Lecture 17. Mechanical waves. Transverse waves. Sound waves. Standing Waves.

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1 Lecture 17 Mechanical waves. Transverse waves. Sound waves. Standing 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. DEMO: Sound in vacuum.

Types of Waves DEMO: Rope, slinky and wave machines

the same direction the wave is moving Sound Slinky Surface:

3 Types of Waves DEMO: Rope, slinky and wave machines Transverse: The medium oscillates perpendicular to the direction the wave is moving. String, rope or wire Longitudinal: The medium oscillates in the same direction the wave is moving Sound Slinky Surface: The medium has both transverse and longitudinal motion (most general case is elliptical motion) Water

4 Forms of waves Continuous or periodic: go on forever in one direction in particular, harmonic (sin or cos) Pulses: brief disturbance in the medium v v Pulse trains, which are somewhere in between. 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. 1 f T Angular frequency: radians ω for a point on the wave in one unit of time. y Amplitude A 2 f x A

7 Connecting all these SHM Wavelength: The distance between identical points on the wave. Speed: The wave moves one wavelength in one period T, so its speed is v f T Amplitude A A Wavelength 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: Dust in front of loudspeaker Consider a small dust particle, suspended in air (due to buoyancy) speaker dust particle When you turn on the speaker, the dust particle A. oscillates back and forth horizontally, and moves slowly to the right B. steadily moves to the right C. oscillates back and forth horizontally

10 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 (3x10 8 ) 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 10 6 (b) About 10 6 (c) About 1000

11 What are these frequencies??? For sound having = 3 m : v 300 m/s f ~ 100 Hz 3 m (bass hum) For light having = 3 m : f = v λ m/s 3m = 100MHz (FM radio)

12 Math form of the harmonic wave Consider a wave that is harmonic in x and has a wavelength : A y v x If y = A at x = 0: y (x)=acos{ 2 π λ x } If this is moving to the right with speed v : y (x, t ) =A cos{ 2π λ (x vt ) }

13 Different forms of the same thing y (x, t ) =A cos{ 2π λ (x vt ) } We knew: v = λ T = λ ω 2 π Define: k = 2π λ Wave number y (x, t ) = Acos (kx ωt ) Or y (x, t ) =Acos { 2π ( x λ t T )} ω =v k 1 T = v λ

14 Wave energy Work is clearly being done: F. dr > 0 as hand moves up and down. This energy must be moving away from your hand (to the right) since the kinetic energy (motion) of the end of the string grabbed by the hand stays the same. P

15 Transfer of energy The string to the left of x does work on the string to the right of x, just as your hand did: x Energy is transferred or propagated.

16 Power Energy for a particle in SHM (attached to a spring k) E = 1 2 k A2 = 1 2 m ω2 A 2 This energy propagates at speed v. the average energy per unit time that flows in the direction of propagation should be proportional to v P v ω 2 A 2 Average power for harmonic waves: remember: ω 2 = k m Average power P = 1 2 ω2 v ρs A 2 = 1 2 ω2 v μ A 2 Sound waves in air (bulk density) Sound waves in rod or wave in rope.. etc (linear density)

17 Intensity I P area Average power (over time) in wave Area of the surface where this power is distributed Example: A siren emits a sound of power 2W at 100 m from you. How much power reaches your ear (eardrum area = 0.7 cm 2 ) Intensity at distance r from source: I R = P at source 2W = 4 πr 2 4 π(100 m) 2 = W/m 2 r Sphere of area 4 r 2 Power absorbed by eardrum: P eardrum = I R (area of eardrum ) = ( W/m 2 ) ( m 2 ) = 1.1 nw

18 Distance and amplitude At distance r from the source, the power is P r I r 1 r 2 We also know that P (Amplitude ) 2 1 Amplitude decreases as r

19 Interference, superposition Q: What happens when two waves collide? A: They ADD together! We say the waves are superposed. These points are now displaced by both waves Constructive interference

20 Interference, superposition Q: What happens when two waves collide? A: They ADD together! We say the waves are superposed.

21 Superposition of two identical harmonic waves out of phase Two identical waves out of phase: y Wave 2 is ahead 1 (x, t ) = Acos(kx ωt ) y 2 (x, t ) = Acos(kx ωt +ϕ ) or behind wave 1 constructive destructive intermediate

22 Reflected waves: fixed end DEMO: Reflection F on wall by string A pulse travels through a rope towards the end that is tied to a hook in the wall (ie, fixed end) The force by the wall always pulls in the direction opposite to the pulse. The pulse is inverted (because of Newton s 3 rd law!) F on string by wall Another way: Consider one wave going into the wall and another coming out of the wall. The superposition must give 0 at the wall. Virtual wave must be inverted: `

23 Reflected waves: fixed vs free end A pulse travels through a rope towards the end that is tied to a ring that can slide up and down without friction along a vertical pole (ie, free end) No force exerted on the free end, it just keeps going Fixed boundary condition Click Me: Super position and reflections Free boundary condition DEMO: Reflection

24 Reflection: The math Reflection at fixed end (x = 0) of a wave initially at x =-a traveling to the right: Incident wave: y 1 Reflected wave: Y 2 Mirror in x and in y Resultant wave: Y = Y 1 + Y 2 v Sum is always zero at reflection point Reflection at free end (x=0) of a wave initially at x =-a traveling to the right: v Incident wave: y 1 Reflected wave: Y 2 Mirror in y only Resultant wave: Y = Y 1 + Y 2 v v

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