1408-002
Principles Physics of Physics
Lecture 30 Chapter 15 November 10, 2008 Sung-Won Lee [e-mail]
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Announcement
Lecture notes
http://highenergy.phys.ttu.edu/~slee/1408/
Announcement II
Exam 3 11/14 Friday
11:00 am 11:50 am Chapters 12, 13, 14, 15
Chapter 12 Chapter 13 Chapter 14 Chapter 15 Static Equilibrium Fluids Oscillations Wave Motion
SI Session (Mitchell Lowery)
Tue. 5:30 7:00 pm, Thu. 4:30 6:00 pm @ Holden Hall 226
HW Assignment #7 (Ch.13) is due by 11:59 PM on Tuesday, 11/11 HW Assignment #8 (Ch.14/15) is now placed on MateringPHYSICS, and is due by 11:59 PM on Tuesday, 11/18
Hopefully! Exam III Average = 100%!
Chapter 15 Wave Motion
The Wave Model
We will begin by distinguishing three types of waves:
1.!Mechanical waves can travel only within a medium, such as air or water. (PHYS 1408) Examples: sound waves, water waves. 2.!Electromagnetic waves are self-sustaining oscillations that require no medium and can travel through a vacuum. Examples: radio waves, microwaves, light, x-rays, !-rays, etc. 3.!Matter waves also can travel in vacuum and are the basis for quantum physics (i.e. quantum mechanics - beyond this course). Examples: quantum wave functions for electrons, photons, atoms, etc.
!Characteristics & Types of Wave Motion !Energy Transported by Waves !Mathematical Representation of a Traveling Wave !The Principle of Superposition, Reflection and Transmission !Interference & Standing Waves
15-1 Characteristics of Wave Motion
Wave characteristics: !Amplitude, A !Wavelength, ! !Frequency, f and period, T !Wave velocity,
Two Types of Wave Motion
A transverse wave is a wave in which the particles of the medium move perpendicular to the direction of wave motion. Examples: waves on a string, electromagnetic waves.
A longitudinal wave is a wave in which the particles of the medium move parallel to the direction of wave motion. Example: sound waves (Compression and Expansion).
Transverse Waves
Waves on a string are produced by transverse motion of each particle of the string, participating in the wave motion by moving in a vertical path as the wave propagates. Note that although the wave moves to the right, the individual particles of the string return to their original positions.
Longitudinal Waves
Longitudinal waves (e.g., sound) are produced in a compressible medium by longitudinal motion of each particle of the medium, participating in the wave motion by moving in a horizontal path as the wave propagates.
vstring =
Ts
(wave speed of a stretched string)
vsound = ! RT
(wave speed of sound); ! = 1.402 for air
Here, R is the ideal as constant and T is the temperature in K
Surface (Water) Waves
+ + +
Earthquake Waves
Earthquakes produce both longitudinal and transverse waves. Both types can travel through solid material, but only longitudinal waves can propagate through a fluidin the transverse direction, a fluid has no restoring force.
! P waves
Water waves are a combination of transverse and longitudinal motion, because each particle of water participating in the wave motion travels in a circular path as the wave propagates. The particles stay in the same average position as the waves move to the right. ! P stands for primary or P for Pressure ! Fastest, at 7 8 km/s ! Longitudinal
! S waves
! S stands for secondary or S for shear ! Slower, at 4 5 km/s ! Transverse
vwater =
g! 2"
(speed depends on wavelength)
1-Dimensional Waves
e.g. Waves on a String
Considering a string of total length L and total mass m String (linear) density: = m/L (mass per unit length) e.g. the linear density of a 2-mlong string with a mass of 4 g:
m (0.0040 kg) = = = 0.0020 kg/m L (2.0m)
Sinusoidal Waves
! The wave represented by the curve is a sinusoidal wave ! Simplest example of a periodic continuous wave
The velocity of a transverse wave on a cord is given by:
m = dL
vstring =
Ts
(wave speed of a stretched string)
! The wave moves toward the right
! The orange wave represents the initial position ! As the wave moves toward the right, it will eventually be at the position of the blue curve
The wave speed on a string depends on both the strings linear density and tension force in the string.
! Each element moves up and down in simple harmonic motion
Amplitude & Wavelength & Period
! The crest of the wave is the location of the maximum displacement of the element from its normal position
! This distance is called the amplitude, A
Summary
! The wavelength, #, is 40.0 cm ! The amplitude, A, is 15.0 cm ! The wave function can be written in the form
! The wavelength, #, is the distance from one crest to the next ! Period, T, is the time interval required for 2 identical points of adjacent wave to pass by a point
y = A cos(kx "t)
15-4 Mathematical Representation of a Traveling Wave
Suppose the shape of a wave is given by:
15-4 Mathematical Representation of a Traveling Wave
After a time t, the wave crest has traveled a distance vt, so we write the wave function:
! This is for a wave moving to the right ! For a wave moving to the left, replace (x vt) with (x + vt)
Or: with
In time t, the wave moves a distance vt.
,
Wave Equations
! We can also define the angular wave number (or just wave number), k ! The angular frequency can also be defined ! The wave function can be expressed as
The Mathematics of Sinusoidal Waves
! The speed of the wave becomes v = # ! If x = 0 at t = 0, the wave function can be generalized to D(x,t) = A sin (kx "t + !) ,where ! is called the phase constant
x $ vt % 'x t( & ' ( D( x, t ) = A sin ) 2! + "0 * = A sin + 2! ) $ * + "0 , # . / -# T . 0
x $ % D( x, t = 0) = A sin & 2! + "0 ' (# )
The Angular Frequency #$ and the Wave k
$ Number &x t D( x, t ) = A sin + 2! ) ( / -# T
" = 2! f =
Analyzing a Sinusoidal Wave
A sinusoidal wave with amplitude A = 1.0 m at and frequency f = 100 Hz travels at v =200 m/s in the +x direction. At t=0, the point at x=1.0 m is on the crest of the wave. 1.! Find A, v, ", k, !, #, T, and !0 for this wave. 2.! Write the wave equation. 3.! Draw a snapshot graph at t =0.
% ' * + "0 , . 0
2! = angular frequency (rad/s) T If x is fixed, D(x1,t) = A sin (kx1 - #t + %) 2! gives a sinusoidal history graph at one k# = wave number (rad/m) point in space, x1. It repeats every Ts. " $ 2! % $ " % v=#f =& '& ' = " / k or " = vk ( k ) ( 2! )
D( x, t ) = A sin [kx # !t + "0 ]
A = 1.0 m; v = 200 m/s; f = 100 Hz; k = 2! / " = ! rad/m = 3.14 rad/m;
! = v / f = 2.0 m; T = 1/ f = 0.010 s;
! = 2" f = 200" rad/s = 628 rad/s;
D(1.0 m, 0) = A = A sin(k (1.0 m) + !0 ) ! ! ! k (1.0 m) + "0 = # "0 = $ ! = $ 2 2 2
Note that: D(0, 0) = A sin !0
D( x, t ) = (1.0 m) sin[(! rad/m) x " (200! rad/s)t ] " ! / 2]
Waves vs. Particles
If two pitching machines simultaneously throw baseballs," they will collide and bounce." Two particles cannot occupy the same space point at the same time.!
Waves vs. Particles
On the other hand, if two loud speakers make sound waves at the same time, they will pass through each other w/o collision. ! Two waves can occupy the same space point at the same time.!
15-6 The Principle of Superposition
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Superposition Example!
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If two or more traveling waves are moving through a medium, the resultant wave function is the algebraic sum of the wave functions of the individual waves!
Two pulses are traveling in opposite directions; (a). The pulses have the same speed but different shapes! When the waves start to overlap (b), the resultant wave function is y1 + y2! When crest meets crest (c) the resultant wave has a larger amplitude than either of the original waves! The two pulses separate. They continue moving in their original directions. The shapes of the pulses remain unchanged; (d)!
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Superposition and Interference!
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15-7 Reflection and Transmission
When the pulse reaches the support, the pulse moves back along the string in the opposite direction. This is the reflection of the pulse. The pulse is inverted
Two traveling waves can pass through each other without being destroyed.!
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A consequence of the superposition principle!
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The combination of separate waves in the same region of space to produce a resultant wave is called interference!
With a free end, the string is free to move vertically The pulse is reflected The pulse is not inverted
15-7 Reflection and Transmission
! Assume a light string is attached to a heavier string (see Fig.) ! Pulse travels through the light string and reaches the boundary ! The part of the pulse is inverted ! The reflected pulse has a smaller amplitude
15-8 Interference
The superposition principle says that when two waves pass through the same point, the displacement is the sum of the individual displacements. In the figure below, (a) exhibits destructive interference and (b) exhibits constructive interference.
A wave encountering a denser medium will be partly reflected and partly transmitted; if the wave speed is less in the denser medium, the wavelength will be shorter.
15-8 Interference
These graphs show the sum of two waves. In (a) they add constructively; in (b) they add destructively; and in (c) they add partially destructively.
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