Standing waves. The interference of two sinusoidal waves of the same frequency and amplitude, travel in opposite direction, produce a standing wave.

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1 Standing waves The interference of two sinusoidal waves of the same frequency and amplitude, travel in opposite direction, produce a standing wave. y 1 (x, t) = y m sin(kx ωt), y 2 (x, t) = y m sin(kx + ωt) resultant wave: y'(x, t) = [2y m sin kx] cos(ωt)

2 y'(x, t) = [2y m sin kx] cos(ωt) If kx = nπ (n = 0, 1, ), we have y' = 0; these positions are called nodes. x = nπ/k = nπ/(2π/λ) = n(λ/2) y 0 π/2 π 3π/2 2π kx

3 y'(x, t) = [2y m sin kx] cos(ωt) If kx = (n + ½)π (n = 0, 1, ), y' m = 2y m (maximum); these positions are called antinodes, x = (n + ½ )(λ/2) y kx 0 π/2 π 3π/2 2π

4 Standing waves and resonance For a string clamped at both ends, at certain frequencies, the interference between the forward wave and the reflected wave produces a standing wave pattern. The string is said to resonate at these certain frequencies, called resonance frequencies. L = λ/2, f = v/λ = v/2l 1 st harmonic or fundamental mode L = 2(λ/2 ), f = 2(v/2L) 2 nd harmonic f = n(v/2l), n = 1, 2, 3 nth harmonic

5 Reflection at a Boundary a b Case a) String tied to wall. The reflected and incident pulses must have opposite signs. A node is generated at the end of the string. Case b) String end can move. The reflected and incident pulses must have same sign. An antinode is generated at the end of the string.

6 Reflection at a Boundary a b Case a) String tied to wall. The reflected and incident pulses must have opposite signs. A node is generated at the end of the string. Case b) String end can move. The reflected and incident pulses must have same sign. An antinode is generated at the end of the string.

7 Reflection at a Boundary a b Case a) String tied to wall. The reflected and incident pulses must have opposite signs. A node is generated at the end of the string. Case b) String end can move. The reflected and incident pulses must have same sign. An antinode is generated at the end of the string.

8 Checkpoint A pulse is traveling down a wire and encounters a heavier wire. What happens to the reflected pulse? 1. The reflected wave changes phase 2. The reflected wave doesn t change phase 3. There is no reflected wave 4. none of the above

9 Checkpoint A pulse is traveling down a wire and encounters a heavier wire. What happens to the reflected pulse? 1. The reflected wave changes phase 2. The reflected wave doesn t change phase 3. There is no reflected wave 4. none of the above

10 Checkpoint A pulse is traveling down a wire and encounters a heavier wire. What happens to the reflected pulse? 1. The reflected wave changes phase 2. The reflected wave doesn t change phase 3. There is no reflected wave 4. none of the above

11 The speed of a traveling wave y(x, t) = y m sin(kx ωt + φ) Wave speed: v = ω/k since ω = 2π/T, k = 2π/λ => v = λ/t = λ f y(x, t) Particle speed: u(x, t) = = ωym cos(kx ωt + φ) t

12 Pipes Open on both ends Antinodes both ends. { { Open on one end, closed on the other end node on closed end, antinode on open end.

13 Chapter 17 Waves (Part II) Sound wave is a longitudinal wave.

14 The speed of sound Speed of a transverse wave on a stretched string τ v = = µ elastic property inertia property

15 The speed of sound Speed of a sound wave (longitudinal): τ v = = µ v = elastic property inertia property Speed of sound in gases & liquids is described by the bulk modulus B ( = p/( V/V)) and density ρ: Speed of sound: B ρ air(0 o C): 331m/s, air(20 o C): 343m/s, water(20 o C): 1482m/s

16 The speed of sound Speed of a sound wave (longitudinal): τ v = = µ v = elastic property inertia property Speed of sound in a solid is described by Young s modulus E and density ρ: Speed of sound in aluminum: E = 7x10 10 Pa, ρ = 2.7x10 3 kg/m 3 => v = 5100 m/s, E ρ In general, v solid > v liquid > v gas

17 Traveling Sound Wave To describe the sound wave, we use the displacement of an element at position x and time t: s(x, t) = s m cos( kx ωt ) s m : displacement amplitude k = 2π/λ ω = 2π/T = 2πf As the wave moves, the air pressure at each point changes: p(x, t) = p m sin( kx ωt ) Pressure amplitude: p m = (vρω)s m

18 Interference Two sound waves traveling in the same direction, how they would interfere at a point P depends on the phase difference between them. θ L = L 1 L 2 If the two waves were in phase when they were emitted, then the phase difference depends on path length difference: L = L 1 L 2 : φ/2π = L/λ

19 Interference For distant points, L = d sinθ d θ D y φ/2π = L/λ L L = d sinθ = mλ y/d = tanθ If φ = m (2π) or L/λ = m ( m = 0, 1, 2 ) We have fully constructive interference L = d sinθ = mλ If φ = (2m +1)π or L/λ = m + ½ ( m = 0, 1, 2 ) We have fully destructive interference d sinθ = (m + ½) λ

21 Checkpoint Today, a lightning bolt is seen in the distance. You start counting when you saw the flash. Twelve seconds elapsed until you heard the thunder. How far away was the lightning? 1) 4.1km 2) 1200m 3) 343m 4) Yikes, head for cover 5) none of the above

22 Checkpoint Today, a lightning bolt is seen in the distance. You start counting when you saw the flash. Twelve seconds elapsed until you heard the thunder. How far away was the lightning? km m m 4. Yikes, head for cover 5. none of the above

23 Checkpoint Today, a lightning bolt is seen in the distance. You start counting when you saw the flash. Twelve seconds elapsed until you heard the thunder. How far away was the lightning? d = v sound t = (343m/s)(12s) = 4.1km = 2.5miles 1) 4.1km 2) 1200m 3) 343m 4) Yikes, head for cover 5) none of the above

24 Intensity and Sound Level Intensity of a sound wave at a surface is defined as the average power per unit area I = P/A For a wave s(x, t) = s m cos( kx ωt ) I = ½ ρvω 2 s 2 m v = ω/k Variation of intensity with distance. A point source with power P s I = P s /4πr 2 Spherical surface area: 4πr 2

25 The figure indicates three small patches 1, 2, and 3 that lie on the surfaces of two imaginary spheres; the spheres are centered on an isotropic point source S of sound. A) Rank the patches according to the intensity of the sound on them, greatest first. I = P s /4πr 2 r 1 = r 2 => I 1 = I 2 r 3 > r 1 => I 3 < I 1 B) If the rates at which energy is transmitted through the three patches by the sound waves are equal, rank the patches according to their area, greatest first. r 1 = r 2 => A 1 = A 2 I = P s /A r 3 > r 1 => A 3 > A 1

26 The decibel Scale The deci(bel) scale is named after Alexander Graham Bell Sound level β is defined as β = (10 db)log (I/I 0 ) Unit for β: db(decibel) I 0 is a standard reference intensity, I 0 = W/m 2 Threshold of hearing: W/m 2 Threshold of pain (depends on the type of music): 1.0W/m 2 An increase of 10 db => sound intensity multiplied by 10.

27 The decibel scale Sound intensity range for human ear: W/m 2 More convenient to use logarithm for such a big range. Sound level β is defined as β = (10 db)log 10 (I/I 0 ) Unit for β: db(decibel) I 0 is a standard reference intensity, I 0 = W/m 2 Examples: at hearing threshold: I = I 0, β = 10 log 10 1 = 0, normal conversation: I ~ 10 6 I 0, β = 10 log = 60 db Rock concert: Ι = 0.1 W/m 2, β = 10 log 10 (0.1/10-12 ) = 110dB

28 Beats Two sound waves with frequencies f 1 and f 2 ( f 1 ~ f 2 ) reach a detector, the intensity of the combined sound wave vary at beat frequency f b f b = f 1 f 2

29 The Doppler effect When the source or the detector is moving relative to the media, a frequency which is different from the emitted frequency is detected. This is called Doppler effect. Case 1: source stationary, the detector is moving towards the source: The detected frequency is: v + vd f ' = f v Note: f ' is greater than f.

30 Case 2: source stationary, detector moving away from the source: v vd f ' = f v f ' is smaller than f. Cases 3 & 4: source moving, detector stationary f ' = f v v ± v S Source is moving toward detector, sign, f ' > f Source is moving away from detector, + sign, f ' < f

31 Rules: General formula for Doppler effect f ' = f v v ± ± v v D S v: speed of sound v D : detector speed v S : source speed Moving towards each other: frequency increases Moving away from each other: frequency decreases For numerator, if detector is moving towards the source, + if detector is moving away from the source, For denominator, if source is moving away from the detector, + if source is moving towards the detector,

Chapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves

Chapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves Chapter 16 Waves Types of waves Mechanical waves exist only within a material medium. e.g. water waves, sound waves, etc. Electromagnetic waves require no material medium to exist. e.g. light, radio, microwaves,