SPH3UW Waes and Sound Mechanical Wae A mechanical wae is a disturbance which propagates through a medium with little or no net displacement o the particles o the medium. A Pulse is a single disturbance which carries energy through a medium or space Water Waes An Introduction to Waes and Wae Properties Wae Pulse People Wae Parts o a Wae 3 : waelength crest equilibrium A: amplitude 2 4 6 x(m) In the drawing, one cycle is shaded in color. The amplitude, A, is the maximum excursion o a particle o the medium rom the particles undisturbed position. The waelength is the horizontal length o one cycle o the wae. The period is the time required or one complete cycle. -3 y(m) trough The requency is related to the period and has units o Hz, or cycles/second. 1 T 1
Velocity o a Wae The elocity o a wae is the distance traeled by a gien point on the wae (such as a crest) in a gien interal o time. Example 1 The Waelengths o Radio Waes AM and FM radio waes are transerse waes consisting o electric and magnetic ield disturbances traeling at a speed o 3.00x10 8 m/s. A station broadcasts AM radio waes whose requency is 1230x10 3 Hz and an FM radio wae whose requency is 91.9x10 6 Hz. Find the distance between adjacent crests in each wae. T T AM 8 3.0010 m s 244 m 3 123010 Hz Problem: Sound traels at approximately 340 m/s, and light traels at 3.0 x 10 8 m/s. How ar away is a lightning strike i the sound o the thunder arries at a location 2.0 seconds ater the lightning is seen? FM 8 3.0010 m s 3.26 m 6 91.910 Hz The speed o light can almost seem instantaneous at these distances, so we need only concern ourseles with the sound component o lightening. d t s m 340 2s s 680m 2
Wae Types Types o Waes Reraction and Relection A transerse wae is a wae in which particles o the medium moe in a direction perpendicular to the direction which the wae moes. Example: Waes on a String A longitudinal wae is a wae in which particles o the medium moe in a direction parallel to the direction which the wae moes. These are also called compression waes. Example: sound http://einstein.byu.edu/~masong/htmstu/waetrans.html Wae types: transerse Wae types: longitudinal Energy transport A transerse wae is a moing wae that consists o oscillations occurring perpendicular (or right angled) to the direction o energy transer. I a transerse wae is moing in the positie x-direction, its oscillations are in up and down directions Longitudinal waes, also known as "l-waes", are waes whose direction o ibration is the same as their direction o trael, meaning that the moement o the medium is in the same direction 3
Other Wae Types Earthquakes: combination (p and s waes) Ocean waes: surace Light: electromagnetic Relection o waes Occurs when a wae strikes a medium boundary and bounces back into original medium. Completely relected waes hae the same energy and speed as original wae. Relection Types Fixed-end relection: The wae relects with inerted phase. Open-end relection: The wae relects with the same phase Reraction o waes Transmission o wae rom one medium to another. Reracted waes may change speed and waelength. Reraction is almost always accompanied by some relection. Reracted waes do not change requency. 4
Sound is a longitudinal wae Sound traels through the air at approximately 340 m/s. It traels through other media as well, oten much aster than that! Sound waes are started by ibration o some other material, which starts the air moing. Hearing Sounds We hear a sound as high or low depending on its requency or waelength. Sounds with short waelengths and high requencies sound high-pitched to our ears, and sounds with long waelengths and low requencies sound low-pitched. The range o human hearing is rom about 20 Hz to about 20,000 Hz. The amplitude o a sound s ibration is interpreted as its loudness. We measure the loudness (also called sound intensity) on the decibel scale, which is logarithmic. Doppler Eect The Doppler Eect is the raising or lowering o the perceied pitch o a sound based on the relatie motion o obserer and source o the sound. When a ambulance siren is sounding when it races toward you, the sound o its siren appears higher in pitch, since the waelength has been eectiely shortened by the motion o the ambulance relatie to you. The opposite happens when the ambulance moes away. The Doppler Eect When a moing object emits a sound, the wae crests appear bunched up in ront o the object and appear to be more spread out behind the object. This change in wae crest spacing is heard as a change in requency. The results will be similar when the obserer is in motion and the sound source is stationary and also when both the sound source and obserer are in motion. 5
Doppler Eect Stationary source Moing source Supersonic source http://www.lon-capa.org/~mmp/applist/doppler/d.htm The Doppler Eect ormula o 0 s s o is the obsered requency. s is the requency emitted by the source. o is the obserer s elocity. s is the source s elocity. is the speed o sound. Note: take s and o to be positie when they moe in the direction o wae propagation and negatie when they are opposite to the direction o wae propagation. Example: A source o sound waes o requency 1.0 khz is stationary. An obserer is traeling at 0.5 times the speed o sound. (a) What is the obsered requency i the obserer moes toward the source? (b) Repeat, but with the obserer moing in the other direction. 6
Example Solution: A source o sound waes o requency 1.0 khz is stationary. An obserer is traeling at 0.5 times the speed o sound. Example Solution: A source o sound waes o requency 1.0 khz is stationary. An obserer is traeling at 0.5 times the speed o sound. (a) What is the obsered requency i the obserer moes toward the source? o is unknown; s = 1.0 khz; o = 0.5; s = 0; and is the speed o sound. (b) Repeat, but with the obserer moing in the other direction. o is unknown; s = 1.0 khz; o = +0.5; s = 0; and is the speed o sound. 05. 0 o o ss s 1. 5 1. 5 khz s 0 05. 0 o o s s 0. 5 0. 5 khz s 0 Pure Sounds Graphing a Sound Wae Sounds are longitudinal waes, but i we graph them right, we can make them look like transerse waes. When we graph the air motion inoled in a pure sound tone ersus position, we get what looks like a sine or cosine unction. A tuning ork produces a relatiely pure tone. So does a human whistle. Later in the period, we will sample arious pure sounds and see what they look like. 7
Sensitiity o the Human Ear We can hear sounds with requencies ranging rom 20 Hz to 20,000 Hz an impressie range o three decades (logarithmically) about 10 octaes (actors o two) compare this to ision, with less than one octae! An Octae is a series o eight notes occupying the interal between (and including) two notes, one haing twice or hal the requency o ibration o the other.. Complex Sounds Because o the phenomena o superposition and intererence real world waeorms may not appear to be pure sine or cosine unctions. That is because most real world sounds are composed o multiple requencies. The human oice and most musical instruments produce complex sounds. Later in the period, we will sample complex sounds. Speakers: Inerse Eardrums Speaker Geometry Speakers ibrate and push on the air pushing out creates compression pulling back creates rareaction Speaker must execute complex motion according to desired waeorm Speaker is drien ia solenoid idea: electrical signal (AC) is sent into coil that surrounds a permanent magnet attached to speaker cone depending on direction o current, the induced magnetic ield either lines up with magnet or is opposite results in pushing or pulling (attracting/repelling) magnet in coil, and thus pushing/pulling on center o cone 8
Principle o Superposition Superposition o Waes When two or more waes pass a particular point in a medium simultaneously, the resulting displacement at that point in the medium is the sum o the displacements due to each indiidual wae. The waes interere with each other. Types o intererence. I the waes are in phase, that is crests and troughs are aligned, the amplitude is increased. This is called constructie intererence. I the waes are out o phase, that is crests and troughs are completely misaligned, the amplitude is decreased and can een be zero. This is called destructie intererence. Constructie Intererence crests aligned with crest waes are in phase 9
Constructie Intererence Destructie Intererence crests aligned with troughs waes are out o phase Destructie Intererence Resonance Certain deices create sound waes at a natural requency. I another object, haing the same natural requency is impacted by these sound waes, it may begin to ibrate at this requency, producing more sound waes. This phenomenon is know as resonance. i.e. opera singer shattering glass with oice 10
Standing Wae Standing Waes A standing wae is a wae which is relected back and orth between ixed ends (o a string or pipe, or example). Relection may be ixed or open-ended. Superposition o the wae upon itsel results in a pattern o constructie and destructie intererence and an enhanced wae. Let s see a simulation. Open and Closed Tubes Many musical instruments depend on the musician moing air through the instrument. Musical instruments like this can be diided into two categories, open ended or closed ended. A Tube or Pipe can be a musical instrument, it is oten bent into dierent shapes or has holes cut into it. An open ended instrument has both ends open to the air. An example would be an instrument like a trumpet. You blow in through one end and the sound comes out the other end o the pipe. A closed ended instrument has one end closed o, and the other end open. An example would be an instrument like some organ pipes, or a lute. Although you blow in through the mouth piece o a lute, the opening you re blowing into isn t at the end o the pipe, it s along the side o the lute. The end o the pipe is closed o near the mouth piece. Closed Ended Pipes Remember that it is actually air that is doing the ibrating as a wae inside the Pipe. The air at the closed end o the pipe must be a node (not moing), since the air is not ree to moe past the sealed end it must be relected back. There must also be an antinode (maximum moement) where the opening is, since that is where there is maximum moement o the air. The simplest, smallest wae that I can possibly it in a closed end pipe is shown Below. L First harmonic = 4L This is ¼ o a waelength. Since this is the smallest stable piece o a wae we can it in this pipe, this is the Fundamental, or 1 st Harmonic this is the lowest possible requency that any instrument can play. 11
Closed Ended Pipes Since the length o the tube is the same as the length o the ¼ waelength We know that the length o this tube is ¼ o a waelength so this leads to our irst ormula: 1 L 4 L is the length o the tube in metres. On it s own this ormula really doesn t help us much. Instead, we hae to sole this ormula or λ and then combine it with the ormula =λ to get a more useul ormula: 1 L 4 4L 4L 4L L When the wae reaches the closed end it s going to be relected as an inerted wae. This does not change the length o the wae in our ormula, since we are only seeing the relection o the wae that already exists in the pipe. Closed Ended Pipes So what does the Next Harmonic look like? I know this name might seem a little conusing (I m the irst to agree with you!) but because o the actual notes produced and the way the waes it in, musicians reer to the next step up in a closed ended pipe instrument as the 3 rd harmonic there is no such thing as a 2 nd harmonic or closed ended pipes. In act all o the Harmonics in closed ended Pipes are going to be odd numbers. Closed Ended Pipes Harm. # # o Waes in Column # o Nodes # o Antinodes Length- Waelength Relationship 1 1/4 1 1 λ= (4/1)*L 3 3/4 2 2 λ= (4/3)*L 5 5/4 3 3 λ= (4/5)*L 7 7/4 4 4 λ= (4/7)*L 9 9/4 5 5 λ= (4/9)*L Check your Understanding The speed o sound waes in air is 340 m/s. Determine the undamental requency (1st harmonic) o a closed-end air column that has a length o 76.5 cm. Since this is the First Harmonic: Now or the Frequency: You could also hae just used: 4L 4L 40. 765m 3. 06m m 340 s 3. 06m 111Hz 12
Open Ended Pipes I know you re probably thinking that there couldn t possibly be any more stu to learn about Resonance, but we still hae to do Open Ended Pipes. Thankully, they re not that hard, and since you already hae the basics or closed pipes it should go pretty easy or you. The undamental (irst harmonic) or an open ended pipe needs to be an antinode at both ends, since the air can moe at both ends. So this is the smallest wae we can it into a open ended pipe: ½ o a waelength L Fundamental First harmonic = 2L Open Ended Pipes So the basis or drawing the standing wae patterns or air columns is that ibrational antinodes will be present at any open end and ibrational nodes will be present at any closed end. I this principle is applied to open-end air columns, then the pattern or the undamental requency (the lowest requency and longest waelength pattern) will hae antinodes at the two open ends and a single node in between. For this reason, the standing wae pattern or the undamental requencies) or an open-end air column looks like the diagrams below. Open Ended Pipes The relationships between the standing wae pattern or a gien harmonic and the length-waelength relationships or open ended tubes are summarized in the table below. Harmonic # # o Waes in Column # o Nodes # o Antinodes Length- Waelength Relationship 1 1/2 1 2 Waelength = (2/1)*L 2 1 or 2/2 2 3 Waelength = (2/2)*L 3 3/2 3 4 Waelength = (2/3)*L 4 2 or 4/2 4 5 Waelength = (2/4)*L 5 5/2 5 6 Waelength = (2/5)*L Check your Understanding A 4 m long organ pipe (open at both ends) produces a musical note at its undamental requency. a) Determine the waelength o the note produced. b) What is the requency o the pipe gien the speed o sound is 346 m/s? Since this is the First Harmonic: Now or the Frequency: 2L 2 4m 8m m 346 s 8m Did you notice, that i we made the pipe longer, the waelength would be bigger, and since waelength and requency are inersely related, that means the requency would be smaller. 43. 3Hz 13
Check your Understanding Example: An organ pipe that is open at both ends has a undamental requency o 382 Hz at 0.0 C. a) What is the undamental requency or this pipe at 20.0 C? b) How long is this organ pipe? At Tc = 0.0 C, the speed o sound is 331 m/s. At Tc = 20.0 C, the speed o sound is 343 m/s. The undamental requency is: 1 L Speed o sound: 1 2 0 0 2L 2L 1 L 1 2 0 20 0 0 20 0 20 m m 331 343 s s 382Hz 20 20 20 2L 2L 20 How long is this organ pipe? 0 2L 0 0 L 2 0 m 331 L s 2 382Hz 0. 433m 20 m 343 382Hz s 331Hz 396Hz 14
Fixed-end standing waes (iolin string) I the string ibrates in more than one segment, the resulting modes o ibration are called oertones 2 n L n Fundamental First harmonic = 2L First Oertone Second harmonic = L Second Oertone Third harmonic = 2L/3 12.7 Beats When two waes with nearly the same requency are superimposed, the result is a pulsation called beats. Beats Beats is the word physicists use to describe the characteristic loud-sot pattern that characterizes two nearly (but not exactly) matched requencies. Musicians call this being out o tune. Two waes o dierent requency Superposition o the aboe waes The beat requency is 1 2 15
Beats I the beat requency exceeds about 15 Hz, the ear will perceie two dierent tones instead o beats. What word best describes this to physicists? Amplitude Answer: beats What word best describes this to musicians? Amplitude Check Your Understanding A tuning ork with a requency o 256 Hz is sounded together with a note played on a piano. 12 beats are heard in 4 seconds. What is the requency o the piano note? Number o Beats Beat Frequency: Total Time 12beats 4s 3 Hz Since Beat Frequency also equals: 1 2 Answer: bad intonation (being out o tune) Then: 3 Hz 256 Hz 2 253Hz or 259 Hz 2 Without urther inormation, there is no way to know which answer is correct... 16
Echolocation Sound waes can be sent out rom a transmitter o some sort; they will relect o any objects they encounter and can be receied back at their source. The time interal between emission and reception can be used to build up a picture o the scene. Example A boat is using sonar to detect the bottom o a reshwater lake. I the echo rom a sonar signal is heard 0.540 s ater it is emitted, how deep is the lake? Assume the lake s temperature is uniorm and at 25 C. Example 1: A boat is using sonar to detect the bottom o a reshwater lake. I the echo rom a sonar signal is heard 0.540 s ater it is emitted, how deep is the lake? Assume the lake s temperature is uniorm and at 25 C. The signal traels two times the depth o the lake so the one-way trael time is 0.270 s. From table 12.1, the speed o sound in reshwater is 1493 m/s. depth t 1493 m/s0. 270 s 403 m The Speed o Sound Waes The speed o sound in dierent materials can be determined as ollows: In luids In thin solid rods B Y B is the bulk modulus o the luid and its density. Y is the Young s modulus o the solid and its density. The bulk modulus (B) o a substance measures the substance's resistance to uniorm compression. While Young s modulus (K) measures the Stiness or elasticity o a solid 17
The Speed o Sound Waes In ideal gases 0 T T 0 Materials that hae a high restoring orce (stier) will hae a higher sound speed. Materials that are denser (more inertia) will hae a lower sound speed. Here 0 is the speed at a temperature T 0 (in kelin) and is the speed at some other temperature T (also in kelin). For air, a useul approximation to the aboe expression is 331 0. 606 T C m/s where T c is the air temperature in C. Example 1: A copper alloy has a Young s Modulus o 1.110 11 Pa and a density o 8.9210 3 kg/m 3. What is the speed o sound in a thin rod made o this alloy? 11 Y 1. 110 Pa 3500 m/s 3 3 8. 910 kg/m Example 2 Bats emit ultrasonic sound waes with a requency as high as 1.010 5 Hz. What is the waelength o such a wae in air o temperature 15.0 C? 331 0. 606 T C m/s The speed o sound in air o this temperature is 340 m/s. The speed o sound in this alloy is slightly less than the alue quoted or copper (3560 m/s) in table 12.1. 340 m/s 3 3. 410 m 5 1.010 Hz 18