1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by

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Marybeth Thornton
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2 result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by amplitude (how far do the bits move from their equilibrium positions? Amplitude of MEDIUM) 1 f period or 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 f wave speed (how fast is the energy transferred?)

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

4 Spherical Waves

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

6 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

to move parallel to the direction of propagation

7 Pulse 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: Tuning Fork Guitar String

8 Types of Waves Sound String

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

10 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

11 Traveling Pulse Wave Form Space Snap t 0 s, y( x,0) ( x) 1 y( x, t) ( x3 t) t 1 s, y( x,1) ( x 3) t s, y( x,) ( x 6) 1

12 Time Plot One position y( x, t) ( x3 t) 1 Changing in x 5, y(5, t) (5 3 t) 1

13 Traveling Waves The media moves in SHM. The wave travels at constant speed. The wave has the same frequency as the shaking source!

14 Traveling Waves The wave represented by the curve shown is a sinusoidal wave It is the same curve as sin q plotted against q 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 y( x, t) Asin( kx t) y( x, t) Asin x vt

15 Wave Functions are Solutions to the Wave Equation y 1 y x v t y( x, t) Asin x vt k T f v f T k Derive these: y( x, t) Asin( kx t) x y( x, t) Asin f ( t) v x y( x, t) Asin t T

16 Speed of wave depends on properties of the MEDIUM v f Speed of particle in the Medium depends on SOURCE: SHM y( x, t) Asin( kx t) v( x, t) Acos( kx t) a x t A kx t (, ) sin( )

17 Wave Speed v 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.

20 Time Plot y( x, t) Asin( kx t) 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. The one piece oscillates in SHM.

21 Space Plot y( x, t) Asin( kx t) Snap shot in TIME. Time is fixed. This is an image of the entire string or the medium s displacement from equilibrium at one instant. Can represent either transverse or longitudinal waves!!

22 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[0.68(x vt)] where v = 1.0 m/s. (a) Sketch y(x, t) at t = 0. y( x, t) Asin x vt (b) Sketch y(x, t) at t =.00 s. When x = 5n, we get a node!!! x = 5, 10, 15, 0

23 Space Snap Shots in Time Note how the entire wave form has shifted.40 m in the positive x direction in this time interval: x= vt =(1.m/s)(s)=.4m!!!

24 Wave Speed is Constant! Medium Accelerates!! String y( x, t) Asin( kx t) v( x, t) Acos( kx t) a x t A kx t (, ) sin( ) y max = A v y, max = A a y, max = A

25 COMPARE: Motion Equations for Simple Harmonic Motion Chapter 15 x is fixed!! Cos or Sin changes phase! x( t) A cos ( t ) dx v Asin( t ) dt a d x dt Notice: Acos( t ) a x

26 Wave Example 16. Consider the sinusoidal wave with a frequency of 8Hz. Find the wave equation. y( x, t) Asin( kx t ) k T f y = (15.0 cm) cos(0.157x 50.3t).

27 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? v t f t t ( vt) kx /3 x k kx x 6.67cm x

28 Wave Function k T f If the wave speed is Find: v 10 cm / s y( x, t) Asin( kx t)

29 Wave Function k T f y( x, t) Asin( kx t) A 4cm 6cm v 10 cm / s f 1.67Hz k f y( x, t) 4sin( x 10.5 t) 3

30 Waves on Strings v f F v ( F Tension) m/ L (linear mass density)

31 Problem: v 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?

32 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 ( m/ L) 5 F (.1 m) (.01 kg / m) 10 N

33 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 F m/ L F m/ L v Wave speed increases by a factor of

34 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 f

35 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? Wave speed depends on the MEDIUM v f Frequency depends on the SOURCE of vibration Wavelength depends on BOTH!

36 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 f v v f f

37 Waves Transmit Energy Wave Energy is proportional to frequency and amplitude! Arrrrf! 1 Av The total kinetic energy in one wavelength is K = ¼ A

38 String Power 1 Av

39 String Power 1 Av

40 Sound Power The power transmitted through a closed surface by a wave is proportional to the amplitude of the wave.

Poynting vector defined as The magnitude of the Poynting vector is

41 Intensity of EM Waves is proportional to Amplitude Squared! The energy flow of an electromagnetic wave is described by the Poynting vector defined as The magnitude of the Poynting vector is The intensity of an electromagnetic wave whose electric field amplitude is E 0 is

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

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