wave 1 Wave Motion A wave is a self propagating disturbance in a medium Waves carry energy momentum information but not matter Examples Sound waves pressure waves in air or in any gas or solid or liquid Waves on a stretched string Waves on the surface of water The Wave at the ballpark stadium The medium is the people Electromagnetic waves light this is the only kind of wave which does not require a medium EM waves can travel in vacuum This was quite a surprise to 19th century physicists who had trouble imagining a wave without a medium The thing that is waving in an EM wave is an electromagnetic field which generates itself as it propagates In a sense an EM wave rolls out it own carpet creating its own medium as it moves forward More on EM waves later We can use a wave to send a signal information without sending any matter Imagine a long line of people holding hands We can send a signal down the line by hand squeezing a disturbance in the people medium and yet no one moves Traveling Waves can be categorized as sinusoidal wave speed v or v impulse Traveling waves can also be categorized as transverse displacement of the medium is perpendicular transverse to the direction of the wave velocity like a wave on water or a string or drumhead or longitudinal displacement of the medium is parallel to the direction of the wave velocity like sound wave or a slinky that has been push pulled Longitudinal Wave Speed 11 10 2009 Wave displacement wave velocity v University of Colorado at Boulder wave 2 For a sinusoidal wave wave speed is distance 1 wavelength v time time for 1 to go by v f T T T period since frequency f 1 T Almost always the wave speed v is a constant independent of and T The wave speed v depends on the properties of the medium not on the properties of the wave Examples medium string properties tension mass per length medium air properties temperature mass per molecule etc em waves in vacuum wavespeed speed of light fixed v f constant increases as f decreases decreases as f increases Mathematical description of traveling sinusoidal waves Sinusoidal waves have a wavelength and a frequency f 1 T Impulse waves have neither y y y x t v x y displacement y displacement from the equilibrium no wave position Snapshot in time freeze time at say t 0 y x y x t 0 A sin 2 A x A sin k x A A is the amplitude of the wave The displacement oscillates between A and A 2 wave number k Now freeze position watch wave go by at position x 0 11 10 2009 University of Colorado at Boulder wave 3 y t y x 0 t A sin 2 T A sin t A t A 2 angular frequency T Wave traveling to the right x t y x t A sin 2 A sin k x t T Notice When position x changes by distance or time t changes by period T the sine function goes through one complete cycle When x increases by one AND t increases by one T then the sine function stays at the same phase we are then riding along with the wave y A x A The argument of the sin function kx t is called the phase A point on the traveling wave traveling along with the wave corresponds to a particular value of the phase kx t As t increases x must increase in order to keep kx t a constant value hence a point of constant phase corresponds to a point moving to the right increasing x Could also have a wave traveling to the left x t y x t A sin 2 T We could have used cosine instead of sine for the form of the wave The only difference between sin and cos here is where we put the zero of time In general you can have anything in between sine and cosine y x t A sin k x t The phase constant is set by the zero of time Wave speed is v v distance 1 wavelength time time for 1 to go by T f T k Another way to see that our formula for the wave y y x t corresponds to a wave moving right with speed v T is to rewrite the formula like so 11 10 2009 University of Colorado at Boulder wave 4 x t 2 y x t A sin 2 A sin x t T T 2 A sin x v t A point traveling with the wave is a point with x v t constant or x vt const This is the equation for a point moving right with speed v graph of x vs t has slope x t v Claim Any traveling wave y y x t traveling with a rigid shape dispersionless has the form y x t f x v t wave traveling to right with speed v or y x t f x v t wave traveling to left with speed v Proof Consider a wave initially with some shape y f x at t 0 y that is moving to the right with speed v x And consider a moving coordinate system x y that is moving along with the wave y y x0 v t Coordinate transform x x v t x x v t y y f x t 0 v f x t1 0 x y f x v x x y x vt y x x y x y x x In the moving x y coordinate system the moving wave is stationary y f x Transfoming to the xy coordinate system the wave y f x becomes y f x v t Done Interference of waves Superposition Principle If two or more waves are present in the same place at the same time the total wave is the sum of the individual waves ytot x t y1 x t y2 x t You get constructive or destructive interference depending on whether y1 and y2 add both have same sign or cancel opposite signs 11 10 2009 University of Colorado at Boulder wave 5 v time 1 v time 2 time 3 time 4 Waves carry energy For a string wave the energy is both KE string moves as wave passes and PE originally straight string must stretch a little as wave passes elastic PE similar to 1 2 kx2 At time 3 above where is the energy Answer in the KE At time 3 the string is moving while instantaneously flat Standing Waves Two sinusoidal traveling waves of the same and therefore the same f v and the same amplitude traveling in opposite directions overlapping in the same region of space make a standing wave L ends fixed nodes anti nodes If the ends of a string of length L are fixed as in a stringed musical instrument then standing waves are only possible at certain resonant frequencies given by the condition …
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