Research On The Acoustic Impedance

Discuss about the effect of the material acoustic impedance on two major scenarios.

 

 

Introduction

The propagation of small sound pressure waves through a medium is termed as acoustics. The area has been of great interest to the scholars as attempts are made to scientifically uncover the key aspects of the propagation phenomenon. Notably, acoustic impedance is among the issues that research scientists have focused their energy to develop the concept further with the intention to explain how the phenomenon occurs and the various parameters that can affect it for a given material. In this brief study, great attempts are made to determine the methods and applications involved. Classical deterministic models are often used to mathematically draw a scenario of acoustic impedance and how it depends on the frequency. In this paper, therefore, two methods with accompanying practical applications will be elucidated; such methods as the ultrasound with applications in the spatial compounding. Now, there are two major applications alongside the methods that are currently in use. Therefore, the aim of this paper is to uncover the fundamental principles and practice in the study of acoustic impedance. Certainly, the paper will greatly serve as a substantive material in the furtherance of acoustic impedance concept in the practical world.

Methods of measuring acoustic Impedance

The 2-microphone transfer function

This method is implemented using an impedance tube with two microphones situated at positions whose distance apart is known. The concept is as follows: An incident wave is propagated from loudspeaker and a reflected wave result. The standing-wave phenomenon can then be used to measure various characteristics like impedance, wavelength and frequency. It should be noted that the tube is in-situ in the set up. The plane wave is made to travel in the duct and the frequency is limited by the geometric properties of the tube such as diameter and speed of sound in air. Mathematically, the upper limit frequency, according to Chalmers (2012) must be:

F_u<0.58C_o/d….(1)

The speaker will act as a source of sound wave and the incident pressure wave is denoted P_R

Notably, the complex sound pressure at the microphone position is given by:

P₁= P₁(x₁) + P_R(x₁)= P₁e^-jkx1 + P_Re^jkx1……(2a)

P₂= P₁(x₂) + P_R(x₂)= P₁e^jkx2 + P_Re^-jkx2

The microphone would then detect the signals (both the incident and reflected pressure waves).

However, due to internal amplitude and phase between the microphones, the microphone interchange method can be used to minimize this mismatch (Chalmer, 2012). Finally, the transfer function of both the incident and reflecting waves between the microphones can be derived using the general equations 3(a) and 3(b):

H₁= P₁(x₂)/P₁(x₁)= P₁e^-jkx2/P₁e^-jkx1= e^jks…3(a)

H_R= P_R(x₂/P_R(x₁)= P_Re^jkx2/P_Re^jkx1= e^-jks…3(b)

Where s=x₂-x₁

The reflection coefficient R can then be determined (which is actually the ratio of the complex reflected and incident pressures.

R_x=x1= H₁₂-H₁/(H_R- H₁₂)…(3)

But note that there is normally a portion of the incident propagating wave that is not reflected but absorbed, that is given by:

    α = 1-/R/²….(4)

The surface acoustic impedance is therefore give as:

Zs= Zo(1+R)/(1-R) where Zo= PoCo acoustical characteristics impedance of air

Effect of frequency on the impedance

The acoustical impedance is affected by the source frequency such that low frequency sources result in lower impedance intensity hence reduced impedance level. However, best performance of the set up occurs at a particular low frequency region as given by equation 1. Besides, the diameter of the tube plays a pivotal role on the frequency range allowable in the method. Higher frequency above the given limit is often constrained by the tube diameter.

Wave decomposition method

In the previous method, consideration was made for plane waves only hence the model is limited to certain applications. However, here the wave decomposition method can cover the oblique-incident and normal pressure waves hence applicable to higher modal order modes (Schultz, Cattafesta & Sheplak, 2006). The method depicts a one-dimensional lined duct with an excitation speaker at one end and impedance boundary on the other end. There is normally a material that will be inserted at the duct-end such that during propagation, the material produces, in part, a reflective and reactive boundary hence complex transfer function results.

Mathematically, dU/dx(L, t)= -k(1/c)dy/dt(L,t)….(5)

Where k= complex acoustic impedance

 X= L,U(L,t)= fluid particle displacement at x=L

C= sonic wave speed in the duct (m/s)

t = time variable

x= spatial variable (m)

L= length of the duct (m)

Hence equation 2 illustrates how the impedance can be determined using the method once the experiment is performed:

Re(T)+iIm(T)= Re(k) cos [w/c(x-L)+i{Im(k)Cos{w/x(x-L)-sin[w/c(x-L)/Re(k) Cos (w/cL)+i{Im(k)cos(wl/c)+sin(wl/c)}…(6)

The acoustic pressure of the system is proportional to the spatial derivative of the particle displacement

P(x,t)= -pc²dU/dx(x,t)

Application of Acoustic Impedance

The turbofan engine acoustic

The two-microphone transfer function method can be used in the turbofan engine to analyze the material acoustics making up the compressor and the turbine.

The model first considers the system to be such that the phenomenon occurs in lined ducts with an axial flow of the sound pressure waves. There is need to know the boundary conditions at the walls and at the treatment surfaces (Malmary & Carbonne, 2001).

According to Malmary & Carbonne (2001) the normalized acoustic impedance is given as:

Zt= 1/PoCo x (P/v’n’) …. (1)

where P= acoustic pressure at a point of the surface of the liner, v’ = acoustic velocity at the same point, n’= normal to the surface of the liner, Co=speed of sound in air and ?= air density

However, a part from the layer impedance, there is cavity impedance which must also be factored in hence the total acoustic impedance from the system is given by:

Zt= Z-jcot(KL) …... (2)

where K= acoustic wave number, K=w/co and L= cavity depth

Therefore, critical parameters that will determine the relevance of the model in the practical sense include: the dimensional property of the attenuating medium and the flow characteristics such as mach number, frequency among others which mathematically can be represented as:

Z= z(e,d,?,w,/v/, M)….(3)

Where w= sound pulsation

/v/= amplitude of the normal acoustic velocity

M= Flow Mach number

However, it should be noted that the above model is linearized where the frequency characteristic relates with the acoustic impedance as follows:

Firstly, for a perforated pipe, according to  Malmary & Carbonne (2001) the impedance is given as: Z= r+jX …..(4)

Where r= (8vw)^0.5/?Co(1+e/d) +1/8?(kd)² and X= w/?Co[e+8d/3π(1-0.7?^0.5) +(8v/w)^0.5(1+e/d)]

However, as pointed out earlier, this model is not sufficient in its entirety so to speak; because the real-world acoustic systems often exhibit non-linear behavior hence a further extension to the method is necessary. However, there are other more specific methods that utilize the acoustic impedance phenomenon with their underlying applications as discussed in the next paragraphs

Spatial compounding

More often, acoustic impedance measurements could be used in the generation of the ultrasound images but problems of occlusions may hinder the performance in situations where the material acoustic impedance is high. Unlike the other scenarios, where the mean of the impedances can be used in a straight fashion; this case requires the beam intensity to be increased so as to overcome the material impedances. This is what is termed as spatial compounding. Hence often the material acoustic impedance will have to be known from the standardized models after which the system design will take root. Nevertheless, spatial compounding ensures that the ultrasound imaging is a near excellent result while overcoming the layer impedances during propagation (Malmary & Carbonne , 2001).

References

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Malmary, C & Carbonne, S. (2001). Acoustic impedance measurement with grazing flow. Available from: https://perso.univ-lemans.fr/~yauregan/publi/AIAA-2001-2193.pdf

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