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