Physics 2210 Fall 2015

smartPhysics 23 Waves

24 Superposition and Interference 12/09/2015

WEB

Final Exam: smartPhysics units 01-24 Day: Tue. Dec. 15, 2015 Time: 3:30-5:30 pm (both sections)

Location Section 01: WEB L104 Section 10: WEB L101 Practice problems for remaining units (21-24) not yet covered in practice problem sheets have been posted on CANVAS and on the class web page http://www.physics.utah.edu/~woolf/2210_Jui/rev5.pdf ***Review for Units 21-24 Friday Dec 11, 1pm-2pm JFB 101

At least ONE, and up to TWO of the six problems on the final exam will be on the materials from units 21-24.

Unit 24

Unit 24

Harmonic waves

If we choose 𝑓(𝑥) = 𝐴 cos𝑘𝑥 as our function to propagate in the +x direction then the result is a harmonic wave:

𝑦 𝑥, 𝑡 = 𝐴 cos𝑘 𝑥 − 𝑣𝑡 = 𝐴 cos 𝑘𝑥 − 𝜔𝑡 Where we have made the substitution 𝜔 = 𝑘𝑣 • With time fixed: the wave advances ONE cycle over length given by 𝑘Δ𝑥 = 2𝜋. But this

is the definition of wavelength, 𝜆: i.e. 𝑘𝜆 = 2𝜋. So we have

𝜆 =2𝜋𝑘 , 𝑘 =

2𝜋𝜆

𝑘 is called the “wave number” and has units m−1 or rad/m. • Pick ONE value of 𝑥 on the string: ONE cycle of the oscillation = one period: ω𝑇 = 2𝜋:

𝑇 =2𝜋𝜔 , 𝜔 =

2𝜋𝑇 , 𝑓 =

1𝑇 =

𝜔2𝜋 , 𝜔 = 2𝜋𝑓

And again we refer to 𝜔 as the “angular frequency”

𝐴cos𝒌 𝒙 − 𝒗𝒗

𝒗

Poll 12-07-02

We have shown that the functional form y(x,t) = Acos(kx-ωt) represents a wave moving in the +x direction. Which of the following represents a wave moving in the -x direction? A. Acos(ωt-kx) B. Acos(kx-ωt) C. Acos(kx+ωt)

Generic Solution to the Equation of Motion Equation of Motion (𝑣 = 𝑇 𝜇⁄ for a string under tension)

𝜕2𝑦𝜕𝑥2

=1𝑣2𝜕2𝑦𝜕𝑡2

Has generic solutions of the form 𝑦1 𝑥, 𝑡 = 𝑓 𝑥 − 𝑣𝑡 , 𝑦2 𝑥, 𝑡 = 𝑓 𝑥 + 𝑣𝑡

Where 𝑓 𝑥 is any single-variable function, for example (𝑏 = 0.3 in the plot)

𝑓 𝑥 =𝑏

𝑥2 + 𝑏2 …𝑇𝑇𝑇𝑇 𝑇𝑇 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂 𝑂𝐸𝐴𝐸𝐸𝑂𝑂

A translational transformation (or just “translation”) of 𝑓 𝑥 − 𝑐 , i.e.

𝑓 𝑥 − 𝑐 =𝑏

𝑥 − 𝑐 2 + 𝑏2 …𝑇𝑇𝑇𝑇 𝑇𝑇 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂 𝑂𝐸𝐴𝐸𝐸𝑂𝑂

Shifts the function to the right/left (+/-x dir.) by 𝑐 for 𝑐>0, and c<0, respectively

𝒇 𝒙 𝒇 𝒙 − 𝟏 𝒇 𝒙 − 𝟑 𝒇 𝒙+ 𝟐

Generic Solutions to the Equation of Motion

𝑓 𝑥 + 𝑣𝑡 , for example 𝑓 𝑥 + 𝑣𝑡 =𝑏

𝑥 + 𝑣𝑡 2 + 𝑏2

Represents a “pulse” of shape given by f(x) that is traveling to the right (+x direction) at speed 𝑣: (𝑣>0). We will now show that 𝑓 𝑥 + 𝑣𝑡 is a solution ***

𝜕𝑓(𝑥 + 𝑣𝑡) 𝜕𝑥 = 𝑓′ 𝑥 + 𝑣𝑡 ∙

𝜕 𝑥 + 𝑣𝑡𝜕𝑥 = 𝑓′ 𝑥 + 𝑣𝑡

𝜕2𝑓(𝑥 + 𝑣𝑡) 𝜕𝑥2 =

𝜕𝑓′(𝑥 + 𝑣𝑡) 𝜕𝑥 = 𝑓′′ 𝑥 + 𝑣𝑡 ∙

𝜕 𝑥 + 𝑣𝑡𝜕𝑥 = 𝑓′′ 𝑥 + 𝑣𝑡

𝜕𝑓(𝑥 + 𝑣𝑡) 𝜕𝑡 = 𝑓′ 𝑥 + 𝑣𝑡 ∙

𝜕 𝑥𝑡𝜕𝑡 = +𝑣 ∙ 𝑓′ 𝑥 + 𝑣𝑡

𝜕2𝑓(𝑥 + 𝑣𝑡) 𝜕𝑡2 = 𝑣 ∙

𝜕𝑓′ 𝑥 + 𝑣𝑡 𝜕𝑡 = 𝑣𝑓′′ 𝑥 + 𝑣𝑡 ∙

𝜕 𝑥 + 𝑣𝑡𝜕𝑡 = 𝑣𝑓′′ 𝑥 + 𝑣𝑡 ∙ 𝑣

= 𝑣2𝑓′′ 𝑥 + 𝑣𝑡

→1𝑣2𝜕2𝑓(𝑥 + 𝑣𝑡)

𝜕𝑡2 =1𝑣2 ∙ 𝑣

2𝑓′′ 𝑥 + 𝑣𝑡 = 𝑓′′ 𝑥 + 𝑣𝑡 =𝜕2𝑓 𝑥 + 𝑣𝑡

𝜕𝑥2 … 𝑞. 𝑒.𝑑.

𝒇 𝒙+ 𝒗𝒗

𝒗

Poll 12-07-03

The pulse in Case A is described by the function y(x,t) = P(x-vt).

Which of the following functions describes the pulse in Case B?

A. y(x,t) = P(x+vt) B. y(x,t) = -P(x+vt) C. y(x,t) = -P(x-vt)

Example 23.2 A wave traveling along the x axis is described mathematically by the equation 𝑦 = 0.17m ∙ cos(0.54𝜋𝑥 + 8.2𝜋𝑡), where 𝑦 is the displacement (in meters), 𝑡 is in seconds, and 𝑥 is in meters. (a) What is the speed of the wave? (b) What is the displacement of the particle at 𝑥 = 0.30 m at time 𝑡= 7.2 s?

Solution (a) (Wave speed)=(frequency)(wavelength) 𝑣 = 𝑓 λ and we know the wave as the form 𝑦 = 𝐴 sin (𝑘𝑥 + 𝜔𝑡 ) ** note the + sign here means the wave is traveling in the –x direction.

so ω = 2𝜋𝑓 = 8.2𝜋 s−1 f = 8.2 s-1 / 2 = 4.1 s-1 and k = 2𝜋/� = 0.54π m-1 λ = 2 / (0.54 m-1) = 3.7 m 𝑣 = 𝑓 = (4.1 s-1)(3.7 m) 𝑣 = 15 m/s (b)

𝑦 0.30 m, 7.2s = 0.17 m cos 0.54𝜋𝑥 + 8.2𝜋𝑡= 0.17 m cos + (0.54π m−1)(0.30 m) + (8.2π s−1)(7.2 s)= 0.17 𝑚 cos 185.9885683 rad

cos(185.9885683 rad)= −0.805 𝑦(0.30 m, 7.2s) = -0.137 m

Note: The phase angles 𝑘𝑥 − 𝜔𝑡 or 𝑘𝑥 + 𝜔𝑡 are measured in radians, not degrees. SET YOUR CALUCLATOR IN RADIAN (RAD) MODE WHEN WORKING ON THIS SORT OF PROBLEM Also: Keep as many digits as you get from the calculator for the phase angle…it’s important to NOT round off before the final result you want

When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses.

In this case, the pulses (both upward) reinforce one another and we have “constructive interference”

Linear Superposition and Interference

http://www.ablongman.com/mullin/AnimaImages/ConsInterf.gif

When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses.

In this case, the pulses (one up, one down) work against one another, and cancel each other at one instant. Here we have “destructive interference”

http://www.ablongman.com/mullin/AnimaImages/DesIntef.gif

Constructive and Destructive interference Video

http://www.youtube.com/watch?v=P_rK66GFeI4

Two waves with the same amplitude and frequency traveling in opposite directions on the SAME medium interfere in such a way as to produce a stationary oscillating pattern This is called a Standing Wave (e.g. the line segment between the two bobs in the ripple tank)

yAgain: the locations where you always have destructive interference are called “nodes”. The locations with maximum constructive interference are “anti-nodes”

[ ])/22sin()/22sin(),(),(),()/22sin(),( ),/22sin(),(

λππλππλππλππ

xftxftAtxytxytxyxftAtxyxftAtxy

++−=+=+=−=

−+

−+

The math of Standing Waves: the rightward and leftward waves are given, respectively:

We again use the identity:

[ ] )2sin()/2cos(2),()2sin()/2cos(2),(ftxAtxy

ftxAtxyπλπ

πλπ=⇒

−=

http://www.kettering.edu/physics/drussell/Demos/superposition/super3.gif

Transverse Standing Waves It is possible to produce self-sustaining standing waves on nearly ideal (very little energy loss as the wave propagates) by taking advantage of reflections There are two special cases of reflections: (1) Reflection from a fixed end (where y=0 always)

Click to start movie http://www.youtube.com/watch?v=LTWHxZ6Jvjs

http://www.kettering.edu/physics/drussell/Demos/reflect/hard.gif

Transverse Standing Waves By providing small impulses at one end, one can generate a large standing wave from the reflections (the hand in this case is very nearly a node), provided that the frequency is right , such that the boundary conditions are satisfied Boundary condition: Two fixed ends (all string musical instruments are made this way)

When you get the right frequencies where small stimuli maintains a large standing wave: we have what is called resonance . All musical instruments depends on resonance to generate sound of a definite pitch/frequency.

http://www.youtube.com/watch?v=jovIXzvFOXo

17.5 Transverse Standing Waves

,4,3,2,1 2

=

= n

LvnfnString fixed at both ends 17

f1 f2=2 f1 f3=3 f1

First Harmonic n = 1

Second Harmonic n = 2

Third Harmonic n = 3

==→=

LvvfL

21

2 11

1

λλ

==→=

LvvfL

22

22

12

2

λλ

==→=

LvvfL

23

23

13

3

λλ

L=length of string The progression of harmonics adds ONE node at a time

http://www.physicsclassroom.com/class/waves/u10l4eani1.gif

Poll 12-09-01 Suppose the strings on your guitar are 24” long as shown.

The frets are the places along the neck where you can put your finger to make the wavelength shorter, and appear as horizontal white lines on the picture.

When no frets are being pushed the frequency of the highest string is 4 times higher than the frequency of the lowest string.

Is it possible to play the lowest string with your finger on any of the frets shown and hear the same frequency as the highest string?

A. Yes B. No

People have discovered that taking videos with iphone cameras leads to interesting effects https://www.youtube.com/watch?v=TKF6nFzpHBU Some claim this is seeing the actual waves traveling down the strings Well…. Not quite These cameras use CMOS sensors and “rolling shutters”

Same type of camera alos takes other really fun videos https://www.youtube.com/watch?v=eTW0rNgMcKk This is a type of “aliasing” which is a big area of study in signal processing See http://nofilmschool.com/2014/02/how-rolling-shutter-affects-our-perceptions-of-the-andromeda-galaxy For an fun lesson on this topic