» physicsforums
  (human.hw.help)
 » hyperphysics
 (detailed ebook)
 » Minds-On-Physics
  (practice like hw)
  press "continue",
  below red words

Plan for Quiz 6 to be Tue.Nov.05 thR.Nov.07
... plan for Exam 3 to be Tue.Nov.12 thR.Nov.14

Topic 6 (Mechanical Waves & wave phenomena)

where disturbances propagate neighbor-to-neighbor

a wave is a disturbance of material medium (from equilibrium) that propagates (moves) from neighbor to neighbor.

Consider a rope stretched out away from you. If you pull the end rightward (to +x), it will pull its neighbor piece-of-rope in the same manner, forming a "wave pulse"
. . . the neighbor piece-of-rope does not immediately follow the pull ... the pull is a Force, which causes the neighbor piece (length L) to begin to accelerate
      depending on how strong the rightward pull is (~ Tension x/L) , but reduced by (divided by) the mass of that length of rope .
      since Δx = ½ a t ² , there's a time delay ~ √(2 x /a) = √(2 x m /Tension×x/L) before it gets to the same x distance as the original was.
=> the speed of the disturbance = L/t = L √ (Tension / m L) = √(Tension/(m/L)) ... depends only on the local environment of the wave.
. . . all waves have formulas for speed that look like √(Force/inertia) . . . the details can usually be determined by making the UNITS match
      speed of sound is an exception ... v = √(P/ρ·7/5)   . . . has an extra factor 7/5 since compression raises air's Temperature, increasing its Pressure.
      . . . (it bugged Newton for 20 years that he didn't know why the ×1.4 was needed)

Transverse: pieces move crosswise ( | ) to the wave propagation direction . . . Longitudinal: pieces move a-long the wave propagation direction
. . . longitudinal waves are Pressure disturbances , with alternating regions of higher Pressure and lower Pressure ... here more compressed and there more rarified than usual.
      sound waves , and Earthquake "P" waves ... in gases and liquids and solids ... also waves in a stretched coil spring.
. . . transverse waves are shear disturbances , with alternating regions here up and there down , or here right and there left.
      whether the wave is up-and-down , or is right-and-left , is called the wave's polarization ... (longitudinal waves are polarized along the wave velocity).
      waves on a rope , and earthquake "S" waves ... in solids only (liquids and gases do not restore shear) ... also light, an Electric and Magnetic Field disturbance.
      water surface waves are transverse (vertical) and longitudinal ; the water molecules move in circles (ellipses at depth)

Very often , wave pulses repeat because the source oscillates with some frequency   f   as it gradually gives Energy to the wave
      the frequency of the wave disturbance (at the source) is the same as the frequency of the source's oscillation.
. . . how far the wave travels (from the source) during one oscillation is the wave length   abbreviated lower-case greek   λ , named "lambda".
=> λ = v · T = v / f     . . . wavelength is the effect of the wave speeding from the source during a source oscillation Time

Doppler effect . . . named for Christian, who found the effect in starlight (binary stars)
. . . if the source is moving through the medium, the wavelength is short in front of the source and long behind the source
      because the wave speed relative to the source is slow in front , but fast behind the moving source.
      that relative velocity would be the wave velocity in the medium minus the source velocity
. . . 100 cycle/sec horn with speed of sound 333 m/s would have   λ = 333 m/s / 100 wave/s = 3.33 m/wave
      but if the beeping car is moving 33 m/s rightward (+x) the sound is only 300m/s faster than it in +x direction => λ_front = 3.00 m/wave
      . . . but the leftward sound recedes from the source at   v = −333 m/s − 33 m/s = − 366 m/s => λ_back = 3.66 m/wave .

. . . if the receiver moves in the medium toward the oncoming waves, it encounters waves more often than a stationary receiver.
      again, it is the velocity of the wave relative to the receiver that goes into that   v = λ f   equation , this time to determine the wave encounter frequency .
      a car rushing 66 m/s headlong into 3.33 meter/wave sound would intercept waves with frequency   f = v / λ = 399 m/s / 3.33 m/wave = 119.8 wave/s .
      . . . but fleeing at 66m/s from a 3.33 m/wave sound (that travels 333 m/s thru the air) would only encounter f = 267 m/s / 80.2 wave/s .
      . . . note that fleeing at 33 m/s from a 3.00 m/wave sound would result in hearing a frequency   f = 300 m/s / 3.00 m/wave = 100 wave/s
           ... must be the same as the source emitted, if the receiver stays the same distance from source.

Huygens: treat each point on wavefront as source of next wave.
faster wave speed comes from longer wavelength ; any bent wave-front will cause the wave velocity to deflect (refract)
. . . shallow water has slower waves ... wavelengths are shorter near the shore, and the wave paths bend to hit the shore closer to  |  .
. . . air slows light waves ... near sunset, the shorter wavelengths on the bottom of sunlight bends the sunlight downward ; we can see the sunlight after the sun is below the horizon!
. . . light is faster in hot air than cold air ... longer wavelengths near the hot pavement can bend blue sky light's path to be rising into your eye ... a mirage .

Two waves interfere where they meet

Diffraction patterns . . . same   f and λ   , but sources are different distances from the receiver
. . . the different path lengths mean that one source is trying to have higher pressure at the receiver
      when the other source is trying to make lower pressure there ... so the pressure there is normal (equilibrium).
. . . half-cycle later, the one is trying to make a lower pressure at that receiver,
      but the other is now trying to make a higher pressure there ... again they add to normal pressure.
you need to draw the two paths then measure the length difference   δ L !
=> δ L = ½ λ   they cancel . . . but if . . . δ L = 1 λ   they reinforce

Waves reflect where there is an abrupt difference in wave speed ... inverted if from "stiffer" medium
. . . the original wave and the reflected wave cancel at the boundary to the stiffer medium => "destructive interference"
      closed end of a box or pipe has air that can't move - so it is a displacement "node" ... (contraction of "no deflection")
. . . the original wave and the reflected wave (not inverted) add up to make double-amplitude wave at boundary to free-er medium => "constructive"
      open end of a box or pipe has air that moves more easily - so it is a displacement "Anti-node" ... (reminder that Amplitude is large there)

Standing waves occur when the reflected wave re-echos from the other side of a cavity (box).
. . . if the wave's round-trip travel time is only a few oscillation times, then resonance can occur
      the wave-form must "fit" in the box correctly, with node or antinode at the appropriate ends.
      closed end & closed-end , or open-end & open-end :   box length fits   ½ λ , 1 λ , 1½ λ , 2 λ , etc.
      closed end & open-end , or open-end & closed-end :   box length fits   ¼ λ , ¾ λ , 1¼ λ , 1¾ λ , etc.