Slinky vs. guitar. W.E. Bailey, APAM/MSE EN1102
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1 Slinky vs. guitar W.E. Bailey, APAM/MSE EN1102
2 Differential spring element Figure: Differential length dx of spring under tension T with curvature is not a constant. θ = θ(x) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 2 / 15
3 Differential spring element At any point x, θ(x) is defined through tan θ(x) = y x W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 3 / 15
4 Differential spring element At any point x, θ(x) is defined through tan θ(x) = y x which for small angles θ 1 (radians), approximated tan θ(x) = sin θ cos θ θ 1 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 3 / 15
5 Differential spring element At any point x, θ(x) is defined through tan θ(x) = y x which for small angles θ 1 (radians), approximated tan θ(x) = sin θ cos θ θ 1 y(x) x θ(x) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 3 / 15
6 Differential spring element: Newton s laws in y Differential length element dx, x to x + dx W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 4 / 15
7 Differential spring element: Newton s laws in y Differential length element dx, x to x + dx If this tiny length has mass m, in ŷ, F y = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 4 / 15
8 Differential spring element: Newton s laws in y Differential length element dx, x to x + dx If this tiny length has mass m, in ŷ, F y = m 2 y t 2 Net forces in ŷ: require some curvature, otherwise: T sin θ down on left and T sin θ up on right W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 4 / 15
9 Differential spring element: Newton s laws in y Differential length element dx, x to x + dx If this tiny length has mass m, in ŷ, F y = m 2 y t 2 Net forces in ŷ: require some curvature, otherwise: T sin θ down on left and T sin θ up on right Newton s law: T (sin θ(x + dx) sin θ(x)) = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 4 / 15
10 Differential spring element: Newton s laws in y Differential length element dx, x to x + dx If this tiny length has mass m, in ŷ, F y = m 2 y t 2 Net forces in ŷ: require some curvature, otherwise: T sin θ down on left and T sin θ up on right Newton s law: T (sin θ(x + dx) sin θ(x)) = m 2 y t 2 Small θ: approximate sin θ θ T (θ(x + dx) θ(x)) = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 4 / 15
11 Differential spring element: Newton s laws in y From the definition of a derivative: ( T θ(x) + θ(x) ) dx θ(x) x = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 5 / 15
12 Differential spring element: Newton s laws in y From the definition of a derivative: ( T θ(x) + θ(x) ) dx θ(x) x T θ(x) x dx = m 2 y t 2 = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 5 / 15
13 Differential spring element: Newton s laws in y For small angles θ(x), T y x x dx = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 6 / 15
14 Differential spring element: Newton s laws in y For small angles θ(x), call dx = δl: T y x x dx = m 2 y t 2 T δl 2 y x 2 = m 2 y t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 6 / 15
15 Differential spring element: Newton s laws in y For small angles θ(x), call dx = δl: T y x x dx = m 2 y t 2 T δl 2 y x 2 = m 2 y t 2 2 ( ) y m x 2 = 1 2 y δl T t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 6 / 15
16 Differential spring element: Newton s laws in y For small angles θ(x), call dx = δl: T y x x dx = m 2 y t 2 T δl 2 y x 2 = m 2 y t 2 2 ( ) y m x 2 = 1 2 y δl T t 2 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 This is a partial differential equation! (PDE). There are partial derivatives in x, t. W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 6 / 15
17 Solving a PDE: wave velocities Wave equation: 2 y t 2 = v 2 2 y x 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 7 / 15
18 Solving a PDE: wave velocities Wave equation: Solved with: 2 y t 2 = v 2 2 y x 2 y(x, t) = y(x vt) a fixed profile of wave y travels with a speed v W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 7 / 15
19 Solving a PDE: wave velocities Wave equation: Solved with: 2 y t 2 = v 2 2 y x 2 y(x, t) = y(x vt) a fixed profile of wave y travels with a speed v To verify: y(x, t) = A cos (kx ωt) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 7 / 15
20 Solving a PDE: wave velocities Wave equation: Solved with: 2 y t 2 = v 2 2 y x 2 y(x, t) = y(x vt) a fixed profile of wave y travels with a speed v To verify: y(x, t) = A cos (kx ωt) where k is the wavenumber, in m 1, and ω is the circular frequency in s 1. These quantities correspond to wavelength, period: λ = 2π k τ = 2π ω W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 7 / 15
21 Solving a PDE: wave velocities The wave moves in the positive x direction with velocity v x = (ω/k) t for same argument: W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 8 / 15
22 Solving a PDE: wave velocities The wave moves in the positive x direction with velocity v x = (ω/k) t for same argument: v = ω k W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 8 / 15
23 Solving a PDE: wave velocities The wave moves in the positive x direction with velocity v x = (ω/k) t for same argument: Or (from wave equation) v = ω k cos (kx ωt) = v cos (kx ωt) t2 x 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 8 / 15
24 Solving a PDE: wave velocities The wave moves in the positive x direction with velocity v x = (ω/k) t for same argument: Or (from wave equation) v = ω k cos (kx ωt) = v cos (kx ωt) t2 x 2 ω 2 = v 2 k 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 8 / 15
25 Solving a PDE: wave velocities The wave moves in the positive x direction with velocity v x = (ω/k) t for same argument: Or (from wave equation) v = ω k cos (kx ωt) = v cos (kx ωt) t2 x 2 ω 2 = v 2 k 2 v is the velocity at ω v = ω k W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 8 / 15
26 How fast on a slinky? 2 y t 2 = v 2 2 y x 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 9 / 15
27 How fast on a slinky? with 2 y t 2 = v 2 2 y x 2 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 9 / 15
28 How fast on a slinky? with 2 y t 2 = v 2 2 y x 2 Thus the velocity is 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 v 2 = ( ) δl m T W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 9 / 15
29 How fast on a slinky? with 2 y t 2 = v 2 2 y x 2 Thus the velocity is 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 v 2 = ( ) δl m T Take total slinky mass M, length L (same ratio) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 9 / 15
30 How fast on a slinky? with 2 y t 2 = v 2 2 y x 2 Thus the velocity is 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 v 2 = ( ) δl m T Take total slinky mass M, length L (same ratio) (k: spring constant) T = k L W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 9 / 15
31 How fast on a slinky? v 2 = ( ) L M k L W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 10 / 15
32 How fast on a slinky? ( ) L v 2 = M k L v = k M L velocity speeds up when the spring gets stretched out. W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 10 / 15
33 How fast on a slinky? ( ) L v 2 = M k L v = k M L velocity speeds up when the spring gets stretched out. transit time t is constant: v = 2 L/ t W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 10 / 15
34 How fast on a slinky? ( ) L v 2 = M k L v = k M L velocity speeds up when the spring gets stretched out. transit time t is constant: v = 2 L/ t check this W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 10 / 15
35 Standing wave profiles Fix spring at either end (x = ±L/2) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 11 / 15
36 Standing wave profiles Fix spring at either end (x = ±L/2) y(x, t) = A cos (kx g ωt) y(x, t) = B sin (kx u ωt) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 11 / 15
37 Standing wave profiles Fix spring at either end (x = ±L/2) y(x, t) = A cos (kx g ωt) y(x, t) = B sin (kx u ωt) restrictions on k u,g : we know that (boundary) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 11 / 15
38 Standing wave profiles Fix spring at either end (x = ±L/2) y(x, t) = A cos (kx g ωt) y(x, t) = B sin (kx u ωt) restrictions on k u,g : we know that (boundary) k g L 2 = π 2 k u L 2 = π W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 11 / 15
39 Standing wave profiles Fix spring at either end (x = ±L/2) y(x, t) = A cos (kx g ωt) y(x, t) = B sin (kx u ωt) restrictions on k u,g : we know that (boundary) Looks like what? (see) k g L 2 = π 2 k u L 2 = π W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 11 / 15
40 Lowest frequency motion Modes: possible ways for string to vibrate W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 12 / 15
41 Lowest frequency motion Modes: possible ways for string to vibrate First mode: L = λ/2, or W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 12 / 15
42 Lowest frequency motion Modes: possible ways for string to vibrate First mode: L = λ/2, or k = π L W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 12 / 15
43 Lowest frequency motion Modes: possible ways for string to vibrate First mode: L = λ/2, or k = π L and for the frequency, ν = 1 2 k M W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 12 / 15
44 Lowest frequency motion Modes: possible ways for string to vibrate First mode: L = λ/2, or k = π L and for the frequency, ν = 1 2 k M very weird! You can t tune a slinky! (guitar strings are different) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 12 / 15
45 What is the wave velocity on a guitar? This is the same as the slinky: 2 y t 2 = v 2 2 y x 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 13 / 15
46 What is the wave velocity on a guitar? This is the same as the slinky: with 2 y t 2 = v 2 2 y x 2 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 13 / 15
47 What is the wave velocity on a guitar? This is the same as the slinky: with 2 y t 2 = v 2 2 y x 2 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 The difference is in the tension T or force F. Recall from the definition of the stress, F = σa W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 13 / 15
48 What is the wave velocity on a guitar? This is the same as the slinky: with 2 y t 2 = v 2 2 y x 2 2 ( ) y(x, t) m 1 2 y(x, t) x 2 = δl T t 2 The difference is in the tension T or force F. Recall from the definition of the stress, F = σa W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 13 / 15
49 What is the wave velocity on a guitar? and the definition of the Young s modulus E, σ = Eɛ, where ɛ is strain (and expressing force as tension T ) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 14 / 15
50 What is the wave velocity on a guitar? and the definition of the Young s modulus E, σ = Eɛ, where ɛ is strain (and expressing force as tension T ) T = ɛ A E W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 14 / 15
51 What is the wave velocity on a guitar? and the definition of the Young s modulus E, σ = Eɛ, where ɛ is strain (and expressing force as tension T ) T = ɛ A E v 2 = ( ) δl m ɛ A E W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 14 / 15
52 What is the wave velocity on a guitar? and the definition of the Young s modulus E, σ = Eɛ, where ɛ is strain (and expressing force as tension T ) T = ɛ A E v 2 = ( ) δl m ɛ A E Recognize the mass density ρ = m/(aδl), W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 14 / 15
53 What is the wave velocity on a guitar? and the definition of the Young s modulus E, σ = Eɛ, where ɛ is strain (and expressing force as tension T ) T = ɛ A E v 2 = ( ) δl m ɛ A E Recognize the mass density ρ = m/(aδl), ( ) Eɛ v 2 = ρ W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 14 / 15
54 What is the wave velocity on a guitar? From the wave velocity v = ω/k, and k = π/l for the lowest mode of the string, with frequency ω 0 W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 15 / 15
55 What is the wave velocity on a guitar? From the wave velocity v = ω/k, and k = π/l for the lowest mode of the string, with frequency ω 0 and from ω 0 = 2πν 0, ω 0 k = Eɛ ρ W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 15 / 15
56 What is the wave velocity on a guitar? From the wave velocity v = ω/k, and k = π/l for the lowest mode of the string, with frequency ω 0 and from ω 0 = 2πν 0, ω 0 k = Eɛ ρ ν 0 = 1 π Eɛ 2π L ρ W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 15 / 15
57 What is the wave velocity on a guitar? From the wave velocity v = ω/k, and k = π/l for the lowest mode of the string, with frequency ω 0 and from ω 0 = 2πν 0, ω 0 k = Eɛ ρ ν 0 = 1 π Eɛ 2π L ρ ν 0 = 1 Eɛ 2 L ρ yes, you can tune a guitar (through strain, ɛ = l/l 0 ) W.E. Bailey, APAM/MSE EN1102 Slinky vs. guitar 15 / 15
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