Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by: Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  09-October-2002

 

Content

 

This second lecture explores my favourite and probably most useful acoustic relationship. 

 

Audio acoustics is fundamentally about the compression and expansion of air when under the influence of an alternating pressure field.

 

 

Change in Pressure  = -(Bulk Modulus) x (Change in Volume)¸(Static Volume)

 

 

This relationship shows how a change in pressure ΔP can cause a change in volume ΔV.  The negative sign indicates that the Pressure increases as the Volume decreases.

 

The Constant B is known as the Adiabatic Bulk Modulus. The significance of the Adiabatic nature of sound will be discussed later during this lecture.

 

On with the Lecture

 

In order to understand Equation [2.0] I will explain the acoustic system alongside a more familiar mechanical system.

 

The Acoustic system will be represented as an open cavity filled with air having a static density ρ and static volume V. The length of the cavity will be denoted as x. 

 

The mechanical system will comprise an ideal spring with one end rigidly attached and the other connected to a circular plate of area A, we will assume the plate to have zero mass.

 

The spring will perfectly obey Hooke’s law and have a stiffness constant K.

The rest length of the spring will be denoted as x. 

 

 

 

 

Case 1

Both the spring and cavity systems are shown below in figure (2.1) in their equilibrium states .

 

Note:  By equilibrium I mean that the Pressure and air Density is the same everywhere, so no external forces are acting on either system.

 

 

( Both systems at rest )

figure (2.1)    

 

Case 2

Now let us consider what happens when a positive pressure change occurs at the front of the acoustic cavity and at the front face of the mechanical spring plate.  In the case of the spring we will assume that the pressure on the back of the plate remains unchanged.

 

Figure (2.2) shows the situation:

 

( Both systems exposed to a positive pressure change )

figure (2.2)    

 

 

In the case of the spring system the positive pressure produces a force F on the plate which causes the spring to be compressed through as distance Δx.

The spring produces an equal and opposite restoring force F_r which is directly proportional to Δx.

 

Thus:

 

We get a very similar affect in the acoustic system, whereby the positive pressure causes the volume of air inside the cavity to be reduced by an amount ΔV and the density of air to increase by an amount Δρ.  The pressure inside the cavity will increase by an amount ΔP to balance the external pressure.  Basically re showing equation [2.0]

 

 

Case 3

Now let us consider what happens when a negative pressure change occurs at the front of the acoustic cavity and at the front face of the mechanical spring plate

 

Figure (2.3) shows the situation:

 

 

( Both systems exposed to a negative pressure change )

figure (2.3)  

 

Be sure to understand the significance of the negative sign in front of the  Bulk Modulus B. In the case of a volume and displacement change the pressure always goes in the opposite direction.  This means that as ΔP becomes more positive, ΔV and Δx become more negative (i.e reduce in value), likewise as ΔP becomes more negative ΔV and Δx become more positive (i.e increase in value). Only the density Δρ has the same polarity as ΔP.

 

Conservation of Mass

 

You will note from the examples given in cases 1,2 and 3 that the Mass of air in the cavity does not change, even when the volume and density is changing.  This is an example of the Conservation of Mass which is a very important concept and is often used as the basis for deriving acoustic relationships.

 

Acoustic Stiffness and Compliance

 

If we compare equations [2.1] and [2.2] we can see that the mechanical stiffness K is directly analogous to the acoustic stiffness as follows:

 

 

Where  B/V  is the acoustics stiffness of the air in the cavity.

Normally in acoustics we talk in terms of compliance which is basically the inverse of stiffness.  Thus the acoustic compliance C_a of the cavity is given by

 

 

 

What do we mean by Adiabatic Bulk Modulus ‘B’  ?

 

Now on to some of the more hidden detail contained within equation [2.0].  The value B is known as the adiabatic bulk modulus, it has different values depending on the type of gas used as the acoustic transmission medium.

 

The Bulk Modulus is a measure of the rigidity of the gas. Rigidity is the ratio Stress (Pressure) divide by Strain (dimensional change ratio).

 

To show this more clearly we need to re-write equation [2.0] as follows:

 

 

In the case of the mechanics of solid objects the equivalent relationship is the Youngs Modulus E.

 

Example of Bulk Modulus at 25°C

 

B_air = 1.49x10⁵ Pa,    B_water = 2.2x10⁹ Pa

 

Note from equation [2.4] for equivalent volumes the Compliance of air is a lot greater than that of water by a factor of :

 

 

Now on to explaining the Adiabatic process and how it relates to Acoustics.

 

To do this we need to go back to the First Law of Thermal Dynamics, which is basically a statement of the law of the conservation of energy. 

 

Heat Energy Added to a system = increase in internal energy + work done on the system

 

This can be expressed mathematically as:

 

Where:

 

DQ =Heat added to the system.

DU =Internal energy gained by the system (Results in Temperature increase)

DW =Work done by the system.

 

Definition of an Adiabatic Process.

Below is the definition according to the Oxford dictionary of Physics.

 

Any Process that occurs without heat entering or leaving the system. In general, an adiabatic change involves a fall or rise in temperature of the system. For example, if a gas expands under adiabatic conditions, its temperature falls (work is done against the retreating walls of the container). If a gas contracts under adiabatic conditions , its temperature rises (work is done by the advancing walls of the system).

 

The above definition clearly indicates that for an adiabatic process the external energy DQ is removed from the system:

 

Thus an adiabatic process can be described as:

 

 

Which can be re-expressed as:

 

 

My own understanding of this process is that for a dynamic system such as a sound wave to exist, it must have been given some initial energy and then become detached from that energy source. The detached energy is then able to propagate away from its source and sustain itself by means of the adiabatic process. This implies that for an ideal adiabatic process the energy contained in a sound wave is never lost from the wave but simply alternates between a state of internal energy and work energy.  No heat energy is ever lost from the wave.

 

The example would be a loudspeaker connected to a power source. The process steps are as follows.

1.                 The  Power source gives electrical energy to the loudspeaker

 

2.                 The electrical energy is converted into the mechanical energy needed to move the loudspeaker diaphragm against the mass and stiffness load presented by the air on front of the diaphragm.

 

3.                 The Mechanical movement of the diaphragm compresses and expands the localised volume of air in front of it, as well as pushing the resulting high and low pressure field away from the diaphragm

 

4.                 The result is that a sound wave propagates away from the loudspeaker diaphragm.

 

5.                 The important point to note is that each cycle of the sound wave was created by the external energy DQ  given to it by the moving diaphragm. As far as that cycle (Compressed and uncompressed air) is concerned the energy was only given to it  at its moment of creation, and then removed as the  wave became detached from its energy source, and propagated.

 

6.                 Once detached from the energy source the wave motion can only be sustained if the initial energy remains trapped within the wave as Internal energy DU and Work DW.

 

7.                 An amount of energy will be lost from the wave if it is given a surface to act upon.  For example a microphone diaphragm will only move if the acoustic wave pressure exerts a force on the diaphragm area and causes it to move. The action of force and displacement is due to the energy released from the wave.

 

8.                 In real life there are many obstacles and moveable surfaces that the wave pressure will act against causing a transfer  of energy from the wave to the surface.

 

9.                 In a perfectly homogenous gas medium with no obstacles or surfaces the wave would propagate for ever without losing any energy.

 

 

I will now show how the Bulk Modulus ‘B’ is connected to the Adiabatic process.  The steps in my proof are as follows:

 

Firstly I must start with the First Law of Thermal Dynamics as applied to an Adiabatic system. Thus we will use equation [2.9].

 

I will now re-express the work done by the gas as a function of Pressure P and volume change DV.  We know from the previous lecture that acoustic work energy can be written as:

 

 

Now from your days of O-level or GCSE physics you should know that the amount of Internal heat energy DU required to increase the temperature of a mass m of gas by DT is related to its Specific Heat Capacity c_v  in the following way.

 

 

 

Note in this case we are using the specific heat capacity for constant volume c_v , Since we have kept P constant and changed the volume from an initial constant V to V+DV.

 

We can now relate equations [2.9], [2.10] and [2.11] giving:

 

 

I now need to introduce the ideal Gas Law (Should have been covered in A-level Physics), thus.

 

 

This law simply relates Pressure P, volume V and Temperature T for any ideal thermal system.  M is the weight of a kilo-mole, m is the actual Mass of gas and R is the Molar Gas Constant (R= 8314.510 JK^-1kmol^-1).

 

Note: As a reminder,  1 mol = number of atoms in 12g of Carbon C12. This quantity is 6.022x10²³ also known as Avogadro’s Number.

 

In the case of an Isothermal Process the temperature remains fixed, so it is easy to see that  PV=constant.

 

However, for an Adiabatic process it is possible for P, V and T to all change together, making the present form of the gas law difficult to work with. So with a slight bit of mathematical manipulation I will make it more user friendly for an Adiabatic process.

 

Here we go….

 

Substituting for P in equations [2.12] and [2.13] we get:

 

 

A little more tidying up gives:

 

 

I will now introduce another simple equation which can be easily derived by considering the fist law of thermal dynamics, the ideal gas equation  and what happens to a system under constant pressure.  I will leave it to the class to work this one out for themselves.

 

 

Substituting into [2.15] we get.

 

 

Dividing by c_v :

 

Writing the ratio of specific heat capacities as g and taking the limit DV®0 and DT®0 we can rewrite [2.18] as follows:

 

 

Evaluating the Integrals:

 

 

Thus:

 

                                               

Thus:

 

This is the first Adiabatic expression relating Temperature and Volume.  By substituting for T in equations [2.13] and [2.22] it can be very easily shown that:

                            

 

Now we are getting somewhere, The reason for expressing the adiabatic equation in terms of (P and V) and not (T and V) is that my fundamental equation [2.0] is in terms of pressure and volume.

 

I will now show that the Bulk Modulus B is in fact derived from the Adiabatic equation [2.23]. to do this I simply take the differential of [2.23]. This may sound very complicated, but it is in fact just a case of applying the differentiation product rule to [2.23].  The fact that the equation equals a constant means that I do not have to concern myself with what the differential is with respect to.

 

Thus:

 

Dividing through by V^(g-1) we have:

 

 

With a little more manipulation I can get equation [2.25] to take the same form as [2.0].

 

 

On comparing [2.0] and [2.26]:

 

 

Thus it should now be clear that the Adiabatic Bulk Modulus B is in fact directly related to the Adiabatic relationship in [2.23].

 

 

 

End of Lecture