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Electro-Acoustics

Lecture 23

Fundamental Wave Acoustics 4

Presented by:

Dave L Moulton

Location: Thales Acoustics Harrow UK

Date: 15-January-2004

Content

In lecture 22 I showed the class how the wave equation could be represented in three ways, namely in terms of displacement, pressure or particle velocity. In this lecture I want to show how the wave equation can be put to use in helping to determine the affect of a wave travelling from one medium of density r₁ to another with density r₂. I will attempt to derive expressions for the reflection and transmission coefficients at the boundary. These coefficients have direct significance to the design and understanding of sound barriers and screens for noise proofing. I will derive each coefficient for each of the three cases of wave equation (pressure, displacement and velocity). Towards the end of the lecture I shall briefly introduce the special case of a standing wave, and explain some of its properties. In summary my lecture will cover the following:

· Define Reflection (a_R) and Transmission (a_T) coefficients.

· Derive expressions for a_R , a_T in terms of the medium density and speed of sound.

· Define Power Reflection and Transmission Coefficients.

· Define Standing Wave Ratio (SWR).

· Explain the conditions for a standing wave and explain some of its properties.

On With The Lecture

Consider the two mediums shown in figure 23.1. If we assume that both mediums are different and have consistent densities r₁ and r₂ , thus both mediums can be considered homogeneous.

Figure 23.1

For the purposes of our analysis to keep things simple we can consider the boundary where the two mediums come into contact to be perfectly flat, that is to say there is no material migration or fusion from one medium into the other. We can set up an x-coordinate axis such that the positive direction of x is from left to right, and the boundary between the mediums occurs at x=0.

For this analysis we will consider the wave to be planar rather than spherical.

Let us now consider the affect of the boundary on each of the three wave parameter variables (Pressure, Displacement and Velocity). We will consider each case individually, starting with Pressure.

Case 1: Pressure

As a reminder the one dimensional wave equation in terms of pressure is given by:

With solutions:

Where p_F is the magnitude of the forward going component of the wave moving in the positive x direction, and p_R is the reverse component moving in the negative x direction.

Let us now consider a pressure wave travelling in medium 1 towards the boundary. We can consider this forward wave as being called an incident wave with peak pressure amplitude p_i. When the wave hits the boundary some of the wave energy will be reflected backwards resulting in a reflected wave with peak pressure amplitude p_r. The remaining energy will result in a wave being transmitted through into medium 2 with peak amplitude pressure p_t. Figure 23.2 illustrates the case. Since both mediums have different densities then the wave number k will also be different for the wave propagating in the mediums, this is due to the fact that the wave speeds will be different in the two mediums.

Figure 22.2

The Pressure at any point within medium 1 can be considered to be the sum of the incident pressure and the reflected pressures as described by the solutions to the wave equation, thus:

Note the use of k₁ for the wave propagating in medium 1. We can consider the wave propagating in medium 2 to be represented as:

This time we use k₂ as the wave number.

We are now in a position to define the reflection coefficient a_RP:

Which is the ratio of the reflected pressure amplitude to the Incident Pressure amplitude.

Similarly the transmission coefficient a_TP is defined as:

This is the ratio of the transmitted pressure amplitude to the incident pressure amplitude.

In order to derive expressions for the coefficients in terms of medium parameters such as density and wave speed, we need to find a condition that uniquely defines the pressure in both mediums. In this case we have such a condition at the boundary. If we consider the boundary to be stationary then the forces acting either side it must be the same. Since these forces are acting over the same effective area, the pressures either side of the boundary must also be the same. We can consider the boundary to be infinitely thin, thus the forces at any given moment in time can be defined at the boundary. In this case we have a unique relationship regardless of time.

Thus at x = 0,we have:

Thus:

Putting x=0 gives:

We can also form another unique relationship by considering the particle velocities of the wave in each medium. To do this we need to be careful to consider the fact that unlike pressure which is a scalar the particle velocity is a vector and thus has direction. We can write Pi, Pr and Pt in the following way.

Where R₁ is the specific acoustic impedance of a plane wave in medium 1. Likewise:

And

One again if we consider what is happening at the boundary, it should be obvious from the fact that the boundary is not moving, that at x=0:

Notice that the particle velocity in the reflected wave is given a negative sign to distinguish its direction from particle velocities in the incident and transmitted waves.

Thus:

Equations 23.9 and 23.14 are all we need to formulate our reflection and transmission coefficients.

Thus substituting for P_t in equation 23.14 we get:

simplifying:

Thus the reflection coefficient can be written as:

If we now substitute for P_r from equation 23.9, we get:

Thus

Resulting in an expression for the transmission coefficient:

Case 2: Displacement

Applying the same methodology as for the pressure case, we can define the solutions to the wave equation in terms of displacement in the following way.

and

Using the definition for the adiabatic bulk modulus, shown in equation 23.23 as a reminder.

We can differentiate equations 23.21 and 23.22 w.r.t x and obtain expressions for the pressure. We can then apply the boundary condition for pressure to determine relationship for the particle displacements at the boundary.

Now:

Thus in medium 1;

and in medium 2:

Using the boundary condition of equation 23.9, we have:

At the boundary x=0, thus:

Using the identities:

and

we get:

Now from 23.13 we know that the velocities at the boundary are in balance: thus:

therefore

Now

giving

Likewise

giving

Using the boundary condition for the velocities, we get.

x=0 at the boundary, thus

Equations 23.31 and 23.39, give us what we need to derive the reflection and transmission coefficients in terms of ratios of displacement. Thus substituting u_t in equation 23.31 we get:

simplifying

Resulting in an expression for the reflection coefficient in terms of displacement:

If we now substitute for u_r from equation 23.39 we get:

Thus

Resulting in an expression for the transmission coefficient in terms of the ratio displacements:

Case 3: Velocity

A quick route to the desired equations can be achieved if we simply consider the Laplace operation for differentiation. Assuming that all initial condition are zero and that the plane wave is not losing amplitude, i.e no damping, thus the Laplace notation can be simplified from (s=jw + s) to s = jw

Giving our velocities in terms of displacement as:

Clearly from these three equations we can see that the equations for the coefficients in terms of velocity ratios is exactly the same as the displacement ratios. Thus:

And

Standing Wave Ratio

Another useful ratio is that of the ratio of the maximum pressure amplitude to the minimum pressure amplitude. This ratio is called the Standing Wave Ratio SWR. The SWR is defined as shown below:

Maximum Pressure in medium 1 = p_i + p_r

Minimum Pressure in medium 1 = p_i - p_r

Thus:

Which can also be written as:

Leading to:

A quick look at the SWR suggests that if there is no reflected wave at all then SWR = 1, and if the wave is 100% reflected SWR = ¥.

Let us now consider the case where we have the condition whereby the wave is 100% reflected. In this case pressure amplitude of the incident wave and reflected wave are the same, resulting in a peak amplitude in medium one of 2 p_i. Figure 23.3 below illustrates the situation:

Figure 23.3

We can now write total pressure in medium 1 as:

Thus

Further manipulation

Thus

Putting this expression into a more convenient form we have

Equation 23.59 above describes what is known as the pressure distribution of a Standing wave.

This is a phasor form of the pressure amplitude distribution along the width of medium 1. The term in the square brackets is the amplitude function and the exponential term represents the rotating phase angle. The standard notation for this type of function is:

Or using Demoivres Theorem.

Where the magnitude r and phase angle q can both be functions of time and position.

Where

As a phase diagram this function is plotted on a complex axis (also known as an Argand diagram) as shown in figure 23.4 below. Notice that the amplitude term is made up of the complex components rcosq and jrsinq.

Figure 23.4

Now let us have a look at what the particle velocity is doing. To establish an expression for particle velocity we must first go back to that old favourite the momentum equation. As a reminder here it is:

Applying this to our expression for pressure described by 23.58,we get:

Thus

Integrating

Giving

Re formatting:

Again this the expression for the particle velocity is expressed in phasor form. We can now plot both the scalar function for pressure and the vector function for velocity in exactly the same format on an Argand diagram.

Figure 23.5

Note that both the amplitude functions for pressure and velocity are functions of sinusoids. One thing to note is that in a phasor diagram the amplitude can never go negative. The exponential function e^jw will always ensure that the amplitude remains positive on the Argand diagram. However the reality is that pressures and velocities in medium 1 can have negative values. These conditions are represented on the Argand diagram by the location of the phasor, i.e what quadrant they happen to be in. sinewave Phasors in the 1^st and 2^rd quadrants will be represented as having a positive amplitude and likewise will have a negative amplitude in quadrants 3 and 4.

As the phasors spin anti-clockwise around the Argand diagram they will pass through each quadrant. And be represented as positive or negative pressure and velocity functions accordingly.

End of Lecture