wae-1 Wae Motion A wae is a self-propagating disturbance in a medium. Waes carr energ, momentum, information, but not matter. Eamples: Sound waes (pressure waes) in air (or in an gas or solid or liquid) Waes on a stretched string Waes on the surface of water "The Wae" at the ballpark stadium. The medium is the people. Electromagnetic waes (light) this is the onl kind of wae which does not require a medium. EM waes can trael in acuum. This was quite a surprise to 19 th centur phsicists who had trouble imagining a wae without a medium. The thing that is "waing" in an EM wae is an electromagnetic field which generates itself as it propagates. In a sense, an EM wae "rolls out it own carpet", creating its own medium as it moes forward. More on EM waes net semester. We can use a wae to send a signal (information) without sending an matter. Imagine a long line of people holding hands. We can send a signal down the line b hand-squeezing (a disturbance in the people medium), and et no one moes. Traeling Waes can be categorized as: sinusoidal: wae speed or impulse: Traeling waes can also be categorized as: transerse: displacement of the medium is perpendicular (transerse) to the direction of the wae elocit (like a wae on water or a string or drumhead). or longitudinal: displacement of the medium is parallel to the direction of the wae elocit (like sound wae or a slink that has been push-pulled). ongitudinal Wae: displacement wae elocit 4/19/2009 Uniersit of Colorado at Boulder
wae-2 Wae Speed Wae speed is distance 1 waelength time time for 1 to go b T, T = period f ( since frequenc f = 1/T ) T Almost alwas, the wae speed is a constant, independent of and T. The wae speed depends on the properties of the medium, not on the properties of the wae. Eamples: medium = string, properties = tension, mass per length medium = air, properties = temperature, mass per molecule, etc = f = constant increases as f decreases, decreases as f increases Mathematical description of traeling sinusoidal waes Sinusoidal waes hae a waelength and a frequenc f = 1/T. (Impulse waes hae neither.) = (, t) = displacement = displacement from the equilibrium (no wae) position Snapshot in time: freeze time at, sa, t = 0 +A A (, t 0) Asin 2 Asin k A is the amplitude of the wae. The displacement oscillates between +A and A. 2 k = wae number Now freeze position, watch wae go b at position = 0: 4/19/2009 Uniersit of Colorado at Boulder
+A A t wae-3 t ( 0, t) Asin 2 T Asin t 2 = angular frequenc T Wae traeling to the right: (, t) Asin 2 Asink t Notice: When position changes b distance or time t changes b period T, the sine function goes through one complete ccle. When increases b one AND t increases b one T, then the sine function stas at the same phase; we are then riding along with the wae. t T +A A The argument of the sin function k t is called the phase. A point on the traeling wae (traeling along with the wae) corresponds to a particular alue of the phase k t. As t increases, must increase in order to keep k t a constant alue, hence a point of constant phase corresponds to a point moing to the right (increasing ). Could also hae a wae traeling to the left: (, t) t Asin 2 T We could hae used cosine instead of sine for the form of the wae. The onl difference between sin and cos here is where we put the zero of time. Wae speed is distance 1 waelength time time for 1 to go b T f ( since frequenc f = 1/T ) T Another wa to see that our formula for the wae = (,t) corresponds to a wae moing right with speed = /T is to rewrite the formula like so: 4/19/2009 Uniersit of Colorado at Boulder
wae-4 t (, t) Asin 2 Asin t 2 2 Asin t T T A point traeling with the wae is a point with ( t) = constant or = t + const. This is the equation for a point moing right with speed (graph of s t has slope / t = ). Claim: An traeling wae = (,t), traeling with a rigid shape ("dispersionless") has the form (, t) f ( t) (wae traeling to right with speed ) or (, t) f ( t) (wae traeling to left with speed ) Proof: Consider a wae initiall with some shape = f() (at t = 0 ) that is moing to the right with speed. And consider a moing coordinate sstem (', '), that is moing along with the wae f(, t=0) f(, t 1 >0) ' '= f(') 0 = t ' Coordinate transform: = ' + t, ' = t = ' t ' ' (, ) = (', ') ' In the moing '' coordinate sstem, the moing wae is stationar: ' = f('). Transfoming to the coordinate sstem, the wae ' = f(') becomes = f( t). Done. Interference of waes Superposition Principle: If two or more waes are present in the same place, at the same time, the total wae is the sum of the indiidual waes: tot (,t) = 1 (,t) + 2 (,t). You get constructie or destructie interference depending on whether 1 and 2 add (both hae same sign) or cancel (opposite signs). 4/19/2009 Uniersit of Colorado at Boulder
wae-5 time 1) time 2) time 3) time 4) Waes carr energ. For a string wae, the energ is both KE (string moes as wae passes) and PE (originall straight string must stretch a little as wae passes elastic PE similar to (1/2)k 2 ). At time 3 aboe, where is the energ? Answer: in the KE. At time 3, the string is moing, while instantaneousl flat. Standing Waes Two sinusoidal traeling waes of the same ends fied (and therefore the same f = / ) and the same amplitude, traeling in opposite directions, oerlapping in the same region of space, make a standing wae. nodes anti-nodes If the ends of a string of length are fied, as in a stringed musical instrument, then standing waes are onl possible at certain resonant frequencies gien b the condition: n 2, n = integer, = fied total length Sound Sound is a pressure wae in air. It is a longitudinal wae. force pressure =, p area F A Air consists mostl of ogen and nitrogen molecules. At room temperature, the molecules hae thermal energ and are moing around rapidl (speed 400 m/s), colliding with each other, and 4/19/2009 Uniersit of Colorado at Boulder
wae-6 with eer eposed surface. The pounding of the air molecules on a surface, like the pitter-pat of rain on the roof, add up to a large force per area. air molecules At sea leel, atmospheric pressure is 1 atm = 14.7 psi = 1.01 10 5 N / m 2 (psi = pound per square inch) This is a big pressure! We are not ordinaril aware of this big pressure because the air pushes on us equall from all sides (een surface from our insides due to the air in our lungs). The big forces on us from all sides cancel out and there is not net force on us. air air acuum forces balanced forces unbalanced (Space Shuttle astronaut heroicall plugs hole with head,) Ordinar sounds are caused b relatiel small changes in the air pressure at our ear drum. A pressure change of a few parts per million is enough to wiggle the ear drum and make the listener perceie a loud sound (the ear is sensitie!). When we here a tone of constant frequenc f = 1 / T, the pressure s. time at the eardrum looks like this: 1 atm pressure at eardrum amplitude A period T amplitude eaggerated frequenc f = 1 / T time t The amplitude A of the pressure wae controls the loudness of the sound. Big A oscillation big wag of eardrum loud sound. big pressure The frequenc is the "pitch" of the tone. Hi f hi notes (soprano) o f lo notes (basso). More on frequenc and music below. Frequenc range of human hearing: f = 20 Hz 20 khz. [frequenc f is usuall measured in hertz (Hz), ccles per second] The upper end degrades as we age. At age 65, most humans cannot hear aboe 10 khz. Dogs can hear up to 35 khz (ultrasonic). 4/19/2009 Uniersit of Colorado at Boulder
wae-7 As with most waes, the speed of sound in air depends on the properties of the medium = air (temperature, mass of molecules, etc) NOT on the properties of the sound wae. sound = 343 m/s (this aries b about 10 m/s, depending on the temperature of the air) Sounds traels about 1 mile in 5 seconds.. If ou see a flash of lighting and then start counting seconds (1-1 thousand, 2-1 thousand, 3-1 thousand...) until ou hear the thunder, ou can estimate how man miles awa the lighting is. The speed of light is er, er fast (light speed c = 3 10 8 m/s) so the light arries almost instantl, while the sound wae takes a while to catch up. Eample: What is the waelength range of human hearing? 343 m/s 343 m/s = f 17 m 0.017 m (17 cm) -1-1 f 20 s 20000 s Music The frequenc is the "pitch" of the tone. "middle C" = 262 Hz, D = 294 Hz, E = 330 Hz, etc. Each "octae" on the musical scale is a factor of 2 change in frequenc, for instance, "C aboe middle C" = 2 262 = 524 Hz. (It's called an "octae", onl because, in the Western musical scale, there are 8 notes in an octae.) A single frequenc sound wae (a pure tone) is an irritating sound to most people. A pleasant musical note is usuall a miture of frequencies = lowest or fundamental frequenc + integer multiples of the fundamental or oertones f 1 = fundamental = 1 st harmonic f 2 = 2 f 1 = 1 st oertone = 2 nd harmonic f 3 = 3 f 1 = 2 nd oertone = 3 rd harmonic f n = n f 1 = (n-1)oertone = n th harmonic How do musical instruments manage to produce this nice miture: f 1, 2f 1, 3f 1, etc?? If a string of length has its ends fied, as in a stringed instrument, then onl standing waes at certain resonant frequencies are possible: 4/19/2009 Uniersit of Colorado at Boulder
wae-8, 2 2 f f1 fundamental or 1st harmonic 2 nd f2 2 f1 2 harmonic 2 3, 2 3 2 rd f 3 3 f1 3 harmonic For the n th 2 harmonic (n = 1, 2, 3,...), n,, fn n n f1. 2 n 2 The frequenc of the n th harmonic is n the frequenc of the fundamental. (In all this, the wae speed is fied b the tension and mass of the string. It turns out that the speed of the wae is gien b string.) FT m /, where F T is the string tension and m/ is the mass per length of the When a guitar string is plucked, all of these resonant frequencies, or normal modes, are ecited all at once. The listener hears all frequencies f 1, f 2 = 2f 1, f 3 = 3f 1, etc It is this miture, which gies the instrument its pleasing qualit. The fundamental, or lowest frequenc, established the "note". The fundamental f 1 = /(2) can be adjusted (tuned) either b changing the tension in the string (changing ) or b holding string at a fret (changes ). 4/19/2009 Uniersit of Colorado at Boulder