Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7)
Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven t, it s time to do it! Everbod has to have met with me before Spring Break
What We Did Last Time! Discussed dispersive waves! Dispersion relation = dependence between k and ω! Determines how the waves are transmitted! Normal modes propagate with different velocities! Waveforms are not conserved dω cg =! Velocit of wave packets dk! Defined group velocit! Represents how fast information can travel in space! Never faster than light! Connection with Quantum Mechanics and Relativit
Goals For Toda! Waves in 2- and 3-dimensions! Etend 1-D wave equation into 2- and 3-D! Normal mode solutions come easil Plane waves! Boundar conditions! Rectangular drum! Chladni plates! Sound in a rectangular room
1-Dimensional Waves! We ve studied 1-D waves etensivel! Non-dispersive wave equation is! Normal modes are! Dispersion relation is! Tr to etend naturall to 2-D or 3-D! It s eas ξ ( t, ) = Ae ω = ck w i( k± ωt) ξ( t, ) ξ( t, ) t 2 2 2 = c 2 w 2 True even for dispersive waves Non-dispersive
1-D to 2-D! Let s etend from 1-D to 2-D! Simplest wa is to ignore! We just declare that ξ(,, t) is constant in! Equation and solution remains the same as 1-D ξ( t,, ) ξ( t,, ) t 2 2 2 = c 2 w 2! e.g. ξ( t,, ) = ξ e i( k ωt)! This isn t the whole stor! We should be able to send waves in -direction too 0 ξ ξ ( t, ) ξ (,, t) Move this wa
Isotrop! We ma define - coordinates as we find convenient! Phsics should not depend on particular direction! That is, unless the medium has a preferred direction! Non-directional media are called isotropic! Etend the 1-D wave equation to make it isotropic 2 2 2 ξ( t,, ) 2 ξ( t,, ) ξ ( t,, ) c 2 w + 2 2 t! Etend the solution as well! Throw it into the equation = ω = c ( k + k ) 2 2 2 2 w 0 i( k + k ωt) ξ( t,, ) = ξ e Dispersion relation Minimal etension
Wavenumber Vector! We can consider (k, k ) as a 2-D vector! Dispersion relation determines the length of k 2 2 2 2 2 2 2 ω = c ( k + k ) = c ( k k) = c k w w w k = ( k, k )! Solution can be written as! Depends on the dot-product! Points on a line perpendicular to k have the same value for ξ! As t increases, this line moves! k points the direction of wave propagation i( k + k ωt) i( k ωt) 0 0 ξ(, t) = ξ e = ξ e k k Move
Rotating the Aes! Rotate the aes to - so that is parallel to k = cosθ sinθ = sinθ + cosθ! The dot-product k becomes k = k+ k = k! Waves look, in the - coordinates i( k ωt) ξ(,, t) = ξ e 0 = ξ e 0 i( k ωt) k k = kcosθ = ksinθ k Move 1-D waves in Ignore it θ
Normal Modes! We now have plane-wave solutions traveling in all i( k ωt) directions ξ (, t) = e ω = ω k = c k ( ) k w! The direction is given b the wavenumber vector k! Dispersion relation determines the length of k! For each ω, there are infinite number of normal modes! It s convenient to use k to specif a normal mode! There is one normal mode for ever 2-D vector k! An arbitrar waves can be epressed b a linear combination of these normal modes! To show this, we need to epand Fourier Transformation
Fourier Transformation! An function f(, ) can be epressed as + + i( k + k ) f (, ) F( k, k ) e dk dk =! Fourier integral F(k, k ) is given b 1 i( k k) Fk (, k) = + + + f( e, ) dd 2 (2 π )! Suppose at t = 0, the wave had a form f(, )! Fourier integral breaks it into i( k+ k ωt)! The travel as e! So we know the complete solution e i( k + k ) + + i( k + k ωt) ξ ( t,, ) Fk (, k) e dkdk = Proof net page NB: ω depends on k
Fourier Transformation + + i( k + k ) f (, ) Fk (, k) e dkdk = 1 i( k k) F(, ) = + + + f(, ) e dd 2 (2 π )! Just throw F into the integral on the left 1 + + + + i( k k + ) i( k+ k) f (, ) e e dd dk 2 dk (2 π ) = 1 1 f (, ) e dk e dk d d 2 π 2 π + + + + ik ( ) ik ( ) + + = f(, ) δ( ) δ( ) dd = f(, ) Isn t δ-function great?
Rectangular Membrane!Imagine a rectangular drum! Elastic film is stretched over a rigid frame!consider a small piece! Mass is m= ρ! It s pulled from 4 edges b tension L T Proportional to the length of the edge L T T " Forces are balanced in the - plane T " Let s make the film vibrate in z
Rectangular Membrane! Vibration makes z = z (,, t)! Viewed from the bottom edge, there is force in the z direction z(, t,) z ( + t,,) Fz = T + T 2 zt (,, ) T 2! Same with the other edges! Total force on this little piece is 2 2 zt (,,) zt (,,) Fz = T + T 2 2 T + Read to write the equation of motion
Wave Equation! The equation of motion is 2 2 2 zt (,,) zt (,,) zt (,,) ρ = T + T 2 2 2 t 2 2 2 z(, t,) T zt (,,) zt (,,) = 2 + 2 2 t ρ! We alread know the normal modes zt (,, ) = e i( k ωt) ω = ck=! Remaining problem: what happens at the edges?! The film can t move there # z = 0 w T k ρ 2-D wave equation!
Standing Waves! Edge of the film is fied z(0,, t) = z( L,, t) = z(,0, t) = z(, L, t) = 0! Guess: 2-D standing waves! Similar to a string with fied ends ξ (0, t) = ξ ( L, t) = 0! Solution: standing waves nπ nπ cw ξ ( t, ) = sin sin t L L i t z( t,, ) = sin( k)sin( ke ) ω! Let s see where this brings us L L
Standing Waves i t z( t,, ) = sin( k)sin( ke ) ω! To satisf the boundar conditions sin kl = 0 sin kl = 0 kl! To satisf the dispersion relation ω = ck w = nπ kl 2 2 n m ω = cwπ + cn w π 2 2 1-D string had ω = L L L = mπ! Frequencies don t come in nice integer ratios! Drums don t have clear pitch! Bottom line: nπ mπ z( t,, ) = sin sin e L L iωt
Standing Waves nπ mπ z( t,, ) = sin sin e L L! Node lines split the film into n m rectangles ( nm, ) = (3,4) iωt ( nm, ) = (1,1) ( nm, ) = (1,2) ( nm, ) = (2,2) z = 0 on these lines
Standing Waves vs. Normal Modes! Standing waves and normal modes don t look related i t z( t,, ) = sin( k)sin( ke ) ω! The are in fact. Just not eas to see! For 1-D waves, 2 normal modes i ( k ±ωt) e have same ω! The move in opposite directions! Adding them gives standing waves! For 2-D waves, infinite normal modes have same ω! The move in all directions in the - plane! Tr adding 4 of them with mied signs zt (,, ) = e i( k ωt) i( k+ k ω t) i( k+ k ω t) i( k k ω t) i( k k ω t) i t + = 4sin( )sin( ) ω e e e e k k e
Chladni Plate Ernst Chladni (1756-1827)! Square plate is held at the center! Vibration of the plate = 2-D waves! Don t worr about the wave equation! Normal mode solution is as usual! Boundar conditions are! No force at the edge! 1-D analog is a pipe with both ends open i t! Guess the solution z(,, t) = cos( k )cos( k ) e ω nπ mπ ka = nπ ka = mπ z(,, t) = cos cos e a a! That sounds eas enough a iωt a
Chladni Plate nπ mπ z(,, t) = cos cos e a a iωt! Etra condition: fied in the middle! It can t move, and can t have a slope nπ mπ i t z cos cos e ω 0 center = = 2 2 z nπ nπ mπ i t sin cos e ω = = 0 a 2 2 center z mπ nπ mπ i t cos sin e ω = = 0 a 2 2 center! Obvious solution is! n and m are odd nπ mπ cos = cos = 0 2 2 a Still quite simple a
Chladni Plate nπ mπ z(,, t) = cos cos e a a iωt c w π 2 2 ω = n + m! Same for n m a! If n m we have two standing-wave solutions with the same frequenc # Linear combination is also standing waves! Think about angular frequenc ω! Suppose n and m are both even! Center is not a node! If we subtract (m, n) from (n, m), we find nπ mπ mπ nπ cos cos cos cos e a a a a! This satisfies all the conditions! Eample: (2, 4) iωt
Chladni Patterns Both n and m are even nπ mπ mπ nπ i t z = cos cos cos cos e ω 0 center = 2 2 2 2 z nπ nπ mπ mπ mπ nπ iωt = sin cos + sin cos e = 0 a 2 2 a 2 2 center z mπ nπ mπ nπ mπ nπ iωt = cos sin + cos sin e = 0 a 2 2 a 2 2 center (2, 4) (4, 2)
Chladni Patterns! Make linear combinations from (odd, odd) solutions! Both sum and difference satisf the constraints (of course) (1, 3) (3, 1) sum! 2 solutions for each (odd, odd) pair! 1 solution for each (even, even) pair! All of them have 4-fold smmetr (= 90 rotation) diff.
3-D Plane Waves! Eas to etend wave equation to 3-D ξ ξ ξ ξ t z 2 2 2 2 2 2 2 = c 2 w + + = c 2 2 2 w! Normal mode is ξ! Dispersion relation ξ i( k + k + k z ωt) i( k ωt) z = e = ω = c w k =,, z e Are we done alread?
Isotrop and Relativit! Wave equation and the normal mode solutions contain 2 onl dot-products of vectors ξ 2 2 = c 2 w ξ! If ou rotate the coordinates, t vectors change, but dot-products don t! Laws of phsics never depend on the coordinate aes! The form k ωt has even deeper meaning! It is called a Lorentz scalar! Does not change with Lorentz transformation! Satisfies Special Relativit ξ = i( k ωt)! Form of equations are constrained b the smmetr principles of nature e
Standing Wave in a Bo! Imagine a rectangular room with rigid walls! Sound in this room makes standing waves i t ξ ( zt,,, ) = sin( k)sin( k)sin( kze ) ω z! Boundar conditions kl nπ kl = nπ kl = z z = z! Dispersion relation n n ω = cwπ + + L L L 2 2 2 nz 2 2 2 z nπ L z L These frequencies resonate in the room L! Can we epress standing waves using normal modes? Tr it!
Concert Hall Acoustics! Standing waves in a room is a Bad Thing! Onl particular frequencies eist! For each frequenc, there are node planes! You can t here the frequenc if ou sit on a node plane! Real walls are not completel rigid! Stone walls (e.g. church) come close! Sound absorbers (soft stuff) ma be attached to the walls! Think of them as termination resistors for sound! Complete absorption makes the room sound dead! You want some echo for musical enjoment! Concert-hall acoustics is a combination of art and science
Roal Festival Hall, London
1/20 th Acoustic Model http://www.cot.freeserve.co.uk/
Summar! Discussed waves in 2- and 3-dimensions! Wave equation and normal modes easil etended ξ ξ ξ ξ t z 2 2 2 2 2 2 2 = c 2 w + + = c 2 2 2 w! Their forms satisf isotrop and relativit! Studied boundar conditions in 2-D and 3-D! Rectangular drum, Chladni plate, sound in a room! Natural etension of the 1-D problems such as a string! Net: spherical waves, shock waves ξ ξ i( k + k + k z ωt) i( k ωt) z = e = e Plane waves