count that 6 waves pass a point in 2 seconds. What would the frequency be? 3 cycles / second 3 Hz we use the term Hertz (Hz) to stand for cycles per second. 16

17. Wave Period The period describes the same thing as it did with a pendulum. It is the time it takes for one cycle to complete. It also is the reciprocal of the frequency. T = 1 / f f = 1 / T 17

18. Wave Speed We can use what we know to determine how fast a wave is moving. What is the formula for velocity? velocity = distance / time What distance do we know about a wave wavelength and what time do we know period 18

19. Wave Speed so if we plug these in we get velocity = length of pulse / time for pulse to move pass a fixed point v = / T we will use the symbol to represent wavelength 19

20. Wave Speed v = / T but what does T equal T = 1 / f so we can also write v = f velocity = frequency * wavelength This is known as the wave equation. 20

21. Wave Behavior Now we know all about waves. How to describe them, measure them and analyze them. But how do they interact? 21

22. Wave Behavior We know that waves travel through mediums. But what happens when that medium runs out? 22

23. Boundary Behavior The behavior of a wave when it reaches the end of its medium is called the wave’s BOUNDARY BEHAVIOR. When one medium ends and another begins, that is called a boundary. 23

24. Fixed End One type of boundary that a wave may encounter is that it may be attached to a fixed end. In this case, the end of the medium will not be able to move. What is going to happen if a wave pulse goes down this string and encounters the fixed end? 24

25. Fixed End Here the incident pulse is an upward pulse. The reflected pulse is upside-down. It is inverted. The reflected pulse has the same speed, wavelength, and amplitude as the incident pulse. 25

26. Fixed End Animation 26

27. Free End Another boundary type is when a wave’s medium is attached to a stationary object as a free end. In this situation, the end of the medium is allowed to slide up and down. What would happen in this case? 27

28. Free End Here the reflected pulse is not inverted. It is identical to the incident pulse, except it is moving in the opposite direction. The speed, wavelength, and amplitude are the same as the incident pulse. 28

29. Free End Animation 29

30. Change in Medium Our third boundary condition is when the medium of a wave changes. Think of a thin rope attached to a thin rope. The point where the two ropes are attached is the boundary. At this point, a wave pulse will transfer from one medium to another. What will happen here? 30

31. Change in Medium In this situation part of the wave is reflected, and part of the wave is transmitted. Part of the wave energy is transferred to the more dense medium, and part is reflected. The transmitted pulse is upright, while the reflected pulse is inverted. 31

32. Change in Medium The speed and wavelength of the reflected wave remain the same, but the amplitude decreases. The speed, wavelength, and amplitude of the transmitted pulse are all smaller than in the incident pulse. 32

33. Change in Medium Animation 33

34. Wave Interaction All we have left to discover is how waves interact with each other. When two waves meet while traveling along the same medium it is called INTERFERENCE. 34

35. Constructive Interference Let’s consider two waves moving towards each other, both having a positive upward amplitude. What will happen when they meet? 35

36. Constructive Interference They will ADD together to produce a greater amplitude. This is known as CONSTRUCTIVE INTERFERENCE. 36

37. Destructive Interference Now let’s consider the opposite, two waves moving towards each other, one having a positive (upward) and one a negative (downward) amplitude. What will happen when they meet? 37

38. Destructive Interference This time when they add together they will produce a smaller amplitude. This is know as DESTRUCTIVE INTERFERENCE. 38

39. Check Your Understanding Which points will produce constructive interference and which will produce destructive interference? 39 Constructive G, J, M, N Destructive H, I, K, L, O

40. WAVE PULSES The x coordinates of two reference frames are related by: So. Thus, the shape of the string in the original frame is: (Wave Function) wave moving in the +x direction. The same line of reasoning for a pulse moving to the left leads to: wave moving in the -x direction.  v is the speed of propagation of the wave. ©2008 by W.H. Freeman and Company Fig.4. Fig.4 shows a pulse on a string at time t=0. The shape of the string can be represented by some function y=f(x). At some later time, the pulse is farther down the string. In a new coordinate system with origin O’ that moves to the right with the same speed as the pulse. The string is described in this frame by f(x’) for all times. …(1) …(2)

41. WAVE PULSES  For waves on a string, the wave function represents the transverse displacement of the string.  For sound waves in air, the wave function can be the longitudinal displacement of the air molecules, or the pressure of the air.  The wave function are solutions of a differential equation called the wave equation, which can be derived using Newton’s laws.

42. SPEED OF WAVES A general property of waves is that their speed relative to the medium depends on the properties of the medium, but is independent of the medium of the source of the wave. For example, the speed of a sound from a car horn depends only on the properties of air and not on the motion of the car. For wave pulses on a rope, we can demonstrate that the greater the tension, the faster the propagation of the waves. Furthermore, waves propagate faster in a light rope than in a heavy rope under the same tension. If F T is the tension and  in the linear mass density (mass per unit length), the wave speed is: For sound waves in a fluid such as air or water, the speed v is given by: Speed of waves on a string …(3) …(4) (  is the equilibrium density of the medium and B is the bulk modulus)  The bulk modulus is the negative of the ratio of the change in pressure to the fractional change in volume:

43. SPEED OF WAVES Comparing Eq.3 and 4, we can see that, in general, the speed of waves depends on an elastic property of the medium (the tension for string waves and the bulk modulus for sound waves) and on an inertial property of the medium (the linear mass density or the volume mass density). For sound waves in a gas such as air, the bulk modulus is proportional to the pressure, which in turn is proportional to the density  and to the absolute temperature T of the gas. The ratio B/  is thus independent of density and is merely proportional to the absolute temperature T. In this case, Eq.4 is equivalent to: T is the absolute temperature (K), which is related to Celsius t C by:  The dimensionless constant  depends on the kind of gas.  The constant R is the universal gas constant: and M is the molar mass of the gas (that is, the mass of one mole of the gas), which for air is: Speed of sound in a gas…(5) (6) …(7)

44. Exercise Surface ocean waves are possible because of gravity and are called gravity waves. Gravity waves are called shallow waves if the water depth is less than half a wavelength. The wave speed for gravity waves depends on the depth and is given by v=sqrt (gh), where h is the depth. A gravity wave in the open ocean, where the depth is 5 km, has a wavelength of 100 km. 1- What is the wave speed of the wave? 2- Is the wave a shallow wave? Information: Tsunamis are known to travel at speeds of 800 km/h in the open ocean.

45. SPEED OF WAVES Derivation of v for waves on a string Eq.3 (v=SQRT F T /  ) can be obtained by applying the pulse-momentum theorem to the motion of a string. Suppose you are holding one end of a long taut string with tension F T and uniform mass per unit length . (the other end of the string is attached to a distant wall). Suddenly, you begin to move your hand upward at a constant speed u. After a short time, the string appears as shown in Fig.5 with the rightmost point of the inclined segment of the string moving to the right at the wave speed v and the entire inclined segment moving upward at speed u. By applying the impulse- momentum theorem ( ) to the string, we obtain: where Fy is the upward component of the force of your hand on the string, m is the mass of the inclined segment, and  t is the time that your hand has been moving upward. The two triangles in the figure are similar, so: Substituting for Fy in Eq.8 gives: where  v  t has been substituted for m. Solving for v gives: Which is the expression for the wave speed that is given in Eq.3. …(8) or

46. SPEED OF WAVES Derivation of v for waves on a string ©2008 by W.H. Freeman and Company Fig.5. As the end of the spring moves upward at constant speed u, the point where the string changes from horizontal to inclined moves to the right at the wave speed v.

47. THE WAVE EQUATION We can apply Newton’s second law to a segment of the string to derive a differential equation known as the wave equation, which relates the spatial derivatives of y(x,t) to its time derivatives. Fig.6 shows one segment of a string. We consider only small angles  1 and  2. Then the length of the segment is approximately  x and its mass is m=  x, where  is the string’s mass per unit length. First, we show that, for small vertical displacements, the net horizontal force on a segment is zero and the tension is uniform and constant. The net force in the horizontal direction is zero. That is, Where  2 and  1 are the angles shown and F T is the tension in string. Because the angles are assumed to be small, we may approximate cos  by 1 for each angle. Then, the net horizontal force on the segment can be written Thus, the segment moves vertically, and the net force in this direction is:

48. THE WAVE EQUATION Because the angles are assumed to be small, we may approximate sin  by tan  for each angle. Then the net vertical force on the string segment can be written The tangent of the angle made by the string with the horizontal is the slope of the line tangent to the string. The slope S is the first derivative of y(x,t) with respect to x for constant t. A derivative of a function of two variables with respect to one of the variables with the other held constant is called a partial derivative. The partial derivative of y with respect to x is written  y/  x. Thus, we have: Then Where S 1 and S 2 are slopes of either end of the string segment and  S is the change in the slope. Setting this net force equal to the mass  x times the acceleration  2 y/  t 2 gives: or…(9)

49. THE WAVE EQUATION ©2008 by W.H. Freeman and Company Fig.6. Segment of stretched string used for derivation of the wave equation. The net vertical force on the segment is F T2 sin  2 -F T1 sin  1, where F is the tension in the string. The wave equation is derived by applying Newton’s second law to the segment.

50. THE WAVE EQUATION In the limit as  x  0, we have: Thus, in the limit as  x  0, Eq.9 becomes We now show that the wave equation is satisfied by any function x-vt. Let  =x-vt and consider any wave function: We use y’ for the derivative of y with respect to . Then, by the chain rule for the derivatives, Because We have Taking the second derivatives, we obtain …(10-a) Wave equation for a taut string. and

51. THE WAVE EQUATION Thus, The same result (Eq.10-b) can be obtained for any function of x+vt as well. Comparing Eq.10-a and 10-b, we see that the speed of propagation of the wave is Wave Equation…(10-b) …(3)

52. The speed of sound is given by: (Eq.4), where B and  are the bulk modulus and density of the medium, respectively. This equation can be obtained by applying the impulse-momentum theorem to the motion of the air in a long cylinder (Fig.7) with a piston at one end with the other end open to the atmosphere. Suddenly, you begin to move the piston to the right at constant speed u. After a short time,  t from the initial position of the piston is moving to the right with speed u. By applying the impulse-momentum theorem ( ) to the air in the cylinder we obtain: Where m is the mass of the air moving with speed u and F is the net force on the air in the cylinder. The air was initially at rest. The net force F is related to the pressure increase  P of the air near the moving piston by: Where A is the cross-sectional area of the cylinder. The bulk modulus of the air is given by: so …(11) SPEED OF WAVES Derivation of v for sound waves

53. Where Au  t is the volume swept out by the piston and Av  t is the initial volume of the air that is now moving with speed u. Substituting for F in Eq.11 gives: Where  Av  t have been substituted for m. Solving for v gives: Which is the same as the expression for v in Eq.4. A wave equation for sound waves can be derived using Newton’s laws. In one dimension, this equation is: Where s is the displacement of the medium in the x direction and v s is the speed of the sound in the medium. or

54. SPEED OF WAVES Derivation of v for sound waves ©2008 by W.H. Freeman and Company Fig.7. The air near to the piston is moving to the right at the same constant speed u as the piston. The right edge of this pressure pulse moves to the right with the wave speed v. The pressure in the pulse is higher than the pressure in the rest of the cylinder by  P.