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ELASTIC WAVES
Elastic Systems● System of particles with stable equilibrium configuration
– When perturbed a small amount: particles undergo oscillations– Normal modes → all particles oscillate in phase (or 180º)– What happens if particle oscillations have phase differences?
● Example: Infinite 1-D mass-spring system
– “Flick” a particle to the right, giving it v0 at t=0– Describe the behavior of the system qualitatively– Inertia of masses → phase difference between neighboring m's– Oscillation “moves” → estimate speed (dimensional analysis)
mk mk mk ......
a0 (equilibrium separation)
State Space: 1-D Mass-Spring System● Generalized coordinates:
– Displacement Δxn of each mass from its equilibrium position
– Define:
● State Vector:
n ≡ xna0
0 ≡ km
mk mk mkΔxn Δxn+1
ω0 defines “time scale” for system
Ψn is dimensionless
| = 1
2
3
...1 /0
2 /0
3 /0
Given a state vector, can calculate:
L = T – U
Equations of motion (one for each mass)
Time Evolution of the State Vector● T and U can be expressed as functions of Ψn
● Equations of motion (L=T–U or Newton's 2nd Law):
T = ∑n
12m a0 n
2 U = ∑n
12k a0 n 1− a0 n
2
T = ∑n
12k a0
2 n
0
2
U = ∑n
12k a0
2 n 1−n2
n = 02 [ n1−n − n−n−1 ]
ddt
| = 1
2
3
...1 /0
2 / 0
3 / 0
= 0 0 0 0 ... 1 0 00 0 0 ... 0 1 00 0 0 ... 0 0 1... ... ... ... ... ... ...−2 1 0 ... 0 0 01 −2 1 ... 0 0 00 1 −2 ... 0 0 0
1
2
3
...1 /0
2 /0
3 /0
This matrix represents the time derivative operator in state space
Example: Normal mode w/ neighboring particles 180º out of phase
Expand | ψ(t) > in a Taylor Series
Elastic Materials● Real materials → “masses/springs” are incredibly tiny
– Individual masses/springs can not be easily distinguished
– But T and U can still be defined → can calculate EOM
● As masses get closer together (a0 → 0 and k → ∞):– Consider system in equilibrium: N masses (m) and springs (k)– Another system: 2N masses (m/2) and springs (2k)– Density and ktotal do not depend on k and a0 individually
– Note: If a0 → 0 then ω0 → ∞ (experiment – size of an atom?)
“Elastic Model” for Solids● Consider a metal rod (length L):
– Atoms are too small to “see” and count → N is unknown– Can elastic properties be determined without k and a0?
● Experiment: apply compression/tension force F to rod – Treating it as one big spring made up of N little springs
– Measure “response” of the rod → ΔL
– ΔL will be “shared” equally among the N springs: Ln= LN
k total=F L
k k
Bulk Modulus● Goal: quantify “elasticity” of a material
– Compression/Tension force F will be uniform in the material:
– Notice ktotal depends on L → can't be intrinsic to the material
● “Bulk modulus” ( in 1-D: ) – This quantity is intrinsic to the material (independent of L)– So k and a0 do not individually affect “bulk” material properties
– Only the product ka0 matters → need not resolve into N atoms
k total =F L
= FN Ln
= kN=
k a0
L
k k
B1D ≡ k total L = k a0
Bulk Modulus Examples● 1-D mass/spring system (total length = 15.0 cm)
– In equilibrium, compressed by force of 200 N– Compression causes length to become 14.8 cm– Calculate the 1-D bulk modulus B1D (has units of force)
● 3-D isotropic material (B is same in all directions)– Compressed by uniform pressure P → has volume V– Now increase pressure to P + ΔP– Causing volume to become V – ΔV (where ΔV << V)– Define a 3-D bulk modulus which is intrinsic to the material– i.e. B would be the same for any shape– Hint: B should end up with units of pressure (in 3-D)
The Continuous Limit● In the limit as a0 → 0:
– Mathematically, treat masses/springs as continuous fields:
– Each point (of width dx) in the material → oscillator (mass dm)
● Looking at the equation of motion:
mn
a0
x ≡ masslength
= dmdx
n t x , t ≡ displacement at x
n = 02 [ n1−n − n−n−1 ]
= 02 a0
2 [ x a0 − x a0
− x − x− a0 a0
a0
] = 0
2 a02 [ ' x − ' x− a0
a0]
∂2 ∂ t 2
= 02 a0
2 ∂2 ∂ x2
The Wave Equation● Plugging in:
– Recall:
– Both B and μ can be measured without knowing a0
● The wave equation (define )
– Work out general solutions using trial functions of the form:– 1) (normal modes or standing waves)– 2) (traveling waves)– v is the wave speed for a given medium– How are solutions of forms 1) and 2) related?
∂2
∂ t 2= 0
2 a02 ∂2
∂ x2=
k a0
ma0∂2
∂ x2
∂2 ∂ x2
− 1v2
∂2 ∂ t 2
= 0v2≡B1D
ka0= B1Dma0= ∂2
∂ t 2=
B1D
∂2
∂ x2
x , t = X x T t
x , t = f k x x± t
Transverse vs. Longitudinal Waves● In 3-D: is a displacement vector
– With components parallel and perpendicular to wave motion– Bulk modulus: handles restoring force for parallel component– Shear modulus: restoring force for perpendicular components
● Longitudinal waves: – Particle motion is parallel to wave motion– Example: What is form of 3-D solution? (with kx, ky, kz)
● Transverse waves:– Particle motion is perpendicular to wave motion– Ideal fluids have S=0 → longitudinal waves only– Wave equation is the same for both types (but with different v)
x , y , z , t
v1D= B1D
v1D= F T
v3D= B
v3D= S
∇ 2 − 1v2
∂2 ∂ t 2
= 0
Example● Earthquake P-waves and S-waves
– P = “primary” = longitudinal– S = “secondary” = transverse– These waves arrive at detectors at different times
● Outer core of Earth is molten (hot fluid)– Which type of wave can travel through the outer core?
● If earthquake has both S and P wave components:– What is the motion of a particle as the wave passes through it?– Surface waves on water also exhibit this behavior
1-D Transverse Waves on a String● Wave speed: (Example: prove this!)
● Traveling waves: – f is an unchanging function of single variable: f(u)– As wave travels, its “profile” is unchanged (in uniform medium)
● Fourier Series: Any well-behaved function f(u) – Which: 1) repeats itself (period=u0) or 2) has finite domain u0
– Can be expressed as a linear combination of sinusoidal functions:
– For any f(u), can calculate An and Bn (“Fourier coefficients”)
v1D= F T
x , t = f k x x± t
f u = A0∑n [An cos n⋅ u
u0 Bn sin n⋅ u
u0]
Fourier Analysis● Fourier coefficients – first calculate the integrals:
● Thus we can calculate An and Bn:
∫0
u0
cos n⋅ uu0 cos m⋅ u
u0 du ∫
0
u0
sin n⋅ uu0 sin m⋅ u
u0 du
An =2u0∫0
u0
f u cos n⋅ uu0 du
Bn =2u0∫0
u0
f u sin n⋅ uu0 du
Fourier Transform from f(u) to An , Bn
f u = A0∑n [An cos n⋅ u
u0 Bn sin n⋅ u
u0]
Inverse Fourier Transform from An , Bn to f(u)
Fourier Analysis: 1-D Traveling Wave
1-D traveling wave can be written as:
x , t = A0∑n [An cos k n , x x±n t Bn sin k n , x x±n t ] k n , x= n k 1, x
n= n 1
Lowest frequency (ω1) and longest wavelength (k1,x) are determined by “time scale” and “length scale” of wave
Example: Waves produced by human voice (60 – 7000 Hz) in long tube. What are λ1 , k1,x?
Any wave is a “recipe” of sinusoidal traveling waves
Wavelength: (repetition period in space)
Period: (repetition period in time)
n≡2 k n , x
x , t = A0∑n[An cos k n , x x± v t Bn sin k n , x x± v t ] v =
n
k n , x
T n≡2 n
(From wave equation)
v=nT n
“Harmonic”
Examples● Fourier series for a wave on a particular string:
– What physically determines lowest and highest frequencies?
● Wave on a string:
– Calculate knx, ωn, An and Bn in terms of n, ψ0, v and λ
– Draw a graph of An vs. ωn
● Taylor Series cut off after few terms– Accuracy of series has limited range (x-x0)– What is limited if a Fourier Series is cut off?
x , t = 0 cos3 2 x− vt
x , t = A0 ∑n[An cos k n , x x−n t Bn sin k n , x x−n t ]
for all (x,t)
Reflection and Refraction● Wave moving through non-uniform medium
– Conservation of energy: wave must reflect and refract– Reflection – some portion of wave energy reverses direction– Refraction – remainder of wave continues through medium change– More dramatic change in medium's wave speed → more reflection
● Example: wave on a non-uniform string
● In 3-D → refraction “bends” direction of wave motion– “Wavefronts” align at a different angle
If traveling wave encounters a decrease in wave speed:
Reflection will be 180º out of phase with incoming wave
Wave Interference● Wave equation – linear PDE
– When 2 waves interact, total wave is sum of individual waves– Not true for non-linear waves! (e.g. surface waves, plasmas)
● Energy in wave interference– For each wave in a linear medium:– When 2 waves interfere:– Behaves differently from the “particle” model used for matter– Constructive / Destructive “fringes” → evidence of waves
● Interference of many waves– With small phase differences → constructive interference– Many waves of random phase → destructive interference
E ~2
E total r , t ~ 12 2 ≠ E1 E 2
Standing Waves on a String● Consider a string with 2 fixed ends (violin, guitar, etc.)
– Plucking or bowing string excites traveling waves– Of many frequencies and wavelengths– Traveling waves reflect from fixed ends– Very quickly, string is filled with many interfering reflections
● Standing Waves (or “harmonics” of the string)– Reflections with appropriate wavelengths interfere constructively– After a short time, only the harmonics remain:
n=2 Ln
For the nth harmonic:
v=n
k n=n freqn
Any vibration on string can be decomposed into “frequency content”
This determines the “timbre” of the instrument
Sound Waves in Fluids
Doppler Effect
“Matter Waves” – Quantum Wavefunction
● Elementary particle– System no longer divisible into smaller pieces → cannot
identify velocity of a “piece”–
– psi_dot and KE must be handled differently
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