1 Mechanical Waves Ch 21-23

2 Waves A wave is a disturbance in a medium which carries energy from one point to another without the transport of matter. The medium allows the disturbance to propagate.

3 Transverse Wave Particles oscillate at right angles to the direction of motion.

4 Longitudinal Waves Particles oscillate parallel to the direction of motion.

5 Periodic Waves & Pulses A wave pulse is a single disturbance. A periodic wave is a series of disturbances or wave train.

6 Transverse Wave Speed Determined by the medium and its properties. elasticity or restoring force inertia

7 Wave on a medium with tension. String, rope, wire, etc… T is the tension, &  is the linear density,  = m/L = mass per unit length.

8 Waves Speed:

9 Wave Terminology Frequency (f) - cycles per second. (Hz) Period (T) - Seconds per cycle. Amplitude (A) - Maximum displacement from equilibrium. The distance that a wave travels in one period is the wavelength ( ).

10 Example 1 A wave travels along a string. The time for a particular point to move from a maximum displacement to zero is s. The wavelength is 1.40 m. What are the period, frequency, and wave speed?

11 Example 1 continued It takes s for one cycle, so T = s f = 1/T, so f = 1.47 Hz

12 Example 2 What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?

13 Example 2 continued

14 Polarization Most transverse waves are linearly polarized They either move just up and down Vertically polarized Or just side to side Horizontally polarized

15 Circular polarization If we combine two perpendicular waves that have equal amplitude but are out of step by a quarter-cycle, the resulting wave is circularly polarized.

16 Polarizing filters Only let through waves that are polarized one way. Like passing a rope through a slot in a board – only waves in the direction of the slot will get through.

17 Longitudinal Wave Speed Depends on the pressure change and the fractional volume change Where  is the density. B is the bulk modulus of a fluid. Y is young’s modulus for a solid. See tables 12-1 and B = 1/k

18 Longitudinal waves Don’t have polarization When the frequency is within the range of human hearing, it is called sound.

19 Sound waves in gases Temperature doesn’t remain constant as sound waves move through air. So, we use the equation Where  is the ratio of heat capacities (ch 18), R is the ideal gas constant (8.314 J/mol∙K), T is temperature in K, and M is the molecular mass (ch 17).

20 Sound waves Humans can hear from about 20 Hz to about Hz. Air is not continuous – it consists of molecules. Like a swarm of bees. Also sort of like wave/particle duality.

21 Mathematical wave description y(x, t) = A sin(  t – kx) (Motion to right) or y(x, t) = A sin(  t + kx) (Motion to left)

22 Reflection When a wave comes to a boundary, it is reflected. Imagine a string that is tied to a wall at one end. If we send a single wave pulse down the string, when it reaches the wall, it exerts an upward force on the wall.

23 Reflection By Newton’s third law, the wall exerts a downward force that is equal in magnitude. This force generates a pulse at the wall, which travels back along the string in the opposite direction.

24 Reflection In a ‘hard’ reflection like this, there must be a node at the wall because the string is tied there. The reflected pulse is inverted from the incident wave.

25 Reflection Now imagine that instead of being tied to a wall the string is fastened to a ring which is free to move along a rod. When the wave pulse arrives at the rod, the ring moves up the rod and pulls on the string.

26 Reflection This sort of ‘soft’ reflection creates a reflected pulse that is not inverted.

27 Transmission When a wave is incident on a boundary that separates two regions of different wave speeds part of the wave is reflected and part is transmitted.

28 Transmission If the second medium is denser than the first the reflected wave is inverted. If the second medium is less dense the reflected wave is not inverted. In either case, the transmitted wave is not inverted.

29 Transmission

30 Transmission

31 Interference

32 Interference The effect that waves have when they occupy the same part of the medium. They can add together or cancel each other out. After the waves pass each other, they continue on with no residual effects.

33 Constructive Interference

34 Constructive Interference out of phase = 360° = 1 cycle = 2  rad

35 Destructive Interference

36 Destructive Interference /2 out of phase = 180° = 1/2 cycle =  rad

37 Superposition of waves If two waves travel simultaneously along the same string the displacement of the string when the waves overlap is the algebraic sum of the displacements from each individual wave.

38 Standing Waves Consider a string that is stretched between two clamps, like a guitar string. If we send a continuous sinusoidal wave of a certain frequency along the string to the right When the wave reaches the right end, it will reflect and travel back to the left.

39 Standing waves The left-going wave the overlaps with the wave that is still traveling to the right. When the left-going wave reaches the left end it reflects again and overlaps both the original right-going wave and the reflected left-going wave. Very soon, we have many overlapping waves which interfere with each other.

40 Standing waves For certain frequencies the interference produces a standing wave pattern with nodes and large antinodes. This is called resonance and those certain frequencies are called resonant frequencies.

41 Standing waves A standing wave looks like a stationary vibration pattern, but is the result of waves moving back and forth on a medium.

42 Standing waves Superposition of reflected waves which have a maximum amplitude and appear to be a stationary vibration pattern. y 1 + y 2 = -2Acos(  t)sin(kx)

43 Standing Waves If the string is fixed at both ends there must be nodes there. The simplest pattern of resonance that can occur is one antinode at the center of the string.

44 Standing Waves on Strings Nodes form at a fixed or closed end. Antinodes form at a free or open end.

45 Standing waves For this pattern, half a wavelength spans the distance L. This is called the 1 st harmonic. It is also called the fundamental mode of vibration.

46 Standing waves For the next possible pattern, a whole wavelength spans the distance L. This is called the 2 nd harmonic, or the 1st overtone.

47 Standing Waves For the next possible pattern, one and a half wavelengths span the distance L. This is called the 3 rd harmonic, or the 2 nd overtone.

48 Standing waves In general, we can write

49 Standing Waves

50 Standing Waves on a String

51 Overtones

52 String fixed at ONE end Note: Only the odd harmonics exist!

53 Example The A-string of a violin has a linear density of 0.6 g/m and an effective length of 330 mm. (a) Find the tension required for its fundamental frequency to be 440 Hz. (b) If the string is under this tension, how far from one end should it be pressed against the fingerboard in order to have it vibrate at a fundamental frequency of 495 Hz, which corresponds to the note B?

54 Example  = 0.6 g/m = 6 x 10 –4 kg/m L = 330 mm = 0.33 m a) F t = ? b) 0.33 m – L 2 = ?

55 Example - A)

56 Example - B) v 1 = v 2 f 1 1 = f 2 2

57 Wave Example 1 The stainless steel forestay of a racing sailboat is 20 m long, and its mass is 12 kg. To find its tension, it is struck by a hammer at the lower end and the return of the pulse is timed. If the time interval is 0.20 s, what is the tension in the stay?

58 Example 1 L = 20 m, t = 0.20 s, m = 12 kg Find: F

59 Example 1 = 2.4 x 10 4 N

60 Note: Wave speed is determined by the medium. Wave frequency is determined by the source.

61 Sound Waves p = BkAcos(  t - kx) If y is written as a sine function, P is written as a cosine function because the displacement and the pressure are  /2 rad out of phase. p max = BkA

62 Waves in 3 Dimensions x A For Spherical Wavefronts: A = 4  r 2

63 Intensity Power per unit area W/m 2

64 Loudness of Sound Also called intensity level Determined by the intensity which is a function of the sound's amplitude. The human ear does not have a linear response to the intensity of sound. The response is nearly logarithmic.

65 Decibel Scale (dB) Where: I o = 1 x W/m 2

66 Common decibel levels Threshold of hearing 0 dB = 1 x W/m 2 Whisper 20 dB = 1 x W/m 2 Conversation 65 dB = 3.2 x W/m 2 Threshold of pain 120 dB = 1 W/m 2

67 EXAMPLE How many times more intense is an 80- dB sound than a 40-dB sound?

68 EXAMPLE

69 EXAMPLE Number of times greater = I 1 /I 2

70 Beats When two sound waves that are at nearly the same frequency interfere with each other, they form a beat pattern. It is an amplitude variation. The beat frequency

71 The Doppler effect When a source of sound is moving towards you, it sounds higher pitched (higher frequency). When it moves away, it sounds lower pitched.

72 The Doppler Effect The S’s stand for the source of the sound. The L’s stand for the listener. v by itself stands for the speed of sound. Be careful with the signs on your velocities!! The direction from listener toward source is positive The direction from source toward listener is negative