CAE334 Lecture 1 : Intro to acoustics

1/13/2014CAE 334/502 - Week 11

CAE 334/502 Lecture 1bBasic Wave Physics Continued

1ObjectivesUnderstanding of phase and wavenumber Understand the basic concept of superposition, beating, and standing wavesUnderstand the basic concept of a phasorUnderstand the difference between acoustic velocity and speed of soundUnderstand the concept of acoustic impedanceUnderstand the concepts of acoustic power and intensity1/13/2014CAE 334/502 - Week 12ReferencesChapter 2 of Longs Architectural Acoustics (LAA)Chapter 1 of Allan D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications, Acoustical Society of America, 1988.

Superposition Java Applethttp://webphysics.davidson.edu/Applets/superposition/GroupVelocity.html

1/13/2014CAE 334/502 - Week 13Phase = [rad or ]The phase defines the value of p at the location x=0 at the time t =0. That meansp(x=0,t=0) = Asin()

The difference in between two different waves is far more important than the exact value of for one alone.When =180 waves can cancel each other outWhen =0 waves reinforce each other

1/13/2014CAE 334/542 - Week 14Phase difference = [rad or ]The phase difference between two waves is related to the time delay between themA positive means the second wave is delayed in time1/13/2014CAE 334/542 - Week 15

Wavenumber = k [m-1] or [ft-1]The acoustic wavenumber, k, is scaled inverse of the wavelength, l and is given by the equation

We use k in acoustics because we find 2/ l all over the equations and using k make for easier notation (kind of like using instead of 2/T )

The product ka , where a is the dimension of an object, is a good measure of whether or not a wave bends around an object.If ka1 sound will mostly reflect and not bend around

1/13/2014CAE 334/542 - Week 166Diffraction (Bending) Around Objecthttp://www.acoustics.salford.ac.uk/feschools/waves/diffract.htm#object

1/13/2014CAE 334/542 - Week 17Speed of Sound = c [ft/s or m/s]c = f l=/k

c is the speed at which the disturbance travels through the medium

It always relates frequency to wavelength

c depends on the medium and medium properties Mediums air, liquid, solid, etc. Properties are temp, density, etc

c is independent of wave amplitude (for linear waves)

Units are [ft/s or m/s]1/13/2014CAE 334/542 - Week 188The speed of sound is the speed at which the wave form propagates i.e. the speed at which the wave peaks travel.The speed of sound is a medium property and is also always the product of frequency and wavelength, i.e. c = f * lambda.

The speed of sound depends upon the chemical composition of the medium of travel, mostly related to the ambient density of the material. Gases have a low speed of sound, liquids higher, and solids higher still. For an ideal gas there is a simple equation that can be used to compute the speed of soundc = sqrt ( gamma * R *T). In this equation gamma is the ratio of specific heats (Cp/Cv), a number between 1 and 2, R is the molar gas constant (the ideal gasconstant Ro divided by the molecular weight M), and T is the absolute temperature of the gas.

For air you can do a series expansion and get the approximations that c ~ 331.5 + 0.6 Tc m/s where Tc is the temperature in Celcius or c~ 1058 + 1.08 Tf ft/s where Tf is the temperature in Fahrenheit. At room temperature (20C or 68F) that means c ~ 343 m/s or 1125 ft/s

This speed of 1125 ft/sec means it takes about 4.5 seconds for sound to travel a mile. This is how you can tell how far away lighting is. Count the number of secondsbetween the lighting and the thunder and divide by 4 or 5. That tells you the distance in miles. (It is probably better to use 4 since the air right around the lighting is very hot and sound will travel faster near the lighting)

It is important to note that c is NOT the speed of the molecular motion. The molecules are typically moving 100 micro-m to 100 mm/s and NOT 343 m/s. The speed of molecular motion is mostly dependent upon the amplitude of the sound wave.Equations for c LAA gives an equation that works for solids, liquids and gasses

where B=Elastic Bulk Modulus of medium, r=total density of medium,For an ideal gas this can be rewritten as

where is the ratio of specific heats ( =cp/cv), R is the molar gas constant, and T is the absolute temp

For temperatures near room temp we can write a Taylor expansion about 20 C to show that

c 331.6+0.6TC m/s or c1053 +1.07 TF ft/s

For T 20 C 68 F c 343 m/s 1125 ft/s1/13/2014CAE 334/502 - Week 19

c in Other materialsTable 2.2 in LAA lists the speed of sound in other common material

Table 2.3 lists a variety of vibrational waves in solids along with formulae for the propagation speeds for many different types of waves.The bending wave in a plate is important in sound transmission and its speed of sound isE = Youngs Modulus, h = thickness = density, = frequency = Poissons RatioNOTE: This is frequency dependent!1/13/2014CAE 334/542 - Week 110

101/13/2014CAE 334/502 - Week 111Pure Tone Sound Wave AgainWe already saw that pure tone plane wave (1-D, one frequency) wave can be described with the equation

Using k=2p/l =w/c we can rewrite this equation to match Eqn. 2.32 of your text

Or, if we let q = + p/2

1/13/2014CAE 334/542 - Week 11212To start seeing how we can describe the physics a bit better, let us think about a piston moving sinusoidally like in the previous animation. We would hear that as a pure tone.

Why alternate math forms?So why show the other math forms? To let you know that there is not just one exact form.You can describe waves with either a sine or a cosine you will just end up with a different phase constant

The important thing to note is that the argument is always of the form kx-t or t-kx.Which term comes first is not important its the + or between them that is really important. With a its a wave going in the +x direction.1/13/2014CAE 334/542 - Week 113131/13/2014CAE 334/502 - Week 114A Pure Tone Sound Wave

14Let us look at this animation of the most simple type of sound wave a pure tone wave that moves in only one direction.In this animation we have a speaker moving at the end of a tube. As the speaker moves backAnd forth it causes the air molecules (shown as dots) to compress (increase in density) andAnd rarefact (decrease in density). This compression and rarefaction is transmitted to the Adjacent molecules and a moving density variation results.

If the speaker motion is at one frequency, the density variations must also occur at that same frequencyand the resulting acoustic wave is called a pure tone.

The graph below the top figure is a plot of the local air pressure. The solid line in the middle isAtmospheric pressure, which does not vary in space. The sine graph is the local air pressure alongThe tube length. Notice that regions of higher molecule density have higher local pressure and areas of low density have low local pressure. The acoustic pressure is the deviation of the local air pressure from the spatially constant atmospheric pressure.

It is important to note that the air molecules just move back and forth a short distance but after one cycle, end back up where they started. The disturbance travels from place to place but the actual molecules do not.Example 1.1 Finding A, T, f, and 1/13/2014CAE 334/502 - Week 115

15Example 1.1: Finding AMeasure peak-to-peak amplitude which will be 2AOn this plot we would find

A1 2.5 PaA2 3 PaA3 2 Pa

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Example 1.1: Finding T and f Find T by drawing a horizontal line and measure the time between wave crossings of the line1/13/2014CAE 334/502 - Week 117

For our plot we would find that for all three waves

T 1.00 msec, f=1/T 1000 Hz, =2f=6280 rad/s17Example 1.1: Finding Look for the time offset, to of the upward zero crossing . That is where the argument of sin(t+) should be zero. So =- to

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181/13/2014CAE 334/502 - Week 1191/13/2014CAE 334/502 - Week 120SuperpositionSuperposition says that the result of any two waves interacting is the sum of the waveforms.Superposition is only a useful concept when the sound sources are coherent Coherent means same frequency and time synchronized with a constant phase differenceIndependent sources are almost always incoherent, (meaning not coherent) and so the phase difference between them is unknown and possibly changing.

20While we can ignore phasors, superposition is quite important. The applet at the URL is a great little program for investigating this phenomenon

1/13/2014CAE 334/502 - Week 121Superposition of two waves at same fThe superposition of two waves of the same f traveling in the same direction is a traveling wave of same f with a new amplitude and phase

Note: The textbook has the more complete equation for superposition of two waves of same frequency but different amplitude that are traveling the same direction

Same sign means traveling same directionA travelling wave results21When we superimpose to waves of the same frequency traveling the same direction the resultant sum is a pure tone with the same frequency but new amplitude and phase. The textbook has the general solution for arbitrary amplitude and phase.

If the direction of the waves are different (i.e. one has t+x/c instead of t-x/c) we get a different solution called a standing wave. Well look more at that later

Lets look at an animation from the java applet at the URL on the previous page.1/13/2014CAE 334/502 - Week 122

1/13/2014CAE 334/502 - Week 123Standing WavesIf the two waves are going opposite directions the superposition result is no longer a traveling wave

This function has sinusoidal time variation and sinusoidal space variation but since they are not coupled, the spatial variation does not move. We call it a standing wave.

Opposite sign means traveling opposite directions23When we superimpose to waves of the same frequency traveling the same direction the resultant sum is a pure tone with the same frequency but new amplitude and phase. The textbook has the general solution for arbitrary amplitude and phase.

If the direction of the waves are different (i.e. one has t+x/c instead of t-x/c) we get a different solution called a standing wave. Well look more at that later

Lets look at an animation from the java applet at the URL on the previous page.1/13/2014CAE 334/502 - Week 124

General SuperpositionThe result for general superposition of two waves in opposite directions is something that looks more like a traveling, standing wave1/13/2014CAE 334/502 - Week 1251/13/2014CAE 334/502 - Week 126

1/13/2014CAE 334/502 - Week 127BeatsIf two wave of close, but not exact, frequency superimpose we get the following: a traveling wave at ave=(1+ 2)/2 and kave=(k1+ k2)/2 , with an amplitude that varies at half the difference frequency =1-2 . We call this phenomenon beating

This is a alternate form of equation 2.30 in LAA.

27If the frequency of the waves is different, we dont get a simple traveling wave. What we get is a traveling wave that has an oscillating amplitude. We call this phenomenon beating. We can hear the amplitude oscillation, also called modulation. Musicians tune instruments by listening for beats and eliminating them.

Lets look at an animation from the previous URL and also listen to a demonstration at 1/13/2014CAE 334/502 - Week 128

1/13/2014CAE 334/502 - Week 129BeatsWhile it may be difficult to see the beating in a visualization, the phenomenon is easily heard

Indeed, the modulation is so clear that it is a commonly used method for tuning instrumentsWe adjust string tension until the beating goes away

400 Hz402 Hz400+402 Hz

29If the frequency of the waves is different, we dont get a simple traveling wave. What we get is a traveling wave that has an oscillating amplitude. We call this phenomenon beating. We can hear the amplitude oscillation, also called modulation. Musicians tune instruments by listening for beats and eliminating them.

Lets look at an animation from the previous URL and also listen to a demonstration at 1/13/2014CAE 334/502 - Week 130Java AppletsA good very general java wave applet is at:

http://webphysics.davidson.edu/Applets/superposition/GroupVelocity.htmlI have added this and several wave other wave apps to the Links and Resources part of the Blackboard page30While we can ignore phasors, superposition is quite important. The applet at the URL is a great little program for investigating this phenomenon

Got to here1/13/20141/13/2014CAE 334/502 - Week 1311/13/2014CAE 334/502 - Week 132PhasorsThe text introduces and briefly discusses the concept of phasors to describe waves and simple harmonic motionPhasors are indeed quite powerful and a standard method of math notation used throughout physics and engineering. They are used extensively for describing waves or really any quantity that undergoes time harmonic variation

You may have seen this in Phys 221 or 224 and will have seen it in CAE 383 if you have taken it.

Luckily, if your math skills are a bit lacking in complex numbers, we dont use them much and full understanding is not essential32The text introduces the concept of a phasor representing a trig function with a complex exponential.Phasors are indeed a very elegant and powerful way to deal with trig functions, but the mathematical analysis used in the text and that we will do in the class do not require it. So, Im going to skip the section.1/13/2014CAE 334/502 - Week 133Phasor BasicsIdea: Represent sinusoidal motion as a component of circular motion and circular motion as a rotating vector described by a complex numberKey math relation: Eulers Equation

Then we could write a complex pressure

33Phasors as ProjectionsA cosine wave is a horizontal projection of a rotating phasor (i.e. the real part)A sine wave is the vertical projection (i.e. the imaginary part, as seen below)1/13/2014CAE 334/502 - Week 134

34Leading and Lagging We may use the terms leading and lagging to describe a phase shift.In the animation, the blue vector is leading the red vector (+) (or, alternately, the red vector is lagging the blue vector)Notice the fixed angle between the red and blue vectors that angle is the phase difference1/13/2014CAE 334/502 - Week 135

351/13/2014CAE 334/502 - Week 136Why Phasors?So why would I want to take nice simple real numbers and make them complex? Because it can actually simplifiy the mathConsider Integration and Differentiation

multiplying by jk and dividing by jw are much easier than converting sines to cosines and vice-versa and trying to remember your +/-s which is what you need to do with trig functions

361/13/2014CAE 334/502 - Week 1371/13/2014CAE 334/502 - Week 138Why u is importantu is important because u is continuous at the interface at the interface between materialsIf a fluid is in contact with a moving surface, the fluid molecules must be at the same velocity as the moving surfaceRecall that frequency of wave is the frequency of the generating vibration, i.e. frequency of surface velocityWhat this means is that a velocity is imposed on the gas, not a pressure. If we can relate that velocity to pressure we can find the pressure amplitude for the waveThere is a simple relation between pressure and particle velocity for pure tones the impedance381/13/2014CAE 334/502 - Week 139Specific Acoustic ImpedanceAnother important property of a medium is the idea of impedance. Impedance is a measure of resistance to motion of the medium.The specific acoustic impedance, z, is the ratio of sound pressure, p, to particle velocity, u,

For a plane wave in a fluid, z=0c where 0 is density. This constant and called the characteristic impedance

39Example 1.2The RMS sound pressure at the surface of a panel of large tractor in the middle of a field is measured as 10 Pa.

What is the RMS surface velocity of the tractor?

Assume plane waves are leaving tractor panel and T = 20 C.

1/13/2014CAE 334/502 - Week 140Example 1.2The RMS sound pressure at the surface of a panel of tractor in the middle of a field is measured as 10 Pa. What is the RMS surface velocity of the tractor?

In the middle of a field there are no walls so there are only waves leaving the tractor. The waves are traveling in air at 20 C so

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1/13/2014CAE 334/502 - Week 142So, why do we care about z?It is a relation between u and pIf we find one we can find the otherAcousic energy is always reflected at a change in zWe can cause sound reflection with changes in z which means We can significantly reduce sound transmission either with very large changes in z (high mass walls) or by constructing a partition with many layers since each layer reflects some energy (multi-layered walls)421/13/2014CAE 334/502 - Week 143Acoustic PowerAcoustic Power (W) [watts=W]It is the time rate of acoustic energy flow i.e. the power required to get medium movingIt is a property of vibrating object has nothing to do with room or observers distance from objectThink of it is analagous to the power rating of a 60W lightbulb. It draws 60W regardless of the room but light level depends upon the location and room reflectanceA typical voice in conversation has W 10 WA loud stereo has W 100 mWA loud car horn has W 1 WTypical sound powers of objects ranges from nW (pin drop) to MW (rocket launch)43We just mentioned we use the RMS level of the wave as its amplitude because the RMS value was related to the energy in the wave.

If there is energy in the wave, it had to get there somehow. The vibrating object can be considered a source of acoustic energy. We characterize that source by its acoustic power. A given source has a given acoustic power. This acoustic power is related to the object vibration and not the medium or the surroundings or enclosurethe object is located in. The acoustic power is simply the energy per unit time the object can put into the medium from its vibration.

We will use the symbol W for acoustic power because thats what your text does. It wouldnt be a problem except the unit for power is watts and that also uses the symbol W, and you may get confused between the power symbol W and the units W. I will try to make the power symbol W italicized to help avoid confusion. But, if you arent sure if W is referring to the power itself or is just a unit, ask. That is not a dumb question. I should note here that the watt is named after James Watt, the 18th century Scottish instrument maker who developed the modern steam engine.

Again, the acoustic power is an object property. The bigger the vibration amplitude, the more acoustic power it will generate. The amplitude of the acoustic wave given off will depend upon the medium, the distance from the object, any reflecting surfaces, etc. the power is a single, immutable quantity.

Typical sound powers of objects range from nano watts (the power in a pin drop) to mega watts (the acoustic power of a large rocket launch). The acoustic power of most everyday objects (people speaking, appliances, stereo equipment) is in the microwatt to watt range.

You should note, right away, that acoustic power of a source is not the same as the electrical power into a source. A person might brag about the 500 W speakers in their pickup truck but that is not 500 acoustic watts. The 500 watts refers to the electrical power into the speakers not the acoustic power generated. A 500 W acoustic source in a pickup truck would cause instant deafness if hearing protection was not worn. Most sources are very inefficient at converting electric power to acoustic power. A typical home or car loudspeaker is about 0.1% efficient. High efficiency speakers like Klipshorns are only 1 to 10% efficient. Even the most efficient horn speakers used in pro audio systems (i.e. concerts) are usually under 20% efficient.

For a comparison, a full orchestra, at its peak, might produce 10 W of acoustic power. 1/13/2014CAE 334/502 - Week 144Acoustic IntensityAcoustic Intensity (I) [W/m2]Acoustic intensity is the acoustic power flow through a unit area and is found from the average product of p and u

Acoustic intensity is dependent upon the source directivity, source location, and room properties

Typical sound intensities range from nW/m2 (pin drop @ 1m) to MW/m2 (rocket launch near platform)For a plane wave we would find that

44Related to acoustic power is the acoustic intensity. The acoustic intensity is important because it is what drives the motion of our ear drum,i.e it is what we hear. A certain acoustic intensity on the eardrum creates a certain force which moves the ear parts well talk about that next week.

The acoustic intensity is the acoustic power that is moving through a unit area. In other scientific words, it is an acoustic power flux. If the power goes through a large area, the intensity is lower than if it is confined to a small area. A 1 W acoustic source creates a high intensity at the end of a short, 2 diameter tube but creates a low intensity at distance of 100m away if there is nothing to confine the sound (like the middle of a field).

The units are power per area so W/m^2. People rarely use W/ft^2 so get used to converting units.

Typical sound intensities range from pico watts/m^2 (a pin drop at 1 m) to W/m^2 (a rocket launch at 0.5 km). When the sound intensity gets muchabove 1 W/m^2 we are in instant deafness levels so there should be no need to talk about it. (i.e. if you are only 5 m away from a rocket launch you will have instant deafness, but along with instant death so I guess going deaf isnt too bad in the big scheme of things)

1/13/2014CAE 334/502 - Week 145Power Intensity relationSince intensity is a power per unit area, we can find the sound power of an object by integrating the intensity on any surface surrounding the object

If we can measure average Ii on each patch of a surface of N patches surrounding a source we can estimate W as

45Powers and Intensities AddFor incoherent sources, powers and intensities add (RMS pressures do NOT)

Wtot = W1 + W2 Itot = I1 + I2 1/13/2014CAE 334/502 - Week 146

W = 0.1 W

W = 30.1=0.3 WExample 1.3The tractor panel of Example 1.2 is 2 m x 1 m in size. How much acoustic power is radiated by the panel?

1/13/2014CAE 334/502 - Week 147Example 1.3The tractor of Example 1.2 has a panel that is 2 m x 1 m in size. How much acoustic power is radiated by the panel?

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