Standing Waves Physics 202 Professor Lee Carkner Lecture 7

PAL #6 Wave Energy  How do you find linear density?  v = f = (  /  ) ½ or  =  /f 2 2  Get frequency from function generator  f =  Get wavelength by measuring on string  =  Get tension from hanging weights  hanging mass is 225g so  = mg = (.225)(9.8) = N   =  Velocity = f  v = (22.54)(1.5) =

Exam #1 Friday  About 1/3 multiple choice  Study notes  Study Quizdom questions  Look at textbook “Checkpoint” questions  About 2/3 problems  Study PAL’s and SuperPALS  Study old homework  Do new practice homework questions  Try to do this with just equation sheet  Need (real) calculator and pencil

Standing Waves   The two waves will interfere, but if the input waves do not change, the resultant wave will be constant   Nodes --  Antinodes -- places where the amplitude is a maximum (only place where string has max or min displacement)  The positions of the nodes and antinodes do not change, unlike a traveling wave

Standing Wave Amplitudes

Equation of a Standing Wave  If the two waves have equations of the form:  Then the sum is:  The amplitude varies with position  e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

Nodes and Antinodes   For kx = n , sin kx = 0 and y = 0  Node: x=n ( /2)   For kx=(n+½) , sin kx = 1 and y=2y m  Antinode: x=(n+½) ( /2)  Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

Resonance Condition  Standing waves occur due to resonance   When do you get resonance?   You must have:   An integer number of “loops”  Since each “loop” is half a wavelength  = 2L/n where n = 1,2,3,4,5 …

Resonance?  Under what conditions will you have resonance?   Solve for n, must be integer   v = (  ) ½ = f   Can find new in terms of old and see if it is an integer fraction or multiple

Harmonics  f=(nv/2L)  For a string of a certain length that will have waves of a certain velocity, this is the frequency you need to use to get strong standing waves   The number n is called the harmonic number   For cases that do not correspond to the harmonics the amplitude of the resultant wave is very low (destructive interference)

Generating Harmonics  Many devices are designed to produce standing waves  e.g.,  Frequency corresponds to note  e.g.,  Can produce different f by  changing v   Changing L 

Next Time  Test #1  For Monday, December 5  Read and do webassign homework

What kind of string propagates waves the fastest? a)Heavy and tight b)Heavy and loose c)Light and loose d)Light and tight e)We can’t know wave speed without knowing the input frequency

How would you modify the wave generator to input the maximum amount of energy? a)Increase frequency, increase amplitude b)Increase frequency, decrease amplitude c)Decrease frequency, increase amplitude d)Decrease frequency, decrease amplitude e)Input energy is independent of frequency and amplitude

What kind of string transmits energy the fastest? a)Heavy and tight b)Heavy and loose c)Light and loose d)Light and tight e)All strings transmit energy at the same rate

Consider a wave traveling along a string that can be combined with three otherwise identical waves with phase shifts of 0.5 , 1.0 , and 1.9  radians. Rank the resulting wave by amplitude, largest first. a)0.5 , 1.0 , and 1.9  b)1.9 , 1.0 , 0.5  c)1.0 , 0.5 , 1.9  d)1.9 , 0.5 , 1.0  e)0.5 , 1.9 , 1.0 