Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  10-December-2003

 

Content

 

In this Lecture I want to continue playing with the wave equation, showing how it can be represented in terms of pressure and particle velocity.  As an example I will go back to the fundamental relationships describing Continuity, Mass Conservation and Momentum Conservation. From these relationships I will show how relatively easy it is to re-express the wave equation in terms of pressure and particle velocity.  I will then move on and explain the difference between Planar and Spherical waves and show the solution to the one dimensional Spherical wave equation. My derivation of the wave equation in terms of pressure will yield some important relationships that reveal important parameter dependencies describing the nature of the wave itself.  Thus my approach will be as follows:

 

·        Show the three versions of the wave equation in terms of Particle Displacement, Pressure and Particle Velocity

 

·        Use a compressed air situation to derive the wave equation. Introducing particle velocity and wave speed into the relationships.

 

·        Explain what is meant by a planar wave.

 

·        Explain what is meant by a Spherical wave showing its solution in one dimension.

 

 

On with the lecture

 

I will start by reminding the class of the wave equation in terms of particle displacement u and also show the solution for a wave travelling in the positive x direction.

 

 

Equation [22.1] can be considered as a general form of the wave equation. In fact this form can also be used to directly describe the wave as pressure function p(x,t) or a particle velocity function v(x,t).  These are shown with their forward wave solutions below in equations [22.2] and [22.3].

 

 

As one can see, all three equations are very similar. Usually in electro-acoustics the wave equation is described in terms of pressure rather than particle displacement or particle velocity. There is no hard set of rule as to which form the wave equations should take, however it is very important that the chosen wave function p, u  or v should be kept throughout any analysis and any changes between functions should be treated carefully. Thus if one is working in terms of a wave equation describing pressure, then it is good practice not to try and convert to a velocity or displacement system, otherwise things can get complicated and confusing.

 

In order to give the class confidence that the wave equation can be written in terms of pressure, I will perform the necessary derivation and at the same time reveal some important acoustic relationships.

 

First let us consider the situation where the movement of a piston creates an area of compressed air. We will let the piston velocity be v and the wave speed be c. Diagram 22.1 shows the situation. The upper tube is filled with air at normal atmospheric pressure with density r and pressure p. In this case the piston is at rest. The lower tube shows the effect of the piston moving forward a distance Dvt. Where Dv is an incremental change in piston velocity and t is time. Notice that as the air in front of the piston gets compressed the air ahead of it propagates away at the speed of sound c.  I have chosen the length of both tubes to be ct. To avoid any confusion I am using the simple relationship between distance, velocity and time to describe the length dimensions of the tube and piston displacement. ‘A’ represents the cross sectional area of the tubes. The delta symbol represents the incremental change function. Thus as the air is compressed the density is increased and thus the pressure in the compressed region is also increased.

 

Figure 22.1

 

We can now set to work analyzing the above situation.  Firstly we can form an expression for the mass of air in the upper tube.  Thus

 

 

We know that the same mass of air exist in the lower tube , except that it is now compressed into a much smaller volume. Thus:

 

 

We can now apply the conservation of Mass and equate these two equations. Thus:

 

 

A little rearranging results in:

 

 

Thus:

Leading to:

 

 

For normal sound waves (Linear) we can make the assumption that r>>Dr leading to:

 

 

Equation [22.10] is very important as it describes the relationship between the incremental change in particle velocity to the incremental change in density.

 

We can now move on to looking at the momentum of the air particles under going compression. For a start we know that the total mass of air is given in equation [22.4]. We also know that all of the mass of air has been moved or compressed into a much smaller volume . The instantaneous momentum can be described as the total mass moved with an instantaneous velocity Dv.  Thus using [22.4] we can write:

 

 

We can now describe the instantaneous change in pressure as a result of the compressed air, by looking at the rate of change of momentum (Force) and dividing it by the area ‘A’. Thus

 

Leading to:

 

Letting the delta limit tend to zero, we get:

 

 

This is a very important equation which relates the pressure to the particle velocity. Notice that the rate of change of pressure with respect to the velocity is the wave impedance. Which we covered in lecture 21 (ref equation 21.45)

 

 

Let us now have another look at what is happening to the pressures inside the compressed region. This time we will introduce an x dimension to the system. Just like figure 21.4 in lecture 21, but using a wave described in terms of pressure. Thus

 

Figure 22.2

 

We have already seen from lecture 21 in equation [21.19] that the forces across the cube can be expressed as:

 

 

We can now treat this equation differently to that of lecture 21 and simply form the differential relationship between particle velocity and pressure. We start by simplifying [21.16] as follows:

 

 

We now divide by Dx resulting in:

 

 

If we now take the delta limits to zero we can write the RHS of equation [21.18] as a negative differential. [22.18] is basically now in the definition form of a fundamental differential. Thus.

 

 

[22.19] is a very important expression and is known in acoustics as the Momentum equation.

 

In order to derive our wave equation in terms of pressure we will need that other very important expression, the Continuity Equation.  As a reminder here it is:

 

We now have all the ingredients to formulate the wave equation. Firstly we can differentiate [22.19] w.r.t x, Thus:

 

 

We can now differentiate [22.20] w.r.t t Thus:

 

 

We can simply rearrange [22.22] so that is looks very similar to [22.21]. Thus

 

Equating [22.21] and [22.23] leads to:

 

 

We are nearly there, all we have to do is convert the density in terms of pressure, which can be done by using the relationship in [22.10], thus

 

 

Taking the delta limit to zero we can write:

Substituting into [22.4] gives the desired form of the wave equation. Thus:

 

 

Making the substitution for velocity in  [22.27] defined by equation [22.10] leads directly to the wave equation being expressed in terms of particle velocity, The link being the impedance factor  ¶p = rc¶v

 

 

That is enough of deriving one dimensional wave equations. I will now move onto something different and describe the two main acoustic wave types, namely the plane wave and the spherical wave. I will not go into any significant detail at this stage on the spherical wave equation, but just state its solution.

 

Let us go back to the general solution of the wave equation in terms of pressure, we will consider the case for a sinusoidal wave moving in the positive x direction. Thus:

 

 

Where P_f is the peak pressure amplitude.

 

Plane Wave

A wave which can be considered to have a constant peak amplitude is known as a plane wave, thus [22.29] describes a plane wave in one dimension.

 

Spherical Wave

A wave which has a peak amplitude which reduces with distance from the sound source can be described as a spherical wave.  Thus:

 

 

[22.30] is a solution to the one dimensional spherical wave equation and shows the peak amplitude decreasing as the distance x from the source increases.

 

A pictorial representation of plane and spherical waves.

Consider a point source of sound, or better described as a very small sphere whose volume changes in a sinusoidal manner. This results in a spherical pattern of acoustic compressions and rarefactions emanating from the sphere. As one moves further away from the sphere the pressure curvature of the wavefront becomes less pronounced. For example consider the case where small microphones are located at points ‘A, B, C and D’. At  point ‘A’ a distance x₁ from the centre of the source , the wave front appears very curved compared to the dimensions of the microphone at ‘A’.  Whereas at point ‘C’ a distance x₃ from the source the wave front appears less curved relative to the dimensions of the microphone.

 

Figure 22.3

As the distance increases from the source the wave front becomes less and less spherical relative to the small dimensions of a microphone say. In fact the wavefront starts to look very straight or plane and shows very little signs of curvature. 

 

Thus we can consider that wavefronts near to the source tend to be Spherical while wave fronts that have propagated a significant distance from the source tend to be Planar.

 

Spherical Surface Energy Density

Now to explain why planar wave fronts tend to have constant amplitude and spherical ones have a reducing amplitude.  Consider that the point source shown in figure 22.3 has an Energy E.

 

Thus the Energy Density E_D  (Energy per unit surface area) of a sphere of radius x₁ is:

 

Also remembering that the Intensity I is equal to the rate of change of energy density. Thus

 

Where W is the total wave power at x₁ and z is the wave impedance. Thus the sound pressure level at x₁ is:

 

Here P_x1  is the actual wave pressure at x₁.

 

Where P₀  is the reference sound pressure, also known as the pressure at the threshold of hearing (P₀ = 20mPa).  These principals should be familiar from Lecture 3.

 

Likewise we can write the sound pressure levels at x₃ and x₄  as

 

and:

 

 

In each case as x increases  P_x reduces and so the sound pressure level reduces as one moves further away from the source. If the separation along the x-axis is such that the change in sound pressure level is significant then the wave affect can be considered to to be spherical.  However consider the case where the separation along the x axis is very small and far enough away from the source, such that the difference in measured sound pressure levels is very small or even insignificant.  Then the wave can be considered to be planar.

 

An example of this is the measurement of the nearfield and far field responses of a microphone. Consider a differential microphone (front and back of diaphragm exposed to the pressure field), where the width of the microphone is 10mm, say.  In the near field the microphone would typically be positioned between 10mm and 15mm from a sound source. Here the generated wave front is very spherical and the wave difference in wave intensities between the front and back of the microphone and is significant. In this situation the microphone is being subjected to a spherical wave (Near field measurements are associated with spherical waves).  If we now locate the microphone a significant distance from the source say 1000mm, then the difference in wave intensity between the front and back of the microphone will be insignificant. It is almost like the wave has traveled the distance of the relatively short the width of the microphone without any insignificant change in peak pressure amplitude. Over such short distances the wave can be considered to have a constant amplitude and appear planar. Thus far field measurements are associated with planar waves.

 

As a summary:

 

·        Spherical waves occur in the Nearfield.

·        Spherical Waves become more planar as one move further away from the source.

·        Spherical waves can be considered planar if there is an insignificant change in intensity between two points along the axis of the wave propagation.

·        Planar waves occur in the Farfield.

·        Planar Waves can be considered to have a constant peak amplitude.

·        Spherical waves have an amplitude that decreases as one moves further from the source.

 

There are a lot more interesting differences between these two wave types, but that will be covered in another lecture.

 

 

 

 

 

 

 

 

End Of Lecture