is a What you Hear The Pressure Wave sets the Ear Drum into Vibration.

Transcription

1

2

3 is a What you Hear The ear converts sound energy to mechanical energy to a nerve impulse which is transmitted to the brain. The Pressure Wave sets the Ear Drum into Vibration.

4 electroencephalogram

5

6 v S v = Mach #

7 Particle Waves Electrons are STANDING WAVES in atomic orbitals. λ = h p

8 result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by 1 f = Τ amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM) periodor frequency (how long does it take for each bit to go through one cycle?) wavelength (over what distance does the cycle repeat in a freeze frame?) v= λ f wave speed (how fast is the energy transferred?)

9 Wavelength and Frequency are Inversely related: The shorter the wavelength, the higher the frequency. The longer the wavelength, the lower the frequency. f = v λ 3Hz 5Hz

10 Problem: v = λ f The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. A) What is the amplitude of the wave? B) What is the wavelength of the wave? C) What is the frequency of the wave? 10 cm / s f = v/ λ = = 1.67Hz 6cm 4cm 6cm 1.67 Hz

11

12

13 Wave speed: Depends on Properties of the Medium: Temperature, Density, Elasticity, Tension, Relative Motion v = λ f

14 Types of Waves Sound String

15 Transverse Wave A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave The particle motion is shown by the blue arrow The direction of propagation is shown by the red arrow

16 Longitudinal Wave A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave The displacement of the coils is parallel to the propagation

17 Sound is a Longitudinal Wave Pulse Tuning Fork Guitar String

18 Spherical Waves

19 Complex Waves Some waves exhibit a combination of transverse and longitudinal waves Surface water waves are an example

20 Example: Earthquake Waves P waves P stands for primary Fastest, at 7 8 km / s Longitudinal S waves S stands for secondary Slower, at 4 5 km/s Transverse A seismograph records the waves and allows determination of information about the earthquake s place of origin

21 v = λ f HO#1

22 Wave PULSE: traveling disturbance transfers energy and momentum no bulk motion of the medium comes in two flavors LONGitudinal TRANSverse

23 Reflection of a Wave Pulse

24 Reflected PULSE: If the end is bound, the pulse undergoes an inversion upon reflection: a 180 degree phase shift If it is unbound, it is not shifted upon reflection. Free End Bound End

25 Traveling Pulse For a pulse traveling to the right y (x, t) = f (x vt) For a pulse traveling to the left y (x, t) = f (x + vt) The function y is also called the wave function: y (x, t) The wave function represents the y coordinate of any element located at position x at any time t The y coordinate is the transverse position If t is fixed then the wave function is called the waveform It defines a curve representing the actual geometric shape of the pulse at that time

26 2 ( x 3 t) + 1 Traveling Pulse yxt (, ) = t = 0 s, y( x,0) = ( x) t = 1 s, y( x,1) = ( x 3) t = 2 s, y( x,2) = ( x 6) 1

27 2 ( x 3 t) + 1 Traveling Pulse yxt (, ) = x= 5, y(5, t) = (5 3 t) 1

28 Linear Wave Equation 2 2 y 1 y = x v t The equation can be written as The linear wave equation is satisfied by any wave function having the form y = f (x ± vt) This applies in general to various types of traveling waves y represents various positions For a string, it is the vertical displacement of the elements of the string For a sound wave, it is the longitudinal position of the elements from the equilibrium position For em waves, it is the electric or magnetic field components

29 Wave Functions are Solutions to the Wave Equation 2π y( xt, ) = Asin x vt λ ( ) 2 2 y 1 y = x v t k = 2π λ 2π ω = = T 2π f v λ = λ f = = T ω k Derive these: yxt (, ) = Asin( kx ωt) x yxt (, ) = Asin2 π f( t) v x yxt (, ) = Asin 2π λ t T

30 Wave Function k = 2π λ 2π ω = = T 2π f y( xt, ) = Asin( kx ωt) A = 4cm λ = 6cm v = 10 cm/ s f = 1.67Hz

31 Wave Function k = 2π λ 2π ω = = T 2π f y( xt, ) = Asin( kx ωt) A= 4cm λ = 6cm v= 10 cm/ s f = 1.67Hz k π = ω = 2π f = π yxt (, ) = 4sin( x 10.5 t) 3

32 Traveling Waves The wave represented by the curve shown is a sinusoidal wave It is the same curve as sin θ plotted against θ This is the simplest example of a periodic continuous wave It can be used to build more complex waves Each element moves up and down in simple harmonic motion Distinguish between the motion of the wave and the motion of the particles of the medium 2π y( xt, ) = Asin x vt λ ( )

33 Space Plot 2π y( xt, ) = Asin x vt λ ( ) Snap shot in TIME. Time is fixed. This is an image of the string or the medium s displacement from equilibrium at one instant. Can represent either transverse or longitudinal waves!!

34 Time Plot 2π y( xt, ) = Asin x vt λ ( ) Snap shot in Space. This is an image of one piece of a string and how it moves as the waves goes by in time.

35

36 Speed of wave depends on properties of the MEDIUM v = λ f Speed of particle in the Medium depends on SOURCE: SHM vt () = Aω sinωt

37 Wave Speed v = λ f This gives the relationship between the wavelength and frequency for constant wave speed. The frequency depends on the source and the speed depends on the properties of the medium. The speed of sound is independent of the frequency. When traveling from one medium to another, if the speed changes, the wavelength changes but the frequency (energy) remains the same.

38 Wave 1 Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function y(x, t) = (0.800 m) sin[0.628(x vt)] where v = 1.20 m/s. (a) Sketch y(x, t) at t = 0. (b) Sketch y(x, t) at t = 2.00 s.

39 Note how the entire wave form has shifted 2.40 m in the positive x direction in this time interval.

40 Wave 2 Consider the sinusoidal wave with the wave function y = (15.0 cm) cos(0.157x 50.3t). At a certain instant, let point A be at the origin and point B be the first point along the x axis where the wave is 60.0 out of phase with point A. What is the coordinate of point B?

41 y = (15.0 cm) cos(0.157x 50.3t). At a certain instant, let point A be at the origin and point B be the first point along the x axis where the wave is 60.0 out of phase with point A. What is the coordinate of point B? π rad cm xb 50.3 rad s t = rad s t± rad 3 ( ) ( ) ( ) π 0.157(0) = 0.157x B ± 3 x B ±π rad = = ± 6.67 cm rad cm ( )

42 Waves on Strings v = λ f v = μ = F (1D string) μ m/ L (linear mass density)

43 Problem: v = λ f The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. D) If the linear density of the string is.01kg/m, what is the tension of the string?

44 Problem: v = F m/ L The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. D) If the linear density of the string is.01kg/m, what is the tension of the string? F = v 2 ( m/ L) 2 5 F = (.1 m) (.01 kg/ m) = 10 N

45 Problem: v = F m/ L The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. e) If the the tension doubles, how does the wave speed change? Frequency? Wavelength? v 2 = F2 m/ L 2F = = m/ L 2v Wave speed increases by a factor of 2

46 Problem: v = F m/ L The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. e) If the the tension doubles, how does the wave speed change? Frequency? Wavelength? v = λ f

47 Problem: v = F m/ L The displacement of a vibrating string vs position along the string is shown. The wave speed is 10cm/s. e) If the the tension doubles, how does the wave speed change? Frequency? Wavelength? v = λ f No Change in wavelength!! Wave speed increases by a factor of 2 Frequency increases by a factor of 2

48 v = λ f HO#3 v = F m/ L

49 Waves Transmit Energy Wave Energy is proportional to frequency: the faster he sends a pulse down the string, the more energy transmitted to the dog! Arrrrf!

50 Wave Energy: E ~ f 6 10 ev 10 ev 4 1 2eV 40eV KeV MeV Energy to ionize atom or molecule: eV

51 Energy The total kinetic energy in one wavelength is K λ = ¼μω 2 A 2 λ The total potential energy in one wavelength is U λ = ¼μω 2 A 2 λ This gives a total energy of E λ = K λ + U λ = ½μω 2 A 2 λ

52 Power Associated with a Wave The power is the rate at which the energy is being transferred: ΔE μω A λ = = = μω Av Δt T 2 The power transfer by a sinusoidal wave on a string is proportional to the Frequency squared Square of the amplitude Wave speed

53 Waves Transmit Energy Power Transmitted on a String: ΔE = = Δt μω A v 2

54 HO#4 ΔE = = Δt μω A v 2

55 HO#5 ΔE = = Δt μω A v 2