Chapter 10 WAVE MOTION - Polytechnic School

1 Chapter 10--Wave Motion341direction of wave motionplane waves (viewed from above)wave crests moving outward from wave sourceFIGURE of water wavesdirection of wave motionChapter 10 WAVE MOTIONA.) Characteristics of waves :1.) A wave is a disturbance that moves through a ) Example: A pebble is dropped into a quiet pond. The DISTUR-BANCE made by the pebble is what moves outward over the water'sonce-still surface. Water molecules are certainly jostled by the wave, butafter the wave passes by, each molecule finds itself back in its original,pre-disturbance position (at least to a good approximation).In other words, water waves are not made up of lumps of water thatmove across the water's surface. They are disturbances that move throughthe water that only temporarily displace water molecules in the 1: A group of waves is called a wave 2: Looking down from above, pebble-produced water waves will looklike a series of crests and troughs moving in ever-expanding circles outward awayfrom the pebble's point-of-entry into the water.

2 If, at a given instant, lines aredrawn on the crests, we find a visual presentation of the waves as shown in Figure shows a side-view of this same pressure regionnormal pressure regionhigh pressure regionhighpressurenormallowpositionPress ure versus Position Graph for SOUND waves (at a given instant in time)direction of wave motionFIGURE 3: Once a wave train has moved far enough away from its source,crests in the immediate vicinity of one another are to a very good approximationparallel to one another (look at the outer sections in Figure ). waves inthis situation are called plane waves and are shown on the previous page inFigure It is not uncommon for plane waves to be assumed when wave-phenomena are being ) A soundwave is a pressuredisturbance thatmoves through airor water, or what-ever the hostmedium happensto vocalsound is generated bythe back and forthvibration of the vocalcords. When thesecords are extended,they momentarilycompress airmolecules togethercreating a highpressure region thatis acceleratedoutward.

3 As the cordspull back, they gener-ate a momentary vac-uum--a low pressureregion (in betweenthese two situations,the air pressure obvi-ously passes througha "normal" pressurecircumstance). Inother words, thevibration of the vocalcords creates regions of high pressure, then normal pressure, then lowpressure, then normal pressure, then high pressure, etc., as they vibrateback and forth (Figure presents a representation of what sound wavesChapter 10--Wave Motion343would look like if our eyes were sensitive to very subtle pressure variations--Figure graphs pressure variation versus position for sound at a givenpoint in time).These pressure disturbances move out into the surrounding air atapproximately 330 meters per second ( , the speed of sound). As theypass a hearing person, the pressure variations motivate tiny hairs in thelistener's ears to vibrate generating electrical signals which, uponreaching the brain, are translated into incoming , sound waves are a disturbance moving through a the medium, there can be no : That's right, the next time you see Star Wars and they show a bigbattle scene viewed from space, you have every right to stand up in the middleof the movie theater and shout at the top of your lungs, "WAIT, WAIT, THISCAN'T BE.

4 THERE IS NO SOUND IN SPACE!!" They'll probably throw youout of the theater for causing a disturbance ( , for making waves --a littlephysics humor), but you will be correct in exposing one of Hollywood's greatestdisplays of scientific misinformation ) waves are important because they carry : If you think about it, this should be obvious. If waves didn't carryenergy, sound waves wouldn't have the wherewithal to wiggle those little ear-hairs that allow you to hear, and tidal waves would not have the ability to blowaway whole island-populations with a single ) There are two kinds of waves , both of which are identified by how thedisturbance-producing force is applied:a.) Transverse waves : These are waves that are created by a forcethat is applied to a medium perpendicular to the direction of the wave'smotion in the ) An example: When a pebble enters a pond, it applies a forceto the water that is perpendicular to the water's surface, henceperpendicular to the wave's direction, as it moves out over the water'ssurface.

5 As such, this is a transverse ) Longitudinal waves : These are waves created by a force appliedto a medium in the same direction as the wave's MOTION in the viewed from side inverted wave after reflectiondoor viewed from sidewave moving towardwall before reflectionFIGURE after reflection (full-wave inversion brings wave back to original orientation)wave before reflection off free endFIGURE ) An example: When sound from a loud-speaker is produced, thespeaker cone applies a force to air molecules that is in the samedirection as the subsequent pressure- waves that move out from ) Wave reflection:a.) Consider a tautrope fixed to a the rope at theunattached end will pro-duce a single wave thatwill travel down the rope(see Figure ). Whenit gets to the door, thewave will bounce off thefixed end, flip 180o ( radi-ans; one-half a cycle;whatever--see ) and proceed backdown the line. This half-wave inversion is typical ofwave-reflection off fixed ) Consider a ropehanging freely from aceiling.

6 A single wavemoving downward (seeFigure ) will bounceoff the free bottom andproceed back up towardits bounce-back flipsthe wave 360o ( , itcomes back to its original position) before the wave proceeds back up therope (see Figure ). This full-wave inversion (net effect--no inversionat all) is typical of wave reflection off free ) Some DEFINITIONS: Chapter 10--Wave Motion345crest to cresttrough to troughany point to where that point repeats itselfnodeanti-nodeFIGURE ) Wave-length ( in me-ters): the dis-tance betweentwo successivecrests, or twosuccessivetroughs, or be-tween two suc-cessive positionsalong the wavethat are exactduplicates of one another (see Figure ).b.) Frequency (" " in cycles/second--this symbol is a Greek "nu"):the number of wavelengths that pass a fixed observer per ) Period ("T" in seconds/cycle): the time required for one full wave-length to pass a fixed observer. As in vibratory MOTION , T = 1/ .d.) Wave velocity ("v" in meters/second): the velocity of a wave dis-turbance as it moves through its medium.

7 Mathematically:v = .(Don't believe me? Check the units.)A consequence of this relationship: for a given wave, high frequencycorresponds to short wavelength and vice ) Nodes and anti-nodes: a node is a null spot on the wave. It cor-responds to a place where the displacement of the wave is zero (seeFigure ). An anti-node is a spot where the displacement is amaximum. It corresponds to a crest or trough (see Figure ).f.) Superposition of waves : when two waves in the same mediumrun into one another, the two disturbances will add to one another in alinear way. Given such a situation, there are a number of outcomes:i.) Constructive superposition: a situation in which the twowaves momentarily produce a single wave that is larger than theoriginal two. For two waves with the same amplitude A, completelyconstructive superposition will yield a displacement of waves moving in opposite directions in same medium completely constructive superposition(the dark line shows the net effect) partially destructive superposition(central superposition is totally destructive)FIGURE ) Destructivesuperposition: a situation inwhich the two waves produce asingle wave that is smallerthan the largest of the originaltwo.

8 For two waves with thesame amplitude A, completelydestructive superposition willproduce a net displacement ) Figure shows twowaves (one denoted with dots,one denoted with dots anddashes) moving in oppositedirections in the samemedium. Figures show the waves atvarious stages of ) Mathematics of Traveling waves :1.) The displacement of atraveling sine wave is a function of bothtime and position. Its displacement willvary at a given time from place to place inaddition to varying at a given place astime ) The function thatcharacterizes this situation is:y(x,t) = A sin (kx + t),where A is the amplitude of the wave, k (the wave number) is defined as 2 / (just as --the angular frequency--governs how fast the function changesrelative to time, k governs how fast the function changes relative to position--itis like a positional-frequency function), and x and t are the two variable-parameters that allow one to zero in on a particular place at a particular time 10--Wave Motion347oscillating mass force appliedmomentarily every secondsFIGURE ) Resonance:1.

9 Vibrating systems usually have at least one frequency at which thesystem will naturally vibrate. A swing, for instance, acts like a pendulum. Inthe last Chapter , we found that for small angles the frequency of a pendulum isdependent upon the acceleration of gravity and the length of the pendulum neither of those changes in a given situation, a pendulum has onlyone frequency at which it will freely, naturally oscillate (in fact, this isapproximately true even for moderately large amplitude oscillations).Although you haven't run into any yet, there are systems that have more thanone natural ) Resonance is a situation in which the frequency of an applied forcematches one of the natural frequencies of the system to which the force is ap-plied. The consequence of such a condition is an increase in the system's energyand amplitude of ) Example: Consider a mass attached to the spring shown in Every such spring/mass combination will have one natural frequency atwhich the system will oscillate (remember = [(k/m)1/2]/2 ?

10 For this ex-ample, assume that the natural frequency is 1/2 cycle persecond. As the period is T = 1/ , the period of thisoscillation will be 2 seconds per cycle. With the mass atrest:a.) A force is briefly applied to the mass,then applied periodically for just a moment seconds (see Figure ). Will the force helpor hinder the oscillatory MOTION ?i.) As the force's frequency is out of syncwith the natural frequency of the system, itdoesn't take a genius to see that the force willfight the natural MOTION of the system. Theconsequence: the mass's MOTION will bedisorderly and ) Another possibility: the frequency of the applied force is now 2seconds/cycle. Under this condition, the push comes each time the massis at its lowest point--each time the mass is naturally ready to startupwards again. The force helps the MOTION because the frequency of theapplied force matches the natural frequency of the oscillating system. As348such, the amplitude of the MOTION gets bigger and bigger with each cycle,and the energy of the vibrating system : You might at first think that because the amplitude of the os-cillation is changing, the frequency must change.