Physics 1 Topic 8: Mechanical Wave Propagation & Phenomena
. . . where disturbances propagate neighbor-to-neighbor
when a medium (material) in equilibrium is disturbed by an Energy source,
that disturbance often propagates (moves) to neighboring places.
. . . such a disturbance is called a wave.
the source does (+) Work to the medium ; the wave is detected when the wave does Work to a receiver.
1-dimensional Waves
Consider a slinky stretched forward (y-direction) from you. If you move the end coil rightward (to +x),
it will pull its neighbor coil in the same manner, forming a "wave pulse"
the (maximum) distance from equilibrium is called the displacement Amplitude, abbreviated A .
. . . You (the source) does Work while displacing the coil against the Tension Force component that would restore the coil to equilibrium:
⇒ Wave Energy = W ~ ½ k A² . . .
. . . the neighbor coil does not immediately follow your hand motion ... coil 1 applies Force (Fx), which causes coil 2 (length Ly away) to accelerate (ax) .
depending on how strong the rightward pull is (~ Tension x/L) , but mitigated by (divided by) coil 2's mass .
since Δx = ½ a Δt² , there's a time delay Δt ~ √(2 Δx /a) = √(2 x m /Tension x/L) before coil 2 gets to the same A (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.
. . . the more "tightly connected" the neighbor coil is, the less time lag before the 2nd one responds to the first one's motion
so the faster the disturbance propagates neighbor-to-neighbor [atom/sec] .
. . . the farther it is from neighbor to neighbor, the faster is the speed of the disturbance [meter/sec]
. . . the greater the mass is that has to be accelerated, the more time lag before the 2nd one matches the motion
so the slower the disturbance propagates
⇒ vwave,material = √F·l/m = √F/(m/l) ... "coupling Force per linear mass density" ... velocity relative to the material.
. . . 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 = √(7·P / 5·ρ) . . . has an extra factor 7/5 since compression raises air's Temperature, increasing its local 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 (fluids do not restore shear) ... also light, an Electric and Magnetic Field disturbance.
water surface waves are transverse (vertical) and longitudinal ; the water molecules move along circular paths (ellipses at depth)
Very often , positive-then-negative wave pulses repeat, because many sources oscillate 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, as the wave speeds from the source during a source oscillation Time
The Energy carried by the wave , like the Energy of each oscillator , is half KE and half PE
. . . the KE , averaged over one wavelength , is ½ · ½ m vmax² = ½ · ½ m (A ω)²
this v is the speed of each moving mass, not the wave propagation speed
. . . the PE , averaged over one wavelength , is ½ · ½ k xmax² = ½ · ½ k A² ... this A is the wave Amplitude .
Waves reflect where there is an abrupt difference in wave speed ... inverted if from "stiffer" medium
. . . the original wave and the reflected wave partially cancel at the boundary to the stiffer medium => "destructive interference"
air molecules can't move wall molecules very easily, so the wall is a displacement "node" ... (contraction of "NO Displacement from Equilibrium")
. . . the original wave and the reflected wave (not inverted) add up to make larger-amplitude wave at boundary to free-er medium => "constructive interference"
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.
. . . musical instruments rely on these re-sonant wavelength conditions, which determines their frequencies
wind instruments' frequencies are f = vsound / λ ...
string instruments' frequencies are f = vstring wave / λ = f1 , f2 = 2 f1 , f3 = 3 f1 , ...
Doppler effect . . . named for Christian, who suggested that light from binary stars would appear to change color
. . . if the source is moving through the medium with velocity vS , wavelengths are short in front of the source but long behind the source.
During each cycle, the source gets vS TS closer to the wave in front ... and vS /fS farther from the wave behind it.
wave velocity relative to the source is the wave velocity thru the medium minus the source velocity thru the medium.
. . . 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 => λfront = 3.00 m/wave
. . . but the leftward sound recedes from the source at v = −333 m/s − 33 m/s = − 366 m/s => λback = 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.
Waves in 2-dimensions and 3-dimensions
Waves travel perpendicular to the wave front , spreading their Energy over a larger region as they travel.
Intensity = Power per Area, so Power spread over a larger Area has less intensity ... Power/A = Energy density · velocity .
. . . water surface waves occur on a 2-dimensional surface, so they spread in rings, along the circumference of the ring;
their Intensity decreases as the distance from the source (Energy/wave is constant, but wave front is longer ~ 2 π r)
. . . sound waves and light waves occur in 3-d volume, so they spread in shells, along the Area of the shell;
their intensity decreases as the distance² from the source since the shell's Area ~ 4 π r ² (for spreading in all directions).
Huygens' Construction: each point on a wavefront is a source for next wave, propagation is ⊥ to wavefront
faster wave speed implies 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, so incoming 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 .
We usually report wave Intensity , which is Power per unit Area : [Watts/m²]
notice that Power/Area = Energy density · velocity [(Joule/m³)·m/s] = Pressure · velocity [(N/m²)·m/s]
For sound wave Intensity, we traditionally report the intensity's power-of-10, in [pico-Watts/m²] = 10−12 W/m²
. . . 100,000 [pW/m²] = 10^5 [pW/m²] => 5 [Bel] = 50 [deci-Bel] = 50 dB
. . . it is handy to notice that a factor ×3 is about ×10^½ => adds +5 [dB]
. . . it is handy to notice that a factor ×2 is about ×10^0.3 => adds +3 [dB]
. . . so that 58 dB = 50 dB (100,000 [pW/m²]) + 5 dB (×3) + 3 dB (×2) = 600,000 pW/m²
or quite loud 67 dB = 70 dB (10,000,000 pW/m²) − 3 dB (means /2) = 5,000,000 [pW/m²] = 5×10−6 W/m²
Two waves interfere where they combine; add ±
Beats . . . if they are different frequency , they sometimes add constructively , but later add destructively
. . . so together they are alternately louder than one, then quieter than one
if one is 256 cycles/sec and the other is 254 cycles/sec , they will take ¼ second (64 cycles) to change from loud to quiet
then ¼ s (64 cycles) to change from quiet to loud again. repeat time ½ second ... modulation frequency is 2 cycles/sec
=> fbeat = fhigh − flow
Diffraction patterns . . . same f and λ , but sources are different distances from the receiver
. . . the different path lengths mean that one source might be 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 will be 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 (zero ΔP)
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
