INDEX

- Scientific Notation
- The nature of sound waves
- Negative Exponents
- The velocity of sound waves
- Speed of sound in solids
- Speed of sound in air
- Speed in transverse waves
- Wavelength and freq
- Pressure velocity and impedence
- Brain Bullets

Because sound embraces such a wide range of values (the ratio of the threshold of pain to the threshold of hearing is nearly a million to one) we need to have an alternate way of writing large numbers otherwise the board will be covered in zeros. The exponent system is a common method for dealing with large numbers and you should take the time to understand it.

Exponents are arrived at by dividing the large number by tens (or any base you choose) and then expressing the result as a N * 10^{x}. One shortcut is that the exponent always indicates the number of zeros that appear after the 1.

And in reverse

In the equations that follow and throughout this course you will encounter negative exponents which may be a new concept for you. They are simply a shorthand for the reciprocal (1/x)

So

- 10
^{2}= 100 - 10
^{-2}= 1/100 = 0.01 - ms
^{2}= meters x seconds^{2} - ms
^{-1 }= meters x (1/seconds) = meters per second - ms
^{-2}= meters x ((1/(seconds) / seconds) = meters /second/second

Propagation in a Medium

**Sound needs something to travel through**

Bell in a vacuum (Video Demo)

**Sound travels in air as a compression wave**

IMPORTANT TERMS: **Compression** and **rarefaction**

**Compression**is the point at which the air molecules are being pushed together and**Rarefaction**is the point at which they are being pulled apart.

It is very common to see sound displayed like this:

which is OK but given the familiarity we all have with waves in the water there is a danger of thinking that the air molecules are going up and down when in fact sound travels as a compression wave with the air molecules going backward and forwards like this.

The text book give the example of golf balls on springs which expresses the reality of a compression wave well.

**Fig 1.1 Golf ball and spring model of a sound propagating material.**

Click on the green button for animation

**Fig 1.2 Golf ball and spring model of a sound pulse propagating in a medium**

This shows a simple one-dimensional model of a physical medium, such as air, which we call the golf ball and spring model because it consists of a series of masses, e.g. golf balls, connected together by springs. The golf balls represent the point masses of the molecules in a real material, and the springs represent the intermolecular forces between them. If the golf ball at the end is pushed toward the others then the spring linking it to the next golf ball will be compressed and will push at the next golf ball in the line which will compress the next spring, and so on. Because of the mass of the golf balls there will be a time lag before they start moving from the action of the connecting springs. This means that the disturbance caused by moving the first golf ball will take some time to travel down to the other end. If the golf ball at the beginning is returned to its original position the whole process just described will happen again, except that the golf balls will be pulled rather than pushed and the connecting springs will have to expand rather than compress. At the end of all this the system will end up with the golf balls having the same average spacing that they had before they were pushed and pulled.

- The region where the golf balls are pushed together is known as a compression
- The region where they are pulled apart is known as a rarefaction
- The golf balls themselves are the propagating medium.

In a real propagating medium, such as air, a disturbance would naturally consist of either a compression followed by a rarefaction or a rarefaction followed by a compression in order to allow the medium to return to its normal state. A picture of what happens is shown in Figure 1.2. Because of the way the disturbance moves-the golf balls are pushed and pulled in the direction of the disturbance's travelthis type of propagation is known as a longitudinal wave. Sound waves are therefore longitudinal waves which propagate via a series of compressions and rarefactions in a medium, usually air.

There is an alternative way that a disturbance could be propagated down the golf ball and spring system. If, instead of being pushed and pulled toward each other, the golf balls were moved from side to side then a lateral disturbance would be propagated, due to the forces exerted by the springs on the golf balls as described earlier. This type of wave is known as a transverse wave and is often found in the vibrations of parts of musical instruments, such as strings or membrane

Longitudinal and transverse waves.

The speed of sound in a solid depends on two factors:

- The mass of the golf balls: the mass affects the speed of disturbance propagation because a golf ball with more mass will take longer to start and stop moving. In real materials the density of the material determines the effective mass of the golf balls. A higher density gives a higher effective mass and so the propagation will travel more slowly
- The strength of the springs: the strength of the springs connecting the golf balls together will also affect the speed of disturbance propagation because a stronger spring will be able to push harder on the next golf ball and so accelerate it faster. In real materials the strength of the springs is equivalent to the elastic modulus of the material, which is also known as the Young's modulus of the material. A higher elastic modulus in the material implies a stiffer spring and therefore a faster speed of disturbance propagation.

The measure of stiffness or elasticity known as Young's modulus is usually expressed as a decimal number multiplied by 10 to the power of some exponent.

To calculate the speed of sound in a material it is only necessary to know the density and the Young's Modulus of the material to arive at a result.

Here are some samples of Young's Moduli.

## Material |
## Young's Modulus (Nm |
## Density (kgm |
## Speed of sound (ms |

Steel | 2.10 x 10^11 | 7800 | 5,189 |

Aluminum | 6.90 x 10^10 | 2720 | 5,037 |

Lead | 1.70 x 10^10 | 11400 | 1,221 |

Glass | 6.00 x 10^10 | 2400 | 5,000 |

Concrete | 3.00 x 10^10 | 2400 | 3,536 |

Water | 2.30 x 10^9 | 1000 | 1,517 |

Air (at 20C) | 1.43 x 10^5 | 1.21 | 344 |

Beech wood(along the grain) | 1.40 x 10^10 | 680 | 4537 |

Beech wood(across the grain) | 8.80 x 10^10 | 680 | 1138 |

For longitudinal waves in solids, the speed of propagation is only affected by the density and Young's modulus of the material and this can be simply calculated from the following equation:

(1.1)

where

- ν = the speed in metres per second (ms
^{-1}) - ρ = the density of the material (in kg m
^{-3}) and *Ε*= the Young's modulus of the material (in N m^{-2})

However, although the density of a solid is independent of the direction of propagation in a solid, the Young's modulus may not be. For example, brass will have a Young's modulus which is independent of direction because it is homogeneous whereas wood will have a different Young's modulus depending on whether it is measured across the grain or with the grain. Thus brass will propagate a disturbance with a velocity which is independent of direction but in wood the velocity will depend on whether the disturbance is travelling with the grain or across it. To make this clearer let us consider an example.

The density of steel is 7800 kg m-3, and its Young's modulus is 2.1 x 1011 N m^{-2}, so the speed of sound in steel is given by:

ν_{steel} = (2.1 x 10^{11})/7800 = 5189 ms^{-1}

The density of beech wood is 680 kg m-3, and its Young's modulus is 14 x 109 N m^{-2} along the grain and 0.88 x 1()9 N m^{-2} across the grain. This means that the speed of sound is different in the two directions and they are given by:

ν_{beech along the grain} =(14 x 10^{9})/680 = 4537 ms^{-1}

and

ν_{beech across the grain} = (0.88 x 10^9)/680 = 1138 ms^{-1}

Thus the speed of sound in beech is four times faster along the grain than across the grain.

This variation of the speed of sound in materials such as wood can affect the acoustics of musical instruments made of wood and has particular implications for the design of loudspeaker cabinets, which are often made of wood. In general, loudspeaker manufacturers choose processed woods, such as plywood or MDF (medium density fibreboard), which have a Young's modulus which is independent of direction.

An interesting application of the speed of sound in solids is testing materials for damage of stress (like bridges and buildings). The speed of sound in concrete is well known so measuring the speed of sound in a concrete bridge will instantly reveal any cracks or decay in the material as a deviation from the normal speed.

So far the speed of sound in solids has been considered. However, sound is more usually considered as something that propagates through air, and for music this is the normal medium for sound propagation. **Unfortunately air does not have a Young's modulus so Equation 1.1 cannot be applied directly,** even though the same mechanisms for sound propagation are involved. Air is springy, as anyone who has held their finger over a bicyde pump and pushed the plunger will tell you, so a means of obtaining something equivalent to Young's modulus for air is required. This can be done by considering the adiabatic, meaning no heat transfer, gas law given by:

(1.2)

where

*P*= the pressure of the gas (in N m^{-2})- V = the volume of the gas (in m
^{-3}) and - γ = is a constant which depends on the gas (1.4 for air)

The adiabatic gas law equation is used because the disturbance moves so quickly that there is no time for heat to transfer from the compressions or rarefactions. Equation 1.2 gives a relationship between the pressure and volume of a gas and this can be used to determine the strength of the air spring, or the equivalent to Young's modulus for air, which is given by:

(1.3)

The density of a gas is given by:

(1.4)

where

- m = the mass of the gas (in kg)
- M = the molecular mass of the gas (in kg mole
^{-1}) - R = the gas constant (8.31 J Kg mole
^{-1}) and - T = the absolute temperature (in K)

Equations 1.3 and 1.4 can be used to give the equation for the speed of sound in air, which is:

(1.5)

Equation 1.5 is important because it shows that the speed of sound in a gas is not affected by pressure. Instead, the speed of sound is strongly affected by the absolute temperature and the molecular weight of the gas. Thus we would expect the speed of sound in a light gas, such as helium, to be faster than that of a heavy gas, such as carbon dioxide, and air to be somewhere in between.

Speed of Sound in a Gas (Video Demo)

For air we can calculate the speed of sound as follows.

The composition of air is 21 % oxygen (02), 78% nitrogen (N2), 1 % argon (Ar), and minute traces of other gases. This gives the molecular weight of air as:

M = 21 % x 16 x 2 + 78% x 14 x 2 + 1 % x 18
= 2.87 x 10-2 kg mole^{-1}

and

- γ = 1.4
- R = 8.31 J K
^{-1}mole^{-1}

which gives the speed of sound as:

ν = (1.4 x 8.31)/(2.87 x 10^{-2}) x *T*

ν = 20.1 x *T*

Thus the speed of sound in air is dependent only on the square root of the absolute temperature, which can be obtained by adding 273 to the Celsius temperature; thus the speed of sound in air at O°C and 20°C is:

ν 0°c = 20.1 V (273 + 0) = 332 ms^{-1}

ν 20°c = 20.1 V (273 + 20) = 344 ms^{-1}

The reason for the increase in the speed of sound as a function of temperature is two fold. Firstly, as shown by Equation 1.4 which describes the density of an ideal gas, as the temperature rises the volume increases and providing the pressure remains constant, the density decreases. Secondly, if the pressure does alter, its effect on the density is compensated for by an increase in the effective Young's modulus for air, as given by Equation 1.3. In fact the dominant factor other than temperature on the speed of sound in a gas is the molecular weight of the gas. This is clearly different if the gas is different from air but the effective molecular weight can also be altered by the presence of water vapour, that is humidity, and this also can alter the speed of sound compared with dry air.

Although the speed of sound in air is proportional to the square root of absolute temperature we can approximate this change over our normal temperature range by the linear equation:

ν = 331.46 + 0.6t ms^{-1}

(1.6)

NOTE: This is misprinted in the book see errata sheet.

where

- t = the temperature of the air in °C

Therefore we can see that sound increases by about 0.6 ms^{-1} for each °C rise in ambient temperature and this can have important consequences for the way in which sound propagates.

- Speed of Sound in Air (Video Demo)

Why should you know how to calculate the speed of sound? Well it was once worth $12,000 to my band. If I may tell a short anecdote: in 1974 my band got booked to play in Youngstown Ohio. For those of you who don't know Youngstown it is the summer watering hole for New York Italians who avoid metal detectors and I got booked into Sabatini's which was where they got their water. The pay was $6,000 a week for two weeks. Sabatini's was immensely long and narrow (1300 feet long) and halfway through the first night Sabatini himself came to me and said "The boys in the back can't hear you turn it up". Happy to oblige I cranked it and at the end of the night Sabatini came to me again and said "The boys in the back can hear OK but you're killing the boys in the front. Fix it." I went out the next morning and rented additional speakers and power amps and positioned them halfway down the club (650 feet from the stage). After setting up the speakers and amps I told the band to hit it and went out and listened hoping to hear something like this:

Normal Band (Audio Demo)

Instead I heard exactly this:

Sabatini Mix (Audio Demo)

Who can tell me what happened?

Lets do the math:

The distance from the stage to the speakers was 650 feet or 200 meters. This meant that the stage sound was taking 581 milliseconds to reach the forward speakers. By purchasing a stereo digital delay, setting it to 100% wet and programming the time to 581 ms I was able to delay the sound to the back speakers by 581 milliseconds. This allowed me to keep the job at Sabatinis and was therefore worth $12,000 to me.

The velocities of transverse vibrations are affected by other factors. For example, the static spring tension will have a significant effect on the acceleration of the golf balls in the golf ball and spring model. If the tension is low then the force which restores the golf balls back to their original position will be lower and so the wave will propagate more slowly than when the tension is higher. Also there are several different possible transverse waves in three-dimensional objects. For example, there are different directions of vibration and in addition there are different forms, depending on whether opposing surfaces are vibrating in similar or contrary motion, such as transverse, torsional and others, see Figure 1.3. As all of these different ways of moving will have different spring constants, and will be affected differently by external factors such as shape, this means that for any shape more complicated than a thin string the velocity of propagation of transverse modes of vibration become extremely complicated. This becomes important when one considers the operation of percussion instruments.

**Fig 1.3 Some different forms of transverse waves **

For transverse waves, calculating the velocity is more complex because for anything larger than a-in principle infinitely thin string the speed is affected by the geometry of the propagating medium and the type of wave, as mentioned earlier. However, the transverse vibration of strings are quite important for a number of musical instruments and the velocity of a transverse wave in a piece of string can be calculated by the following equation:

(1.7)

where

- μ = the mass per unit length (in kg m
^{-1}) and *T*= the tension of the string (in N)

This equation, although it is derived assuming an infinitely thin string, is applicable to most strings that one is likely to meet in practice. But it is applicable only to pure transverse vibration: it does not apply to torsional or other modes of vibration. However, this is the dominant form of vibration for thin strings. Its main error is due to the inherent stiffness in real materials which results in a slight increase in velocity with frequency. This effect does alter the timbre of percussive stringed instruments, like the piano, and gets stronger for thicker pieces of wire. However, Equation 1.7 can be used for most practical purposes.

Let us calculate the speed of a transverse vibration on a steel string.

The mass per unit length is given by:

μ steel = ρ steel (πr^2) = 7800 x (3.14 x (0.8 x 10-^3)/2= 3.92 X 10-3 kg m^{-1}

The speed of the transverse wave is thus:

ν steel transverse = 627/(3.92 x 10-3) = 400 ms^{-1}

This is considerably slower than a longitudinal wave in the same material and generally transverse waves propagate more slowly than longitudinal ones in a given material.

So far we have considered the speed of a single pulse through various materials however if there were repeated pulses we would create a periodic wave called a sine wave.

Sine waves have three parameters

- amplitude
- frequency
- phase

Amplitude is a measure of the maximum pressure created by the wave but is better known to most of us as the volume of a sound. The measurement of amplitude is a complex subject which we will get to later by decibels (or dB) are the most common units of measurement.

[demonstrate **Amplitude** using Signal Scope and Signal Suite]

**Frequency** is how many times the wave repeats a second and is measured in Hz.

[demonstrate **Frequency** using Signal Scope and Signal Suite]

**Phase** is a measure of the starting position of the wave and only has meaning relative to time or another wave. Especially another wave.

[demonstrate **Phase** using Signal Scope and Signal Suite]

Because a wave propagates at a fixed velocity a length can be assigned to the distance between the repeating cycles of compression and rarefaction. This length can be calculated from the following equation:

(1.8)

At this point you are probably asking yourself "What does all this have to do with me and music?"

Good question, let's have a "for instance". You graduate from the College of Santa Fe and seek your fortune in LA or New York where you rent a one bedroom apartment with a large living room for $1200 a month. You set up your MIDI studio in the living room because the bedroom is too small and you set up facing the short wall which is 14 feet away. Because you are broke you apply no acoustic treatment to the walls of the living room. After hitting the bricks you finally get a gig doing a commercial and you put it together with great skill and speed. You stay up mixing it all night and run it down to Ad agency the next morning eager for your first check. However when you play back the CD in the tiny office it sounds terrible, no low end and all midrangey.. Its a disaster and they let you know that they are using someone else and they will call you.

What happened?

Lets do the math. Your studio was 14 feet long or 4.2 meters. Plugging this value into the Frequency = Velocity/wavelength equation we can conclude that your room is same length as a wave of 82Hz...the heart of Bass land. Your room would therefore have been **resonant** to this frequency which means that the low end of your bass guitar and bass drum would have appeared louder to you than they actually were causing you to mix them softer than they needed to be.

The little 6*10 studio at the ad agency had a resonant frequency of 191Hz (the heart of woof land) which would have not only failed to reinforce your bass but would have significantly boosted the lower midrange muddying everything you did.

What do you do? Take this course, pay attention during the "Environment part", play to the long wall, install bass traps, put diffusers on the back wall to break up the standing wave and be aware that the condition exists. Learn to mix the bass a little higher than sounds normal to you and the next time you will keep the gig.

In acoustics the wavelength is often used as the 'ruler' for measuring length, rather than metres, feet or furlongs, because many of the effects of real objects, such as rooms or obstacles, on sound waves are dependent on the wavelength.

Impedence is a word that you will hear everywhere in sounds. Impedence is a measure of resistance to motion through a medium. The impedence of an electric circuit is a measure of how much the circuit resists the passage of electricity and the impedence of an acoustic system is a measure of how much the system resists the passage of sound. This is an important subject in acoustics because a sudden drop in impedence has the ability to reverse an acoustic wave and send it back the way it came. It is this quality of impedence that allows all wind instruments to work. The flare of a bell on a trombone is a way of rapidly lowering the impedence of the tube. This causes the wave to be reflected back up the horn and allows oscillation to take place. Similarly opening a tone hole on a flute causes a rapid drop in impedence which effectively shortens the tube.

Another aspect of a propagating wave to consider is the movement of the molecules in the medium which is carrying it. The wave can be seen as a series of compressions and rarefactions which are travelling through the medium. The force required to effect the displacement, a combination of both compression and acceleration, forms the pressure component of the wave. In order for the compressions and rarefactions to occur, the molecules must move closer together or further apart. Movement implies velocity, so there must be a velocity component which is associated with the displacement component of the sound wave. This behaviour can be observed in the golf ball model for sound propagation described earlier. In order for the golf balls to get closer for compression they have some velocity to move towards each other. This velocity will become zero when the compression has reached its peak, because at this point the molecules will be stationary. Then the golf balls will start moving with a velocity away from each other in order to get to the rarefacted state. Again the velocity between the golf balls will become zero at the trough of the rarefaction. The velocity does not switch instantly from one direction to another, due to the inertia of the molecules involved, instead it accelerates smoothly from a stationary to a moving state and back again. The velocity component reaches its peak in between the compressions and rarefactions, and for a sine wave pressure component the associated velocity component is a cosine. The force required to accelerate the molecules forms the pressure component of the wave. As this is associated with the velocity component of the velocity of the wave it is in phase with it. That is, if the velocity component is a cosine then the pressure component will also be a cosine. Figure 1.6 shows a sine wave propagating in the golf ball model with plots of the associated components.

**Fig 1.6 Pressure, velocity and displacement components of a sine wave propagating in a material**

Therefore a sound wave has both pressure and velocity components that travel through the medium at the same speed.
Pressure is a scalar quantity and therefore has no direction; we talk about pressure at a point and not in a particular direction. Velocity on the other hand must have direction; things move from one position to another. It is the velocity component which gives a sound wave its direction.
The velocity and pressure components of a sound wave are also related to each other in terms of the density and springiness of the propagating medium. A propagating medium which has a low density and weak springs would have a higher amplitude in its velocity component for a given pressure amplitude compared with a medium which is denser and has stronger springs. This relationship can be expressed, for a wave some distance away from the source and any boundaries, using the following equation:

Pressure component amplitude
/ Velocity component amplitude = Constant = Z_{acoustic} = *p /U *

where

- p = the pressure component amplitude
- U = the volume velocity component amplitude and
- Z
_{acoustic}= the acoustic impedance

This constant is known as the acoustic impedance and is analogous to the resistance (or impedance) of an electrical circuit.

The amplitude of the pressure component is a function of the springiness (Young's modulus) of the material and the volume velocity component is a function of the density. This allows us to calculate the acoustic impedance using the Young's modulus and density with the following equation:

Z_{acoustic} = ρ*E*

However the velocity of sound in the medium, usually referred to as *c*, is also dependent on the Young's modulus and density so the above equation is often expressed as:

(1.9)

Note that the acoustic impedance for a wave in free space is also dependent only on the characteristics of the propagating medium. However, if the wave is travelling down a tube whose dimensions are smaller than a wavelength, then the impedance predicted by Equation 1.9 is modified by the tube's area to give:

Z_{acoustic tube} = ρ*c/ S*_{tube}

where *S*_{tube} = the tube area

This means that for bound waves the impedance depends on the surface area within the bounding structure and so will change as the area changes. As we shall see later, changes in impedance can cause reflections. This effect is important in the design and function of many musical instruments as discussed in Chapter 4.

Before next class please read section

- 1.2.n Sound Intensity power and pressure level

pages 14 to 32 of Acoustics and Psychoacoustics.

**Newton:**- Force which will change the velocity of a one kilogram (2.2 lbs) object by one metre per second over a period of one second.
- Symbol: N
- Units: kg sec per sec (kg ms
^{-2})

**Young's Modulus**- A measure of stiffness or the force needed to compress a material
- Symbol:
*E* - Units: Newtons per square meter (Nm
^{2)}

**Pressure**- Mass/area
- Symbol:
*P* - Units: kilograms per square meter (kg m
^{-2})

**Velocity**- A measure of speed
- Symbol: ν (nu) or
*c* - Units: meters per second (ms
^{-1})

**Density**- Mass per unit volume
- Symbol: ρ (rho)
- Units: Kilograms per cubic meter (kg m
^{-3)}

**Cycle**- One period of a wave oscillation

**Frequency**- The number of cycles per second of a periodic wave
- Symbol:
*f* - Units: Hertz (Hz)

**Wavelength**- The length of one cycle of a wave
- Symbol: λ (lambda)
- Units: meters (m)

**Impedence**- Measure of resistance to the passage of sound
- Symbol:
*Z* - Units: Kilograms per sqare meter per second (kg m
^{-2}s^{-1)}