Musical Acoustics Spring 2017 Prof Jim Thomas Acoustics

Musical Acoustics Spring 2017 Prof. Jim Thomas Acoustics The Nature of Sound Its Production Propagation Perception Everything should be as simple as possible, but no simpler.

Class webpage – via http: //panda. unm. edu. First assignment: On one page, write down what is important to you in life [examples - having a supportive and loving family, “my dog”, having a rewarding career, being generous to others, getting rich, playing music…keep it to a couple sentences. ] Then, write down what you’d like to learn about in this class. [examples – trumpet, violin, voice, pitch, equal temperament, hearing. You can choose something else!] Put your name on it. . .

Field Trips: We will have field trips to the Cathedral of St. John (Downtown ABQ) and to the UNM Violin Workshop. The class is large enough that we should split into two groups. If you can go on a 10 -11 am field trip, please write that on your paper. If you can go on a 12 -1 pm field trip, please write that on your paper. Then turn it in.

There will be a lot of reading required in this class. You may “skim” parts of the required reading, but I will try to emphasize the important parts. For Thursday: read Chapter 1 “How Sound Propagates” Key points: Impedance ~ Force/velocity for a wave (Velocity of what? ) Impedance changes (“mismatches”) cause reflections

Heller put the concept of impedance before the chapter on sound waves! (That’s how important it is. ) Sound waves are pressure waves (usually in air, but also in other gases, liquids, even solids. ) Orange = higher than ambient pressure Blue = lower than ambient pressure These look like ripples on a pond… but in 3 D sound spreads out spherically

What is pressure? Pressure is the force the gas exerts on a surface, divided by the surface area. (A surface twice as large feels twice the force, from the same pressure. ) The force is caused by the collisions of the molecules of the gas with the wall. The pressure is defined even if there is no wall present… it is the force that a wall would feel if it were there! Ambient air pressure is about 14 lbs per square inch. My book has an area of ~ 100 square inches. So when the bottom is exposed, air pushes up on it with 1400 lbs of force! Demo!

Usually, forces from air pressure balance out. But in a sound wave, you might have high pressure on one side of you and low pressure on the other! Why don’t I get pushed back and forth as sound goes by? Moderately loud sounds have pressures that are 1 part per million (ppm) above/below ambient!

There is another “half” to the picture of the sound wave. (In fact, there is another half of the picture of any wave!) • Higher pressure regions have a higher density of air molecules Demo – putting air molecules in a smaller space increases pressure • The difference in pressure between orange and blue regions causes air molecules to move into the low pressure (blue) region. The main reason this happens is that in the “gradient” region, molecules will collide more with molecules on the left (where there are molecules) than on the right. In other words, pressure differences cause air to move, which causes pressure differences…which causes air to move. . . which causes pressure differences. . .

Let’s take a look at Dan Russell’s Most Excellent webpage! http: //www. acs. psu. edu/drussell/Demos/waves-intro. html Some key concepts: • Compression • Rarefaction • Actual wave motion of air molecules is only microns ! • The power in a sound wave is very small, < 1 Watt. Every wave motion involves the interplay between two physical quantities. In sound, it is pressure and displacement. In light, it is electric and magnetic fields.

When a pressure difference pushes molecules into the rarefaction, they overshoot! This is because of inertia. Any mechanical wave (sound, string vibration, plate vibration etc. ) involves a restoring force and inertia. Important concept – transverse vs. longitudinal waves In any mechanical wave, atoms or molecules move. If they move in the direction that the wave moves in, the wave is “longitudinal”. If they move perpendicularly, the wave is called “transverse”.

If I whack the rubber rope, the wave that results is a) Transverse b) Longitudinal Sound waves moving through air are a) Transverse b) Longitudinal Sound waves in gases and liquids are longitudinal. But in solids you can also have transverse sound waves! http: //www. acs. psu. edu/drussell/Demos/wavemotion. html

There’s a serious oversimplification of the picture of the longitudinal sound wave… All of the molecules are moving only left and right. . . and some of them change direction without colliding with anything! The molecules actually have random, thermal motion in addition to their back-and-forth motion. And they only “communicate” through collisions. Without molecular collisions, there can be no wave! The speed with which the wave moves (in a gas) cannot be faster than thermal motion. It is slightly slower.

When you heat up a gas, the random motion of the molecules gets faster. What do you think will happen to the speed of sound? a) It will get faster b) It will get slower c) It will be unchanged At the same temperature, a gas composed of heavier molecules will have slower-moving molecules. Would you predict that sound in a heavy gas would move a) Faster than b) Slower than c) The same speed as …sound in a lighter gas?

Here’s a little detail regarding the relationship between pressure and density of molecules. You may (or may not… no problem if not) know about “the ideal gas law”. . . PV = n. RT. We can rewrite this P = (n/V)RT= number density x RT. According to the ideal gas law, if I reduce the volume of a gas by half, what happens to the pressure? a) It’s halved b) It’s doubled c) It’s unchanged Let’s do an experiment and see!

In fact, the pressure went up more than we predicted! This is because, when I rapidly compressed the gas, it got a little hotter. (The ideal gas law is still true, but we wrongly assumed that compressing the gas wouldn’t change its temperature. ) Sound waves also compress gas rapidly. (We call this “adiabatic compression”. ) So there’s a different law we must use to describe the relation between pressure and density in a sound wave. Don’t worry about the math!

“Impedance” Impedance is not really what it sounds like. Waves move equally well through gas (or regions of gas) with high or low impedance. (“Absorption” will prevent a wave from propagating. ) The book refers to impedance as “push and push-back”.

Musical Acoustics – Lecture 2 • Last time: Mechanical waves (including sound) are a propagating, cooperative motion of a medium. The motion of individual molecules is not the same as the overall wave motion. For sound in air, molecules move back and forth a tiny bit (in addition to their thermal motion). Because the direction of motion is in line with the wave direction, this kind of wave is called longitudinal. Associated with the molecular motion is a pressure variation. Molecules communicate through collisions. Hotter gases have faster moving molecules that collide more frequently , so sound waves move faster also.

Clicker Question #1: I remembered to bring my clicker today. A) True B) False

Impedance We can see an example of impedance to a wave-like motion using the Newton Cradle. Demo! Why does the energy not flow freely past the ball with different mass? When one ball hits another identical ball, it pushes on it and gives it a velocity (speed). Because the balls are identical, it can give the second ball the same speed it originally had, and then stop. (This is an example of a “symmetry” argument. ) With balls of different mass, it can’t do that. With a lighter target ball, it pushes too hard, given the “push back” (inertia) of the small ball. (Notice that the original ball keeps moving forward!) With a heavier target ball, it can’t push hard enough. The original ball gets rejected and moves backward!

In the assigned reading, different kinds of impedances are discussed. What was not discussed? a) b) c) d) Impedance of transverse waves on a string Refractive index as impedance for light Acoustical impedance Viscosity of liquids as impedance

Impedance for Sound Waves When you “push” on air (compress it), it “pushes back” in a wave, displacing (moving) neighboring air molecules. Recall that any wave has two “married” physical quantities… in sound, they are pressure and displacement.

To help you understand impedance, we need to talk a little about the most special wave shape – the sine-shaped wave. These waves are called “sinusoidal”. (A cosine-shaped wave is also sinusoidal. ) How can sound waves have a sinusoidal shape? ? We mean that a plot of pressure vs. position is a sine wave.

Orange = High pressure, Blue = Low pressure The wave shown is so loud it would kill you. The wave can move left The wave can move right The wave can “stand” – oscillate in place This pressure variation cannot persist indefinitely… Air will move from the high pressure regions to the low pressure regions. There are 3 ways it can do that!

P These are pressures P will see in the future These are the pressures P experienced in the past The mathematically perfect sine wave goes forever in both directions. If I sit at one spot, the pressure change over time is described by the same curve. If the wave is moving to the right, I need to reverse the xaxis. Do you see why? Pressure at P Future Now Past Increasing time

Here’s an excellent simulation of a sound wave. The thermal motion has been removed, but you can better see molecules pushing on their neighbors. http: //physics. bu. edu/~duffy/semester 1/c 20_disp_pressure. html Notice how a sinusoidal pressure wave has an accompanying sinusoidal displacement wave. But the peaks are not at the same time (at a specific place) or at the same place (at a specific time. ) For a given medium (gas) and environment, if I double the pressure amplitude (height of the pressure wave), I will double the displacement amplitude. I will also double the speed with which the molecules move. Everything scales “linearly”.

We can now be a little more precise about what impedance means. “Push” means force, or pressure. (They are not exactly the same, of course, but close enough. ) “Push back” is determined by how fast the molecules move. Heavier molecules push back more (like Newton cradle balls) and acquire a slower velocity for a given pressure. Impedance of sound in a large volume of gas, Z = P/v for a sinusoidal wave, with P and v taken to be their peak (largest) values. Acoustical impedance does change with the frequency of a wave. Higher frequencies have higher impedance, because v is lower.

Impedance differences (“mismatches”) give reflections There a precise mathematical rule for the energy reflected. It’s basically the impedance difference (divided by the impedance sum) squared. Z 1 is the impedance of the medium the wave is coming from, Z 2 is the impedance of the medium the wave is going into.

Suppose ZA=1 and ZB=2 (in appropriate units). Which wave would have a stronger reflection? a) A wave from A to B b) A wave from B to A c) Both would be equally reflected Answer: c. The same reflection occurs regardless of whether the impedance goes up or down. You can see reflections off the water surface (water has higher impedance than air. ) But, if you are under water, you can also see reflections on the other side!

Computational Sound Computers can calculate sound propagation, using the known laws about gas pressure vs. density. The region to be studied must be broken up into small “cells”, each of uniform pressure. The cells push on each other. To be of uniform pressure, the cells must be small compared to the wavelength of sound to be simulated. A piston makes a sound pulse. If you are really “with it”, you will understand that the computer needs to know not just the pressure in each cell, but also the displacement and the velocity! The sound wave is the marriage of pressure and displacement.

Sound in Pipes – Reflection off a Closed End 1. The pressure gradient tries to push molecules through the wall. Instead, they bounce off the wall. So the positive pressure pulse is reflected.

By the way, the impedance of a hard wall is infinity. What does our formula predict for the fraction of energy reflected? When we are interested in pipes that change diameter, or that are open, we need to use a slightly different definition of impedance… We replace the velocity (distance/time or meters/second) with “volume velocity” (volume/time or m 3/s) = v * cross sectional area. Rationale: in a pipe, the wave can’t spread out in all directions. Some of its push is against the side walls. This push is ineffective at moving the wave along… we need more total push for the same v. Smaller pipe. . . higher impedance!

The volume velocity works as long as the pipe diameter is small compared with the wavelength. If the pipe diameter is large, our lumped impedance should approach some constant, independent of the diameter. Otherwise, the impedance out in the open (an “infinitely large pipe”) would be zero! OK – think about this statement “If the pipe diameter is large”… What does it mean? a) b) c) d) Diameter is bigger than 1 m Diameter is bigger than 100 m Diameter is quite a bit bigger than the wavelength of sound Answer d. In physics, when we say something is large or small, we always mean compared to something else!

For a very large pipe, then, we basically ignore the radius (and cross sectional area) of the pipe, and use an “effective” cross section. Essentially, we are saying that the pipe doesn’t matter if the wavelength is small. The exact expression for impedance is not very important to us. If you like, you can use 2 pl for the effective radius of a large pipe. We can now talk about reflections at the open end of a pipe! Wait… sound reflects off of –nothing- ? ? ?

When a sound wave confined to a pipe reaches the open end, the lower impedance outside causes the “push” from the wave to be more than what is needed to sustain the wave. The molecules “overshoot” as they leave the pipe, and create a rarefaction that propagates back up the pipe! Before After The reflection actually appears to come from just beyond the physical end of the pipe. This is called an “end correction. ”

The figure shows all of the sound being reflected. That’s not right. How much sound is reflected depends on the impedance mismatch. Open space is just a really (really) big pipe! So which pipe has a bigger impedance mismatch with open space? Pipe A Or choose C: the mismatch is the same for both Or choose D: neither has a mismatch with open space Pipe B Answer B. Smaller pipe looks more different from open space!

Which wave will be more strongly reflected from the open end of the pipe? Pipe A Pipe B Or choose C: They will be equally reflected Or choose D: Neither will be reflected, because the pipe is open!

We know: Before After A hard wall (infinite impedance) reflects a + pressure pulse as + i. e. a reflection off a higher impedance is not inverted Open space (very small impedance) reflects a + pressure pulse as – i. e. a reflection off a lower impedance is inverted Pipe impedance increases as the pipe gets smaller. Which pulse will have an inverted reflection at the change in pipe diameter? Pipe A Before Pipe B Or choose C: both D: neither

There is a wave simulation Java applet on the “Whyyouhearwhatyouhear” website, chapter 1. Let’s use it to examine a wave reflecting off a change in tube width. The buttons I will push are listed on our class webpage, in a file called ‘Impedance Mismatch in Ripple. pdf’

Homework: In groups of 2, run Ripple to study the reflected and transmitted pulse in a pipe that changes width suddenly. Turn in one data sheet per pair. (Some settings notes and the data sheet are on the class webpage. ) Read Chapter 2