Content of the course 3NAB0 (see study guide)

Content of the course 3NAB0 (see study guide) 17 November diagnostic test! Week 1 : 14 November Week 2 : 21 November Introduction, units (Ch1), Circuits (Ch25,26) Heat (Ch17), Kinematics (Ch2 3) Week 3: 28 November Newton, Energy (Ch4 6) Week 4: 5 December Energy, Momentum (Ch7 8) 8 December Intermediate assessment Week 5: 12 December Week 6: 19 December Week 7: 9 January (2016) Rotation, Elasticity, Fluid mechanics (Ch9 12) Harmonic oscillator and Waves (Ch14 15) Sound (Ch16) Light (Ch33) 25 January Final assessment

Het basisvak Toegepaste Natuurwetenschappen http://www.phys.tue.nl/nfcmr/natuur/collegenatuur.html Applied Natural Sciences Leo Pel e mail: phys3nab@tue.nl http://tiny.cc/3nab0

Chapter 15 Mechanical Waves PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Copyright 2012 Pearson Education Inc.

LEARNING GOALS What is meant by a mechanical wave, and the different varieties of mechanical waves How to use the relationship among speed, frequency, and wavelength for a periodic wave. How to interpret and use the mathematical expression for a sinusoidal periodic wave. How to calculate the speed of waves on a rope or string. How to calculate the rate at which a mechanical wave transports energy. What happens when mechanical waves overlap and interfere. The properties of standing waves on a string, and how to analyze these waves. How stringed instruments produce sounds of specific frequencies. 4

Simple Harmonic oscillator 5 5

Harmonic oscillator and waves 6

Types of waves Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: Transverse if the direction of displacement is perpendicular to the direction of propagation Earth quake waves Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 7

Types of waves Depending on the direction of the displacement relative to the direction of propagation, we can define wave motion as: Longitudinal if the direction of displacement is parallel to the direction of propagation Sound Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 8

Types of waves But also a combination of Transverse and Longitudinal Water waves Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State 9

Transverse and longitudinal waves Mechanical (part I) materials Sound (part II) air 10

Waves have in common: 1. In each case the disturbance travels or propagates with a definite speed through the medium: the wave speed. 2. The medium itself does not travel through space. 3. To set any of these systems into motion, we have to put in energy by doing mechanical work on the system. Waves transport energy, but not matter, from one region to another. 11

MATHEMATICAL 12

Harmonic oscillator and waves 13

Have a description One position: SHM x( t) Acos( t ) One time shot: position in space NEED ONE 14

Wavelength (lambda), 16

Harmonic wave Shake end of string up & down with SHM period = T wavelength = Wave speed =v Wave speed = v = distance time wavelength period = = T = f V=for f=v/ but 1/T=f this is the Golden Rule for waves 17

Example: wave on a string 2 cm 2 cm 2 cm A wave moves on a string at a speed of 4 cm/s A snapshot of the motion shows that the wavelength, is 2 cm, what is the frequency,? v =, so = v / = (4 cm/s ) / (2 cm) = 2 Hz T = 1 / f = 1 / (2 Hz) = 0.5 s 18

Movement of wave Look at: 2 y( x) Acos ( x 0) x=5 x=10 2 y( x) Acos ( x 5) 2 y( x) Acos ( x 10) x( t) Acos( t) x( t) shm position = 5 t= vt Waves 2 Acos t f 2 y( x, t) Acos ( x vt) move right 2 y( x, t) Acos ( x vt) move left 19

) ( 2 cos ), ( vt x A t x y vt x A t x y 2 2 cos ), ( f v t f x A t x y 2 2 cos ), ( t kx A t x y cos ), ( 2 k Wave number Alternate notation + if wave travels toward x - if wave travels toward +x 20

Definitions Amplitude (A, y m ) Maximum value of the displacement of a particle in a medium (radius of circular motion). Wavelength () The spatial distance between any two points that behave identically, i.e. have the same amplitude, move in the same direction (spatial period) Wave Number (k) Amount the phase changes per unit length of wave travel. (spatial frequency, angular wavenumber) Period (T) Time for a particle/system to complete one cycle. Frequency (f) The number of cycles or oscillations completed in a period of time Angular Frequency Time rate of change of the phase. Phase kx t Time varying argument of the trigonometric function. Phase Velocity (v) The velocity at which the disturbance is moving through the medium 21

The Wave Equation If we see an equation that looks like:... we can write down the amplitude, frequency, velocity, and wavelength of the wave it describes. y -1-1 0.15m cos 1570.8 s t 4.5796 m x y A cos( ) cos( ) Acos2ft 0.15 m 2x 2f 1570.8 s 2-1 4.5796 m 1.372 m v f 343 m/s, in the x direction. -1 f 250 Hz 22

General description SHM The force described by Hooke s Law is the net force in Newton s Second Law. m d 2 x( t) dt 2 kx( t) x( t) where = Acos( t) k m 1. The acceleration is proportional to the displacement of the block 2. The force is conservative. In the absence of friction, the motion will continue forever. 23

General description waves Wave Equation: 2 y(x,t) x 2 position 1 v 2 2 y(x,t) t 2 time Partial Differential Equation (PDE) A solution is: y( x, t) Acos kx t We will not try to solve this equation 24

General description waves Wave Equation: y t 2 y 2 t y 2 y(x,t) x 2 ( x, t) 1 2 y(x,t) v 2 t 2 Acos kx t Asinkx t y kasinkx t x 2 A coskx t 2 y 2 k A coskx t 2 x 2 2 k A coskx t A coskx t 2 v And k 2 v f v k 25

Sinusoidal Wave on a String, final The maximum magnitudes of the transverse speed and transverse acceleration are v y, max = A a y, max = 2 A T v y v golf getal k The speed v is constant for a uniform medium, whereas v y varies sinusoidally. 26

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Rope: mass + springs x ( t) Acos( t) where = k m How fast will it be transmitted 1) Mass rope: m larger > slower 2) More tension > k larger > faster 29

Speed of a Wave on a String The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected. tension force v tension mass / length F (string mass / string length) This assumes that the tension is not affected by the pulse. This does not assume any particular shape for the pulse. 30

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Energy in a Wave The average power: P ave 1 2 F 2 A 2 Duck swimming When a duck swims, it necessarily produces waves on the surface of the water. The faster the duck swims, the larger the wave amplitude and the more power the duck must supply to produce these waves. The maximum power available from their leg muscles limits the maximum swimming speed of ducks to only about 0.7 m/s (2.5 km/h) 32

Waves Intensity Wave intensity for a three dimensional wave from a point source: I P in units of W/m 2 4r 4r r I 2 1 I1 4 2 2 2 2 I I 1 2 r r 2 2 2 1 Example: earthquake, sound waves 33

Wave boundary/ interference 34

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The principle of superposition When two waves overlap, the actual displacement of any point at any time is obtained by adding the displacement the point would have if only the first wave were present and the displacement it would have if only the second wave were present: y( x, t) y ( x, t) y ( x, t) 1 2 36

Superposition of equal amplitude waves Constructive Destructive 37

Reflection (from a fixed end) Animations courtesy of Dr. Dan Russell, Kettering University 38

Reflection (from a loose end) Animations courtesy of Dr. Dan Russell, Kettering University 39

Superposition of equal amplitude waves Constructive Destructive 40

Standing Wave 42

Adding waves harmonic waves in opposite directions incident wave reflected wave resultant wave (standing wave) Animations courtesy of Dr. Dan Russell, Kettering University 43

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Standing Waves Resonance y1( x, t) y 2 ( x, t) Acos Acos kx t kx t y( x, t) Acos kx t Acoskx t cos x - cos y = -2 sin ((x - y)/2) sin( (x + y)/2 ) y( x, t) 2Asin( kx)sin( t) 45

Nodes and Antinodes Node position of no displacement Antinode position of maximum displacement y 2Asin kx cos t Nodes 2 kx x 0,,2,3... 3 x 0,,,,... 2 2 46

Confined waves Only waves with wavelengths that just fit in survive (all others cancel themselves out) 2 kx x 0,,2,3... 2L n 3 x 0,,,,... 2 2 v 2L L n n f n n nf1 2 47

Allowed frequencies = 2L =L f 0 =V/ = V/2L Fundamental tone f 1 =V/ = V/L=2f 0 =(2/3)L =L/2 1 st overtone f 2 =V/=V/(2/3)L=3f 0 2 nd overtone f 3 =V/=V/(1/2)L=4f 0 3 rd overtone =(2/5)L f 4 =V/=V/(2/5)L=5f 0 4 th overtone 48

VIDEO + UITWERKING L = 1.8 m String mass= 188 gr Tension = 3N v f n n n 4. 7 2L µ =0.00810 kg/m T v 16.92 m / 1 2 3 4 5 6 7 8 9 10 11 12 13 14 4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.8 Hz s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 4.7 9.4 14.1 18.8 23.5 28.2 32.9 36.6 42.3 47.0 51.7 56.4 61.1 65.8 50

Fundamental, 2nd, 3rd... Harmonics n 2 L 2nd harmonic 3rd harmonic Fundamental (n=1) Fig 14.18, p. 443 Slide 25 51

Loose Ends L 2n 1 4 L 4 n 4L (2n 1) L 3 4 f n (2n (2n 1) 1) v 4L f 1 L 5 4 (Organ pipes open at one end) 52

Harmonics with a bridge 53

Stringed instruments Three types Plucked: guitar, bass, harp, harpsichord Bowed: violin, viola, cello, bass (Bowing excites many vibration modes simultaneously mixture of tones (richness) Struck: piano All use strings that are fixed at both ends Use different diameter strings (mass per unit length is different) The string tension is adjustable tuning f 1 1 2L T 54

A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string? 1. 220 m/s 2. 440 m/s 3. 660 m/s 4. 880 m/s 5. 1.10 km/s 55

A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz. What is the speed of the waves on this string? 1. 220 m/s 2. 440 m/s 3. 660 m/s 4. 880 m/s f n n nf1 5. 1.10 km/s v 2L v f 2L 1 56

A stretched string is fixed at points 1 and 5. When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points 1. 1 and 5. 2. 1, 3, and 5. 3. 1 and 3. 4. 2 and 4. 5. 1, 2, 3, 4, and 5. 57

A stretched string is fixed at points 1 and 5. When it is vibrating at the second harmonic frequency, the nodes of the standing wave are at points 1. 1 and 5. 2. 1, 3, and 5. 3. 1 and 3. 4. 2 and 4. 5. 1, 2, 3, 4, and 5. 58

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration? 1. L/6 2. L/4 3. L/5 4. L/2 5. L/3 59

A stretched string of length L, fixed at both ends, is vibrating in its third harmonic. How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration? 1. L/6 2. L/4 3. L/5 4. L/2 5. L/3 60

Drum: 2D standing waves Mode (0,1) Mode (1,1) Mode (2,1) Mode (0,2) Animations courtesy of Dr. Dan Russell, Kettering University 61

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Summary 63

Summary 64

Applied Natural Sciences Thank you for your attention Merry Christmas Have a happy holiday and see you next year 65