Waves - UMD Physics

1 WavesOscillations and Waves1/23/132 Physics 132 What s the difference?Oscillations involve a discrete set of quantities thatvary in time (usually periodically).Examples: pendula, vibrations of individualmolecules, firefly lights, currents in between twoatoms in a molecule r(t) Waves involve continuous quantities the vary in both space and time. (variationmay be periodic or not)Examples: light Waves , sound Waves , elastic Waves , surface Waves , electro-chemical Waves on neuronstr(t)Oscillationsy(t)Wavesy(x,t)x Waves involve continuous quantities the vary in both spaceand time. (variation may be periodic or not)Mechanical WavesExamples:Transverse or Longitudinal?WaveWave Type, , , Both= on stretched Waves in Liquids/ (Seismic) Waves in Waves on Waves : Disturbance (pulse) moves to of pulse determined by mediumShape of pulse determined by source of wave. v=TensionLinearMassDensityMaterial in string moves upand downWave speed versus Material SpeedOscillationsy(t)Wavesy(x,t)xWhat is the restoring Force?

2 Md2y(t)dt2=Fy= ky(t)Newton says:Newton says: d2y(x,t)dt2=(Fy/l)=?mass per unit lengthforce per unitlengthHooke s lawxConsider three segments of a string under tension vertical displacement of the string at some instant of timeis y(x, t).Cy(x)Tension TAB1. Which segment isfeeling a verticalforce from thestring?2. Is the ? ?y(x)xConsider three segments of a string under tension vertical displacement of the string is y(x).Tension TABC3. Which bestexpresses the lawgiving the restoringforce?4. What else shouldthe restoring forcedepend on?Fy y(x)Fy dy(x)dxFy dy(x)dxFy d2y(x)dx2Fy d2y(x) ,TA & LAThe wave equation: d2y(x,t)dt2=Td2y(x,t)dx2 2y(x,t) t2=v2 2y(x,t) x2 StringSoundElastic wavesEM wavesv=T/ v= P/ v=E/ v=cTension/Linear mass densityPressure/mass densityYoung s Modulus/mass density11 How do the beads move?yxPulse moving to the right Sketch the y position of thebead indicated by the arrowas a function of timeWhiteboard,TA & LAIf this is the space-graph (photo atan instant of time) what does thetime-graph look like for the beadmarkedwith a red arrow?

3 FouryxPulse moving to the FiveChoice SixChoice SevenChoice EightChoice EightDescribing the motion of the beadsyxPulse moving to the rightvyx Sketch the velocity ofeach bead in the topfigure at the time shownA pulse is started on the string moving to the right. At a time t0 aphotograph of the string would look like figure 1 below. A point on thestring to the right of the pulse is marked by a spot of paint. (x ishorizontal and right, y is vertical and up)Which graph would look most like a graph of the y displacementof the spot as a function of time?7 None of theseA pulse is started on the string moving to the right. At a time t0 aphotograph of the string would look like figure 1 below. A point on thestring to the right of the pulse is marked by a spot of paint. (x ishorizontal and right, y is vertical and up)Which graph would look most like a graph of the y velocity ofthe spot as a function of time?7 None of theseA pulse is started on the string moving to the right.

4 At a time t0 aphotograph of the string would look like figure 1 below. A point onthe string to the right of the pulse is marked by a spot of paint. (x ishorizontal and right, y is vertical and up)Which graph would look most like a graph of the y forceon the spot as a function of time?7 None of theseA pulse is started on the string moving to the right. At atime t0 a photograph of the string would look like figure 1below. A point on the string to the right of the pulse ismarked by a spot of paint. (x is horizontal and right, y isvertical and up)Which graph would look most like a graph of the x velocity of thespot as a function of time?7 None of these4/5/1320 Physics 132 What controls the widths of theWhat controls the widths of thepulses in time and space?pulses in time and space?ytyx t_L21 Physics 132 Width of a pulseWidth of a pulse The amount of time the demonstratorThe amount of time the demonstrator sshand was displaced up and downhand was displaced up and downdetermines the time width of the t-pulse,determines the time width of the t-pulse, The speed of the signal propagation on theThe speed of the signal propagation on thestring controls the width of the x-pulse, string controls the width of the x-pulse, The leading edge takes off with some speed, The leading edge takes off with some speed, The pulse is over when the trailing edge is pulse is over when the trailing edge is done.

5 The width is determined by The width is determined by how far the leadinghow far the leadingedge got toedge got to before the displacement was over. before the displacement was = 0 What Controls the Speed of the Pulseon a Spring?To make the pulse go to the wall your hand up and down more quickly(but by the same amount). your hand up and down more slowly(but by the same amount). your hand up and down a larger distance in the your hand up and down a smaller distance in thesame a heavier string of the same length under the a string of the same density but decrease the a string of the same density but increase the more force into the wave, less force into the 132 Foothold principles:Foothold principles:Mechanical wavesMechanical Waves Key concept: We have to distinguish between themotion of the bits of matter and the motion of thepattern. Mechanism: the pulse propagates by each bit of stringpulling on the next.

6 Pattern speed: a disturbance moves into a mediumwith a speed that depends on the properties of themedium (but not on the shape of the disturbance) Matter speed: the speed of the bits of matter depend onboth the Amplitude and shape of the pulse andpattern 132 Dimensional analysisDimensional analysis Build a velocityusing mass (m),length (L), andtension (T) ofthe string: [v] = L/T [T] = ML/T2 [T/m] = L/T2 [TL/m] =L2/T2 Square bracketsSquare bracketsare used toare used toindicate aindicate aquantitiesquantitiesdimensions dimensions mass (mass (MM), length), length((LL), or time (), or time (TT)) [[mm]]= = MM [[LL]]= = LL [[tt]]= = TT [[FF]]= = ML/TML/T22 TvLmmTLv===020 using or, 4/5/1325 Physics 132 Foothold principles:Foothold principles:Mechanical wavesMechanical Waves Key concept: We have to distinguish themotionof the bits of matter and the motion of thepattern. Mechanism: the pulse propagates by eachbit of string pulling on the next.

7 Pattern speed: a disturbance moves into amediumwith a speed that depends on the propertiesof the medium (but not on the shape of thedisturbance) Matter speed: the speed of the bits of matterdepend on both the size and shape of thepulse and pattern speed. Tv=0v0 = speed of pulseT = tension of spring_ = mass density of spring(M/L)We now want to expand the picture in the following way:EM Waves propagate in 3D not just 1D as we have Diffraction - Waves coming from a finite source spread Waves propagate through material and are Dispersion - Waves are slowed down by media, differentfrequency Waves travel with different speeds- Reflection - Waves encounter boundaries between energy is Refraction - wave trajectories are bent when crossing fromone medium to Waves can take multiple paths and arrive at the same point. - Interference - contributions from different paths add 132 Speed of a beadSpeed of a bead The speed the bead movesThe speed the bead movesdepends on how fast the pulse isdepends on how fast the pulse ismoving and how far it needs tomoving and how far it needs totravel to stay on the stringtravel to stay on the (x,t)=dy(x,t)dt=dydx dxdt =dydx v0speed of beadspeed of pulseslopeof pulsedx = how far pulsemoves in time dtdy = how far beadmoves in time dtProperties of electromagnetic Waves in vacuum: Waves propagate through vacuum (no medium is required likesound Waves )All frequencies have the same propagation speed, c in and magnetic fields are oriented transverse to the directionof propagation.

8 (transverse Waves ) Waves carry both energy and emanating from apoint source4/3/1330 Physics 132 Displacements onDisplacements onan elastic string / springan elastic string / spring Each bit of the string can move up or downEach bit of the string can move up or down(perpendicular to its length) (perpendicular to its length) transverse transversewaveswaves Each bit of string can also moveEach bit of string can also movetoward/away along the string length if thetoward/away along the string length if thestring is elastic (most notable on verystring is elastic (most notable on verydeformable strings such as slinky, rubberdeformable strings such as slinky, rubberband). band). longitundinal longitundinal waveswaves