WaveEquation

WaveEquation - µ F v = F tension in string N =m/L linear...

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Liu UCD Phy9B 07 15 Graphical Representation y y vs t: Motion of a single point in the medium y vs x: Snap shot of the whole wave at one instant
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Liu UCD Phy9B 07 16 y y =Acos(- ω t) = Acos ω t x=0 y =Acos( π −ω t ) x= λ /2 y =Acos( 3 π /2 −ω t ) x=3 /4 y=Acos(kx- ω t+ φ) y =Acos ( 2π− ω t ) = Acos ω t x= General expression: use specific points to determine φ Phase
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Liu UCD Phy9B 07 17 Particle Velocity and Acceleration ) cos( ) , ( t kx A t x y ω = Transverse velocity of a particle ) sin( ) , ( t kx A t t x y v y = = Generally true Acceleration of a particle ) , ( ) cos( ) , ( ) , ( 2 2 2 2 t x y t kx A t t x y t x a y = = = Recall wave velocity: v= λ f= /k Proportional to displacement Opposite direction
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Liu UCD Phy9B 07 18 Wave Equation Since ) , ( ) , ( 2 2 2 t x y t t x y ω = ) , ( ) , ( 2 2 2 t x y k x t x y = 2 2 2 2 2 2 ) , ( ) , ( t t x y k x t x y = 2 2 2 ) , ( 1 t t x y v = Valid for waves on a string that have any shape.
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Liu UCD Phy9B 07 19 Wave Snapshots v 2 4 7
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Liu UCD Phy9B 07 20 15-4. Speed of a Transverse Wave string a on wave e transvers a of speed
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Unformatted text preview: , µ F v = F : tension in string, N =m/L : linear mass density Please read text on your own. Method 1: Impulse-momentum theorem Liu UCD Phy9B 07 21 Speed of a Transverse Wave µ F v = F : tension in string, N =m/L : linear mass density Method 2: Newton’s second law ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = + = ∆ + x x x y y y x y x y F F F F 2 1 2 2 t y x ma F y y ∂ ∂ ∆ = = 2 2 2 2 lim t y F x y x x y x y x x x x ∂ ∂ = ∂ ∂ = ∆ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∆ + → ∆ 2 2 2 2 2 1 t y v x y ∂ ∂ = ∂ ∂ Wave equation:...
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This note was uploaded on 02/17/2009 for the course PHY na taught by Professor Richardscalettar during the Spring '09 term at UC Davis.

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WaveEquation - µ F v = F tension in string N =m/L linear...

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