Standing Waves + Reflection

Transcription

1 Standing Waves + Reflection Announcements: Will discuss reflections of transverse waves, standing waves and speed of sound. We will be covering material in Chap. 16. Plan to review material on Wednesday and Friday. Web page:

2 Announcements Have decided to increase from 1 -> 2 crib sheets 8.5 x 11 front and back for the final. Smart Physics will be done today there was a mix-up on the fluids section which had the wrong due date and it will be dropped from counting. Last CAPA assignment is due tomorrow (Wednesday at 8am). Will put the CAPA summary on D2L by end of week. Today is the last day that Clicker response will count.

3 Energy Transport of Transverse Wave y(x,t) = y m sin(kx ωt + φ) P avg = 2 dk dt avg = 1 2 µv ω 2 y m 2 µ is the mass per unit length Depends on amplitude squared and angular frequency squared

4 Wave Speed on a Stretched String v = τ µ τ is the tension in the string µ is the mass per unit length The speed of a wave along a stretched ideal string depends only on the tension and the linear density of the string and not on the frequency.

5 Reflection at a wall A pulse reaching the end of a medium becomes inverted whenever it either reflects off a fixed end, or is moving in a less dense medium and reflects off a more dense medium.

6 Reflection at an open end A pulse reaching the end of a medium does not invert in reflection if it either reflects off an open end, or is moving in a more dense medium and reflects off a less dense medium.

7 Reflection Dense to Less Dense A pulse in a more dense medium is traveling towards the boundary with a less dense medium. 1. The reflected pulse in medium 1 (will, will not) be inverted. 2. The speed of the transmitted pulse will be (greater than, less than, the same as) the speed of the incident pulse. 3. The speed of the reflected pulse will be (greater than, less than, the same as) the speed of the incident pulse. 4. The wavelength of the transmitted pulse will be (greater than, less than, the same as) the wavelength of the incident pulse. 5. The frequency of the transmitted pulse will be (greater than, less than, the same as) the frequency of the incident pulse.

8 Reflection Less Dense to Dense A pulse in a less dense medium is traveling towards the boundary with a more dense medium. 1. The reflected pulse in medium 1 (will, will not) be inverted. 2. The speed of the transmitted pulse will be (greater than, less than, the same as) the speed of the incident pulse. 3. The speed of the reflected pulse will be (greater than, less than, the same as) the speed of the incident pulse. 4. The wavelength of the transmitted pulse will be (greater than, less than, the same as) the wavelength of the incident pulse. 5. The frequency of the transmitted pulse will be (greater than, less than, the same as) the frequency of the incident pulse.

9 Clicker question 1 Set frequency to BA A B C D E

10 Clicker question 1 Set frequency to BA A B C D E

11 Interference (Standing Waves) y(x,t) = y m sin(kx ωt) + y m sin(kx + ωt) sinα + sinβ = 2sin 1 2 (α + β)cos 1 (α β) 2 Wave moving left and right α = kx ωt β = kx + ωt α β = (2ω)t α + β = (2k)x so y(x,t) = [2y m sinkx] cosωt In a standing wave the amplitude varies with position. The place where amplitude is zero is when k = 2π λ kx = nπ for n = 0, 1, 2,... so x = nλ for n = 0, 1, 2,... (nodes) 2 Places where string doesn t move are called nodes. Places where the string moves the maximum are called antinodes. x = (n ) λ 2 for n = 0, 1, 2,... (antinodes)

12 Interference (Standing Waves) y(x,t) = [2y m sinkx] cosωt Amplitude at position x Oscillating Term In a standing wave the amplitude varies with position. The place where amplitude is zero is when k = 2π λ kx = nπ for n = 0, 1, 2,... so x = nλ for n = 0, 1, 2,... (nodes) 2 λ = 2L n for n =1, 2, 3,... f = v λ = n v 2L for n =1, 2, 3,... n is called the harmonic number

13 Clicker question 2 Set frequency to BA A string on an instrument plays an A (440 Hz) when plucked. If you put your finger down in the middle of the string, and then pluck, you are mostly likely to hear A: A an octave higher (880 Hz) B: A, an octave lower (220 Hz) C: Same tone D: Some entirely different note

14 Clicker question 2 Set frequency to BA A string on an instrument plays an A (440 Hz) when plucked. If you put your finger down in the middle of the string, and then pluck, you are mostly likely to hear A: A an octave higher (880 Hz) B: A, an octave lower (220 Hz) C: Same tone D: Some entirely different note If you put your finger in the middle, you don't allow the "fundamental" vibration mode to happen. Putting your finger there requires a NODE there, and the next higher standing wave, the one with a FULL wave length fitting into the length, is now allowed. That's half the wavelength, or twice the frequency, 880 Hz.

15 Clicker question 3 Set frequency to BA A string is clamped at both ends and then plucked so that it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points along the string is... A: zero everywhere. B: positive everywhere. C: negative everywhere. D: depends on the position. a b

16 Clicker question 3 Set frequency to BA A string is clamped at both ends and then plucked so that it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points along the string is... A: zero everywhere. B: positive everywhere. C: negative everywhere. D: depends on the position. a b Zero everywhere! Every point on the string has reached its "extreme value. When you reach the end of your motion, your speed is instantaneously zero.

17 Clicker question 4 Set frequency to BA A string is clamped at both ends and then plucked so that it vibrates in a standing mode between two extreme positions a and c. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points along the string is... A: zero everywhere. B: positive everywhere. C: negative everywhere. D: depends on position. a b c

18 Clicker question 4 Set frequency to BA A string is clamped at both ends and then plucked so that it vibrates in a standing mode between two extreme positions a and c. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points along the string is... A: zero everywhere. B: positive everywhere. C: negative everywhere. D: depends on position. a b c It depends on position. At the point above "b", the string is on its way from a to c and thus moving DOWN. But 1/2 wavelength to the right, it's on its way up! node: never moves.

19 The Speed of Sound v = τ elastic property = µ inertial property Speed of any wave depends: On the inertial property of the medium (to store kinetic energy) On the elastic property of the medium (to store potential energy) For sound it is v = B ρ = Bulk Modulus density The bulk modulus determines the extent to which an element of the medium changes in volume when the pressure on it changes. B = (Units: Pascals) Δp ΔV /V

20 Table of Speeds of Sound Molecules in a solid are much closer together than those in a liquid or gas, allowing sound waves to travel more quickly through it. Sound waves travel 17 times faster through steel than through air. The speed of sound in steel is 5,960 meters per second (13,332 mph)!

21 So why does sound travel slower in cold air than hot air? The speed of sound depends on wind conditions, temperature, and humidity, but not on frequency. All notes travel at the same speed. Water vapor slightly increases the speed. Sound travels faster through warm air than through cold air because the faster-moving molecules in warm air bump into each other more often and transmit a pulse in less time. For each degree rise in temperature above 0 degrees C, the speed of sound in air increases by 0.6 meters per second.

22 Clicker question 5 Set frequency to BA A source of frequency of 500 Hz emits waves of wavelength 0.4 m, how long does the waves take to travel 600 m? A 3 s B 6 s C 9 s D 12 s

23 Clicker question 5 Set frequency to BA A source of frequency of 500 Hz emits waves of wavelength 0.4 m, how long does the waves take to travel 600 m? A 3 s B 6 s C 9 s D 12 s v = fλ = (500Hz)(0.4m) = 200m /s t = d v = 600 m 200 m /s = 3 s

24 Clicker question 6 Set frequency to BA A. λ < λ' and f < f ' B. λ = λ' and f = f ' C. λ > λ' and f = f ' D. λ < λ' and f = f ' E. λ > λ' and f > f '

25 Clicker question 6 Set frequency to BA A. λ < λ' and f < f ' B. λ = λ' and f = f ' C. λ > λ' and f = f ' D. λ < λ' and f = f ' E. λ > λ' and f > f ' The time interval between two successive wave fronts remains the same. The distance between two successive wave fronts decreases since the waves travel slower.