Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  26-November-2003

 

Content

 

In this Lecture I want to extend our understanding of wave mechanics. To do this I want to look just a little deeper into the one dimensional wave equation and show some of the important parameters that form the heart of the equation. In doing so I will introduce the idea of Mass Conservation and the Continuity Equation, I will attempt to explain these using a simplistic rather than very rigorous mathematical approach.  From the exponential form of the solution to the wave equation I will show how simple acoustic parameters like Characteristic Acoustic Impedance of air are derived. I will use the following steps during this lecture:

 

·        Explain the effect of a passing wave on a fixed Mass of air.

·        Show the relevance of the Conservation of Mass.

·        Derive the Continuity equation as a forerunner to a later lecture

·        Introduce the Adiabatic Bulk Modulus and the Taylor series.

·        Show the parameters that affect the speed of sound.

·        Derive an expression for the Characteristic Acoustic Impedance of air

 

 

 

On with the lecture

 

Let us consider a homogeneous region of air, that is to say the air density is the same throughout every point within the region. If we were to consider a cube of air within that region then we can quite rightly deduce that the density of air ‘r’ within the cube is the same as the density outside of the cube. Also we can calculate the mass of air within the cube simply from the cube dimensions, width ‘Dx’ ,cross sectional area ‘A’ and air density ‘r’. Figure 21.1 shows the situation.

Figure 21.1

Now let us express the mass of air within the cube as follows:

 

 

Let us now assume that our cube is in the path of a traveling sound wave moving in the positive x direction. I will show the wave in its exponential form describing an instantaneous particle displacement ‘u’

 

Figure 21.2

 

As the wave energy passes through our cube the effect will be to expand and contract the cube volume. This is as a result of the air in the cube experiencing the pressure Compressions and Rarefactions caused by the moving sound wave.  If we consider the case where the cube is experiencing a rarefaction then we can consider its dimension along the +ve x direction to increase by an amount ‘Du’, alternatively when it experiences a compression the dimension will decrease by an amount ‘Du’.  Both cases are shown in figure 21.3 below:

 

Figure 21.3

 

Now let us first consider the case where the cube volume is increased by the rarefaction. In this case we can express the mass of air within the expanded cube as:

 

 

Now I have some explaining to do. You may be wondering why I have used the term (r + Dr) when clearly the cube is expanding and a rarefaction is associated with a region of low pressure and hence low air density. You would be right to wonder this. However, we are dealing here with mathematical convention and calculus. The fact that I have made Dr positive and not negative is to ensure that all of my increments are moving in the same direction. Thus if mass is a function of r and x, usually written M(r,x) then M+DM must be a function of (r + Dr) and (x + Dx). Likewise M-DM must be a function of (r - Dr) and (x - Dx). These are simple rules of calculus and they do no contradict the validity of the system, note in this case I have substituted Du for Dx. Depending on what I equate these functions to, the end result will be that Dr and Du will be forced to take up the appropriate sign. So in the case of a compression I can write:

 

 

Now for the crucial bit. Without going into the realms of continuum mechanics it is important to accept that as the cube is contracted and expanded its total mass remains the same as it was before the wave passed through it. Thus the mass in each of the cubes shown in figures 21.2 and 21.4 are the same. This is known as the Conservation of Mass.

 

Let us first consider applying the conservation of Mass to equation [21.2]. To do this we simply equate [21.1] and [21.2].

 

Expanding:

 

Rearranging:

Dividing by Dx

 

Taking the limit as Dx, Du and Dr tend towards zero. We can write the equation in its partial derivative form and let the last term in [21.7] go to zero. I am adopting the partial derivative symbol because we are now dealing with functions of more than one variable.

 

This equation makes sense because it clearly indicates that as the volume increases the density reduces, hence the negative sign.  Hopefully this can be seen to validate the argument about the incremental sign changes all having to be in the same direction. To prove that I am not bluffing I will apply the conservation of mass to the case where there is a reduced volume. Thus equating equations [21.1] and [21.3] we have:

 

Expanding:

Rearranging:

Dividing by Dx

Taking the incremental limits to tend towards zero and neglecting the last term results in:

Which is exactly the same as equation [21.8]. Thus wether or not the volume is contracting or expanding the end result is the same.

 

I would now like to make a slight detour into the world of continuum mechanics and look at what equation [21.13] represents. Firstly let us look at an expression for the rate of change of density within the cube, thus:

 

Now we can define the particle velocity as:

Thus [21.4] can be written as:

and more commonly as:

 

 

This is known as the Continuity Equation in one dimension. This is one of the most important equations in wave acoustics. It has a formal description as:

The conservation of the mass of a body of moving air, relating the net mass flux into a fixed volume to the rate of accumulation of air mass within the volume.

 

I suggest you don’t get to bogged down in the meaning of this definition. I will explain it more clearly in another lecture. The only reason that I have introduced the continuity equation at this stage is because it is fundamental to the laws of physics and acoustic wave motion.

 

We will now move on and look at the forces that are being exerted on our cube of air when the wave energy passes through it. Now we know that the cube has mass and that the mass is displaced by the expansion or contraction of the tube. In our case the displacement can be described as Du. Thus in order for the whole mass to remain constant and also be displaced, there must be a force exerted on the mass of the cube. This force can be easily described as:

But these forces must be the result of the pressure changes across the width of the cube caused by the changing pressure introduced by the moving wave. Figure 21.4 shows the pressure difference at either end of the cube caused by the traveling wave.

 

Figure 21.4

 

Thus we can write the force equation as balancing the inertial forces with the pressure forces:

Simplifying:

 

Now we must refer back to Lecture 2 where I introduced the Adiabatic Bulk Modulus. Let us remind ourselves of equation [2.2] from lecture 2. This expression defines the change in pressure due to a change in Volume.

 

 

In our case we are dealing with pressure changes on two opposite faces of a cube. I can rearrange this expression in terms of x and u as follows.

 

Resulting in:

Now DP is a function of Du and Dx. Using the partial notation I can write the force on face abcd as:

What about the force on the face efgh. To find an expression for this force we need to take time out and remind ourselves of the Taylor series, the sort of thing most people are introduced to during their A-level mathematics course.  The Taylor series is an excellent way of predicting the nature of a function at some small displacement from a datum point.  Thus if I have a function of x, f(x), and I now its derivatives at x, then I can interpolate the the value of the function at x+Dx by using the Taylor series.

 

Thus:

So if I apply the Taylor series to interpolate the Force on face efgh we get:

 

Normally it is appropriate to just go to the second term and not bother with the higher order derivative terms. Thus:

 

 So using equation [21.24] we can write:

 

Simplifying:

Now applying expression [21.23] we get:

We can now re-express our Force Balance equation [21.20] as:

Leading to:

Simplifying:

 

 

You should recognize [21.33] as the wave equation. It has the same form as the wave equation derived in lecture 20. Shown again below in [21.34]

 

 

The important point to note here is that we now have a feel for the factors that affect the speed of wave propagation ‘c’.  From [21.33] and [21.34] we can deduce that:

Thus:

Indicating that the speed of wave propagation (speed of sound) is directly proportional to the adiabatic bulk modulus of the medium and inversely proportional to the medium density.  The table below shows some calculated wave speeds in different mediums.

 

Medium	Bulk Modulus Kgm-1s-2	Density kgm-3	Wave Speed ms-1
Air	141,032	1.22	340
Water	2,239,515,992	998	1498
Hydrogen	148,379	0.09	1284
Glass	65,000,000,000	2600	5000
Brass	104,125,000,000	8500	3500
Mercury	28,651,851,360	13590	1452

 

From Lecture 2 we also deduced that the adiabatic bulk modulus is the product of the Atmospheric pressure and the ratio of the specific heat capacities for constant pressure and constant volume. Thus:

 

 

Characteristic Acoustic Impedance Of Air (Za)

We will now look at using the solution to the wave equation to establish the Characteristic Acoustic Impedance of Air. Firstly we must go back to our analogies and see that the Impedance to a plane sound wave (amplitude remains constant) is the ratio of the instantaneous pressure to the particle velocity. Thus

 

 

Now from the exponential solution to the wave equation we can describe a wave moving through a medium in the positive x direction as:

Thus we can express the particle velocity as:

We know from equation 21.23 the expression for the change in pressure and we can calculate the Strain as:

Thus:

Thus our expression for the Characteristic impedance of air is:

 

 

Using the definition for k and B in terms of density and speed we can simplify equation [21.43] to the following:

 

Thus the Characteristic Acoustic impedance of air is simply expressed as:

In fact this expression represents the characteristic impedance of any gaseous medium. For Air with density 1.22kgm^-3  and speed of sound 340ms^-1, the characteristic impedance is 414.8 Rayls.

 

End Of Lecture