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Lecture 05 mechanical waves. transverse waves.

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- 1. Lecture 5 Mechanical waves. Transverse waves.
- 2. What is a wave ? A wave is a traveling disturbance that transports energy but not matter. – – – – Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Light waves (what moves??) Waves exist as excitations of a (more or less) elastic medium.
- 3. Types of Waves Transverse: The medium oscillates perpendicular to the direction the wave is moving. •String •Water Longitudinal: The medium oscillates in the same direction the wave is moving • • Sound Slinky DEMO: Rope, slinky and wave machines
- 4. Forms of waves • Continuous or periodic: go on forever in one direction v → in particular, harmonic (sin or cos) • Pulses: brief disturbance in the medium • Pulse trains, which are somewhere in between. v v
- 5. Harmonic waves Each point has SHM
- 6. A few parameters Amplitude: The maximum displacement A of a point on the wave. Period: The timeT for a point on the wave to undergo one complete oscillation. Frequency: Number of oscillations f for a point on the wave in one unit of time. Angular frequency: radians ω for a point on the wave in one unit of time. y Amplitude A A 1 f = T ω = 2πf x
- 7. Connecting all these SHM Wavelength: The distance λ between identical points on the wave. Speed: The wave moves one wavelength λ in one period T, so its speed is λ v = = λf T Wavelength λ Amplitude A A y x
- 8. Wave speed The speed of a wave is a constant that depends only on the medium: → How easy is it to displace points from equilibrium position? → How strong is the restoring force back to equilibrium? Speed does NOT depend on amplitude, wavelength, period or shape of wave.
- 9. ACT: Frequency and wavelength The speed of sound in air is a bit over 300 m/s, and the speed of light in air is about 300,000,000 (3x108) m/s. Suppose we make a sound wave and a light wave with a wavelength of 3 m each. What is the ratio of the frequency of the light wave to that of the sound wave? (a) About 106 (b) About 10−6 (c) About 1000 v f = λ vlight v sound ~ 10 6 flight fsound ~ 10 6
- 10. What are these frequencies??? For sound having λ = 3 m : For light having λ = 3 m : v 300 m/s (bass hum) f = ~ = 100 Hz λ 3m v 3 × 108 m/s (FM radio) f = ~ = 100 MHz λ 3m
- 11. Mathematical Description y Suppose we have some function y = f(x) : x y f(x − a) is just the same shape moved a distance a to the right: x a Let a = vt Then, f(x − vt) will describe the same shape moving to the right with speed v. y v vt x
- 12. Math for the harmonic wave y λ Consider a wave that is harmonic in x and has a wavelength λ: A If y = A at x = 0: If this is moving to the right with speed v : v x 2π y ( x ) = A cos x÷ λ 2π y ( x ,t ) = A cos ( x − vt ) ÷ λ
- 13. Different forms of the same thing 2π y ( x ,t ) = A cos ( x − vt ) ÷ λ λ ω We knew: v = = λ T 2π Define: k = 2π λ Wave number y ( x ,t ) = A cos ( kx − ωt ) ω v = k
- 14. ACT: Wave Motion A harmonic wave moving in the positive x direction can be described by the equation y(x,t) = A cos ( kx - ωt ) Which of the following equations describes a harmonic wave moving in the negative x direction? (a) y(x,t) = A cos (kx + ωt) (b) y(x,t) = A cos (−kx + ωt) (c) Both y ( x ,t ) = A cos ( kx − ωt ) came from y ( x ,t ) = A cos ( kx + ωt ) cos ( −kx + ωt ) = cos ( kx − ωt ) ↔ ↔ 2π y ( x ,t ) = A cos ( x − vt ) ÷ λ − x direction + x direction It’s the relative sign that matters.
- 15. The wave equation General wave: y = f ( x −vt ) ∂y ∂f =v ∂t ∂u 2 ∂2y 2 ∂ f =v 2 ∂t ∂u 2 Let u = x −vt ∂y ∂f = ∂x ∂u ∂2y ∂2f = ∂x 2 ∂u 2 2 ∂2y 2 ∂ y =v 2 ∂t ∂x 2 ∂2y 1 ∂2y − 2 = 0 Wave equation 2 2 ∂x v ∂t
- 16. ACT: Waves on a string A single pulse is sent along a stretched rope. What can the person do to make the start of the pulse arrive at the wall in a shorter time? A. Flick hand faster B. Flick hand further up and down C. Pull on rope before flicking hand Pulling on rope increases tension, and propagation speed depends only on medium, not on how you start the wave.
- 17. Faster flick up/down → narrow pulses Slower flick up/down → wider pulses Large flick up/down → higher pulses Once pulse leaves your hand, you cannot influence it. Propagation speed down string is ~ same for all these pulses
- 18. What determines the wave speed? Back to 221… Problem: A pulse travels in the +x direction in a string with mass per unit length of the string is µ (kg/m) subject to a uniform tension F . What is the speed of the pulse?
- 19. Consider the segment of length ∆x when the string is relaxed: ∆m = µ∆x y F2 F2y F F1x ∆m F2x F µ F1 x x + ∆x F1y |F1x | = |F2x| because ax = 0 (transversal wave, no displacement in the x direction) Fx must also be equal to the tension in the string when there is no wave, ie, |F1x | = |F2x| = F
- 20. y F2 F2y F ∆m = µ∆x F µ F1 At x : At x + ∆x : F1y F F2y F x x + ∆x F1y ∂y = − ÷ ∂x x Fy net = F2 y − F1y ∂y = ÷ ∂x x +∆x ∂2 y ∂y ∂y F − ÷ ÷ = µ∆x ∂t 2 ∂x x +∆x ∂x x ∂y ∂y = F − ÷ ÷ ∂x x +∆x ∂x x = ∆m ay ∂2 y = µ∆x ∂t 2
- 21. ∂2 y ∂y ∂y F − ÷ ÷ = µ∆x ∂t 2 ∂x x +∆x ∂x x ∂y ∂y − ÷ ÷ µ ∂2 y ∂x x +∆x ∂x x = ∆x F ∂t 2 ∂y ∂y − ÷ ÷ ∂2 y ∂x x +∆x ∂x x lim = ∆x →0 ∆x ∂x 2 Wave equation! ∂2 y µ ∂2 y = ∂x 2 F ∂t 2 1 v2 Wave speed in a string F v = µ
- 22. In-class example: Wave speed A string under a certain tension has transverse waves with a wave speed v. A second string made of the same material but half the cross-sectional area is under twice the tension of the first string. What is the speed of transverse waves in the second string? A. v/4 B. v/2 C. v F ' = 2F µ µ' = (half the mass in the same length) 2 D. 2v E. 4v v'= F' 2F F = =2 = 2v µ' µ µ 2

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