fzx physics lecture notes reserved fzx: personal lecture...

FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved

FZX: Personal Lecture Notes from Daniel W. Koon

St. Lawrence University Physics Department

CHAPTER 11-12

Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.

11/12. Wave Motion and Sound Wave Collisions Standing waves in strings Standing waves in pipes Beats Doppler effect Digging Deeper: Harmony

page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


FZX, Chapter 11/12: WAVE MOTION and SOUND We’ve finally escaped mechanics, more or less. We will still be studying motion, although it will be motion of things called waves, which behave much differently than matter. In mechanics we have methodically built up a substantial, many-layered structure of knowledge, consisting of many quantities and equations -- velocity, acceleration, force, work, energy, power, torque, etc. The good news is that we will not continue to build up this structure, and we will not need to carry the baggage of sustaining such a large structure. In other words, we can leave behind much of that material, and the math will be much easier. The bad news is that, as we study new phenomena, it will seem like there is not much connecting what we learn: physics might seem like a novel without a plot. Let’s consider waves. We can define waves as periodic (repeating) disturbances in some material. The material that is being disturbed, and that thus supports the wave, is called the ‘medium’. It is always useful to consider what is being ‘waved’ when a wave moves by. Consider an ocean wave. The height of the water at any point is the thing that’s varying. That’s what’s ‘waving’. Consider a group of sports fans doing ‘the wave’ in a stadium. The height of their hands is what is waving. There are three ways (at least) of looking at a wave. Each has its own charm or usefulness. Consider ocean waves coming toward the beach. Let’s consider three ways of looking at the waves. FIRST, consider the waves as viewed by someone at the dock. This person sees the height of the water at her feet going up and down, over and over. The water behaves a lot like a mass on a spring, executing harmonic motion. This person could measure the frequency, ‘ ’, the number of times the wave repeats its motion (up and down) per unit time, or equivalently, measure its period, .

fT f/1=

NEXT, consider the waves as viewed by someone sitting on a boat with a camera. The camera captures the wave at one instant in time. The photographer may note the distance between the crests of the wave. This distance, the distance during which the wave repeats itself, is called the wavelength, and is designated by a lower-case Greek lambda, λ . LAST of all, consider a surfer riding one of the crests. Surfer Bob’s interest is in how quickly the crests move, because that tells him how fast he will move when he’s riding the wave. The speed with which this crest moves is the speed, ‘ ’ of the wave.

v

If we compare the three descriptions of the wave given by our three observers, we find that the three quantities they measure are related. If the wavelength, for example, is kept the same but the speed of the wave is slowed down, then the up and down motion observed at the dock will slow down too. Considering all of the possible permutations, we arrive at λfv = [ Speed, frequency, wavelength of wave ] Waves consist of disturbances that travel in some direction. Our convention will be to use the x-axis for the direction of travel, and we will measure our disturbance in terms of the displacement, y, of whatever is waving from its equilibrium point. This does not mean that the x- and y-axes are necessarily perpendicular to each other. If they are, we call the wave a ‘transverse wave’. If the displacement is in the direction of travel, we have a ‘longitudinal wave’. Consider the sports fans doing the wave. Their wave is transverse because their hands go up, but the wave moves horizontally. I can think of two waves to invent a new, ‘longitudinal wave’ for sporting events. In one version, the fans stretch their arms out in the direction of their neighbors to the left or right. In the other version, best suited for the ‘seventh-inning stretch’ at a baseball game, the fans shuffle one step toward their neighbors, who then shuffle toward their neighbors, and so on. Try these in class, or at the next football game, or wherever. Just remember where you heard this idea first. This last longitudinal wave is not unlike what happens every time you talk, grunt, or make any other kind of noise. Sound is a longitudinal wave, usually propagated in air, in which the position of the individual molecules in the medium is what’s moving. A typical human ear may be able to hear sound waves between f=20Hz and about 16kHz (textbooks usually just say 20Hz-20kHz). Light, on the other hand, is a transverse wave of sinusoidally-varying electric and magnetic fields. Since we haven’t looked at electricity or magnetism, we will save the details for later. Your eye is sensitive to light between about 400nm and 700nm, which corresponds to just under 1015Hz, a much higher frequency than audible sound.



WAVE COLLISIONS: While two objects cannot occupy the same space at the same time, two waves can. We also know that it is possible to talk at someone who is speaking at us, and that our voices cross each other, and don’t seem to suffer for having occupied the same space as the other voice. What happens when two waves occupy the same space? If we describe a wave in terms of displacement, y, we can say that, when two waves are in the same place, the net displacement is just the sum of the displacements due to the individual waves. This may sound innocuous enough, but consider what it means when two waves look identical, but one is upside down. If the two waves occupy the same space, the sum of the two waves is zero displacement: the two waves have disappeared. Have we lost the energy that was there in these two waves? Are they gone forever? No. We can understand how two waves can cancel each other out without destroying each other by considering the energy in the two waves. When a wave pulse travels in a string, for example, the string has potential energy wherever the string has been displaced from equilibrium. It has kinetic energy wherever the string is moving. If two waves cancel each other out as they pass, there is no potential energy because the entire string may be at its equilibrium position. However, since the string had to move to get to that equilibrium, it still has kinetic energy, and that means that the two waves have not



disappeared forever. A flat string may appear to have no energy, just as a snapshot of a mass on a spring at its equilibrium point will appear to have no energy, whether the mass is moving or not. STANDING WAVES IN STRINGS If we start with a wave pulse in a very long string, it travels along the string until it comes to the end. If the end is rigidly held, the wave reflects off the end and returns, upside down. We can think of the fact that the fixed end of the string does not move up or down as being a consequence of these two waves, one upside-down relative to the other, cancelling out at the end of the string by superposition. If we have a wave pulse that is as long as the string, the superposition of the original and reflected waves will not appear to be a travelling wave at all, but a sine wave shape that simply gets larger, goes to zero, becomes negative, goes to zero again, and so on. Such a pattern is called a ‘standing wave’ pattern, and occurs in many places in physics. Any place where the string (or whatever the medium is) is always stationary is called a ‘node’, and the places in between nodes, where the medium ‘waves’ most intensely are called ‘antinodes’. The lowest frequency oscillation of the medium is called the ‘fundamental’ or ‘first harmonic’. In it, there are two nodes at the end of the medium, with a single antinode inbetween. We can show that for this oscillation, the wavelength of the wave is twice the length of the string. λ1 = 2L [Fundamental] Higher frequency oscillations are known as ‘overtones’ or ‘harmonics’. The first overtone -- or second harmonic -- has two antinodes, sandwiched inbetween three nodes. The second overtone -- or third harmonic -- has one more node and one more antinode. We can summarize the relations between harmonics by the following expression: λn = 2L/n [Standing waves on a string] where the n subscript tells us which overtone we are dealing with, and L is the length of the string. Using v=fλ, we see that the frequencies are related by fn = nf1 [Frequencies of harmonics on string] In sketching node patterns, draw a straight string, then superimpose a sketch of the string as it is oscillating at its maximum amplitudes. For a string with say three antinodes, it should look like a string with three [sinusoidal] beads on it. The diagram is important for visualizing the relationship between the length of the string and the wavelength of the oscillation. Each ‘bead’ you have drawn represents half a wavelength.



STANDING WAVES IN PIPES Standing waves can occur in pipes too, which explains how all wind instruments and organ pipes produce the tones they do. Pipes are a bit more difficult to analyze because either end of the pipe can be either open or closed, and these two options give rise to much different relations between the wavelengths of the harmonics and the length of the pipe. If we think of the position of air molecules as being the thing that ‘waves’, then the closed end of a pipe is bound to have a node, since the molecules cannot leave that end, while an open end will have an antinode at resonance. Sketching such a pipe, we see that L= λ/4 for the fundamental. For a pipe with two open ends, we see that there will be a node at either end, so that L= λ/2. DIY: Calculate the harmonics of a partly-filled soda bottle, given the length of the column of air between the liquid inside and the mouth of the bottle. Will the next harmonic be twice the frequency or three times? BEATS This phenomenon is a lot easier to understand mathematically than to understand conceptually, unless you have experienced it and recognized it for what it is. The best way to do this is to take a guitar and play the lowest two strings, with the lowest string fretted so that it is nearly in tune with the other string. If it is slightly out of tune, and you pluck the two strings together, you will FEEL the strings reverberate as they go in and out of synch with each other. Do this right now if you can get your hands on a guitar! Your understanding will be much better if you know what we are talking about here. The rate at which the strings go in and out of synch with each other is called the beat frequency and is related to the frequencies of the two strings by 21 fffb −= [ Frequency of beat pattern ] Another way to simulate beats is to do what mouth-organ (‘harmonica’ is a more common name) players sometimes do for effect. Clasp your two hands in front of your mouth so that you have a nearly airtight compartment in front of your mouth, and flap one of your hands in such a way as to open and close the compartment. The variation in loudness you get is just what you hear when you ‘beat’ two notes together. DOPPLER EFFECT Can you recreate the sound of an Indy 500 racecar speeding around the track? If you find yourself recreating the changing pitch of the engines as the pass by you, you are simulating the Doppler effect. This is a shift in the frequency (pitch) of a sound emitted by a source by the time it gets to an observer because of motion of either the source or the observer or both. The frequency which the observer hears is given by



fv vv v

fo

so=

±11

//m

[Doppler effect]

where vo is the velocity of the observer, vs is the velocity of the source, v is the speed of sound, and fo is the frequency emitted by the source. To use this equation, use the upper symbol of ± (the +) if the observer is moving toward the source, and the lower symbol (the -) if it is moving away from the source. Likewise use the upper sign of the if the source is moving toward the observer, and the lower symbol if it is moving away.

m

Upper symbol if approaching [ Rules for applying Doppler effect equation ] Lower symbol if receding APPLICATION: Radar speed detection A policeman is aiming a gun at your car, but it fires microwaves, not bullets. Why? The microwaves are meant to bounce off your car, returning to the ‘gun’ with a slight Doppler shift, proportional to your speed. The return signal is ‘beat’ against the original signal to produce a signal whose beat frequency is equal to the amount of the Doppler shift. This provides for a fast, accurate measurement of your speed. Microwave radar guns are now being replaced with infrared guns, which operate exactly the same way. In calculating how Doppler shift and your speed are related, the thing to beware of is that there are two parts to this problem: a microwave pulse is sent by the police and received by you, then it is sent by you (like an echo) and received by the police. The radar arrives at your car Doppler upshifted in frequency because of the listener’s (your) velocity, and then it returns to the police Doppler upshifted because of the sender’s (your) velocity. It gets Doppler shifted twice -- once by source (you) and once by receiver. So, we can write f v c

v cfo=

+−

11

//

[ Double-Doppler shift of a signal bounced off a moving object ]

where v is your speed (assumed to be approaching the radar), and c is the speed of light, the relevant speed for the waves in this problem. The radar gun will take the beat frequency of and , which will be f 0f 0fffb −= .




DIGGING DEEPER: Harmony Why do we use the 12-tone scale that we use? It seems to depend on a number of coincidences. Plucking a guitar string, for example, does not just produce the harmonic, but little bits of the 1st, 2nd, 3rd, ... overtones. These overtones blend in because they are either identical to, or almost identical to, other notes in the 12-tone scale. Consider the harmonics generated by a guitar string. Let’s start with a 1kHz string, just to make the math easy. Its harmonics are 1kHz, 2kHz, 3kHz, 4kHz, 5kHz, (You get the picture.) Now 2kHz, 4kHz, 8kHz, and so forth are one, two, and three octaves above the fundamental. They will all sound like the same note, just ‘higher’. What about 3kHz, 5kHz, and so forth? Well, 3kHz will be between 2kHz and 4kHz. It will be 1.5 times higher than 2kHz. Likewise, 5kHz will be 1.25 times higher than 4kHz. If we use a 12-halftone scale, 1.5 corresponds to 7/12 of the way up the octave, or 7 half-steps. Wait a minute: 7/12?! Why not 6/12?! Because the steps are geometrical, not additive. To go up one halfstep, your frequency is multiplied by some constant, not added to by a constant. Now 27/12 = 1.498. Meanwhile, 1.25 is just four halfsteps above the base note: 24/12 = 1.260. Okay, the numbers don’t exactly work out, but it’s close enough for most ears. The seventh harmonic is a problem, though. It is 7/4 = 1.75 above the fourth harmonic, which is two octaves above the original tone. This is not even close to being an integer number of halfsteps above 4kHz. 29.688/12 = 1.75. Well, at least this system gets us through the first few harmonics, and the string doesn’t produce a lot of the 7th harmonic anyway.

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