Moulton  Lectures

 

On

Electro-Acoustics

 

 

 

Presented by:

Dave L Moulton

 

 

 

 

 

Location:  Thales Acoustics Harrow UK

Date:  18-February-2004

 

Content

In this lecture I want to show the class the various ways in which the wave equation can be represented.  So far we have only looked at the standard form of a one dimensional wave equation, however in practice it is usual to work in more than one dimension and to also represent the wave equation in an appropriate coordinate system. Two dimensional and Three dimensional forms of the wave equation are more suited to real life situations and the coordinate system is chosen to best describe the environment in which the wave operates. For example the excitation modes produced by a circular membrane are best described and analysed by using a two dimensional wave equation represented  in polar coordinates. Whereas the wave motion inside of a cylindrical tube is best analysed by using a three dimensional wave equation represented in cylindrical coordinates. As a rule of thumb the analysis is made easier if the coordinate system is chosen to best suit the geometry of the wave environment. Due to the mathematical complexity and amount of white board space required, I will only state the solutions to all of the wave equations in this lecture rather than derive them from first principals.

 

In this lecture I want to cover the following:

 

·        Show details of important representations of the wave equation without deriving them from first principals.

·        Two and Three dimensional wave equations in rectangular (Cartesian) coordinates.

·        Two dimensional wave equation in polar coordinates.

·        Three dimensional wave equation in cylindrical coordinates.

·        Three dimensional wave equation in spherical coordinates

·        Simplified wave equation in spherical coordinates after making assumptions about the symmetry of the wave.

 

On With The Lecture

Firstly we will remind ourselves of the format for the one dimensional wave equation.

 

 

Which has solutions of the form:

 

 

 

Equation 25.1 is one dimensional by the fact that we only refer to the x-axis as a means of locating the position of the wave.  If however we had a square membrane in two dimensions and we wanted to analyze its modes of vibration, then we would have to introduce another coordinate, y into the wave equation.

 

 

 

Figure 25.1

 

Thus the two dimensional wave equation in rectangular (Cartesian) coordinates can be described as:

 

 

Where z is the resulting motion of the membrane perpendicular to the xy plane.

 

This two dimensional equation has a solution of the form:

 

 

Where A_i, B_i, C_i, D_i, E_i and F_i are constants.  Notice from this solution there are two modal angular frequencies ω_i and k_i, and a variety of possible coefficients. Applying boundary conditions to this solution will greatly simplify things. A typical boundary condition could be that the square membrane is rigidly fixed along it outer edge, thus at the edges the displacement is always zero.

 

Now let us suppose that we are trying to analyze the motion of a circular membrane with radial component r and angular component θ. See figure 25.2 below:

 

Figure 25.2

 

In this case we take our two dimensional wave equation [25.2] and make the polar coordinate substitutions for x and y.

 

and

 

 

The substitution results in the polar coordinate form of the two dimensional wave equation.

 

 

Again z is the motion of the membrane perpendicular to the xy axis.

 

In the case of a circular membrane this equation can be simplified by considering that the radial component r is independent of angle θ. Thus :

 

 

Which has a solution of the form:

 

 

In this case J_o is a zero order Bessel Function with solutions of the first kind. The indices indicate that there are many solutions to this equation, thus many modes of vibration.

 

Now let us consider the motion of a wave confined to a cylindrical tube, where we are interested in determining the pressure at any given point within the tube. In this case the tube represents a three dimensional space having a circular cross section of radius r and a length z. Thus we can use polar coordinates to describe a point anywhere on the circular base of the tube and the z coordinate to locate the pressure a some point directly above the given base coordinate.

 

Figure 25.3

 

The wave equation in polar coordinates can be simply modified to become the wave equation in cylindrical coordinates as follows:

 

 

The solution to this equation takes the form:

 

 

Where J_ω3 is Bessel function of the first kind with order ω₃ and its associated Bessel solutions, the A’s and B’s are constants.

 

Now let us consider the 3 dimensional wave equation in spherical coordinates. First we must recognize that the 3 dimensional wave equation in rectangular (Cartesian) coordinates is simply an extension of the two dimensional wave equation shown in [25.2], except we now have to use the z axis as our 3^rd coordinate, thus:

 

 

If we now consider the spherical coordinate system in figure 25.5, we see that the coordinate transformations required to convert [25.12] into spherical coordinates are:

 

and

 

 

Figure 25.4

 

The resulting Spherical Wave Equation takes the form:

 

 

This equation has quite a long and complex solution. However if we consider the simple example of a small sphere with a radius that can change in a steady state sinusoidal manner, then we have a spherical source of sound waves.  The symmetry of our spherical source will allow us to simplify equation [25.16]. Thus the pressure will not be dependant upon the angular parts of the coordinate system, so all the terms which indicate a rate of change with angle will reduce to zero.  Thus:

 

 

Now this simplified version of the wave equation (only applies to perfectly spherical source) can be further simplified to:

 

 

expression 25.18 can be easily proved by making use of the product rule for differentiation. A little more manipulation leads to:

 

 

Which is a comparable to the format of our original wave equation depicted in [25.1].  Thus it is reasonable to expect the solution to take the form:

 

 

 

Thus dividing by r gives the more appropriate format:

 

 

So for a spherical wave the pressure amplitude reduces at a rate of 1/r as the wave propagates further away from the symmetrical point source.  Equation 25.21 is a general equation and describes the wave propagation up to an infinite distance from the source.

 

You may be wondering what is happening here, since in previous lectures I have said that at significant distances from the source the wave behaves as a plane wave and obeys the solutions shown in equation [25.2]. Nothing has changed, the plane wave solution is a special case of the more general spherical wave solution. Equation [25.2] is only valid when comparing the amplitude of the wave front between two close points along the radial axis, providing that those points are at a significant distance from the source. Under this condition the separation distance between the two points is very small compared to the radial distance of each point from the source. Thus the difference in pressure amplitude at both points is insignificant. Thus, over very short separations the amplitude can be considered constant.

 

Figure 25.5

 

The significance of the spherical and planar wave equation solutions will become more clear during my future lecture on differential noise cancelling microphones.

 

 

 

 

End of Lecture