The acoustic impedance o Specific acoustic impedance is the resistance of material media to the passage of sound waves. It is constant for a certain medium, which goes from a rocky layer in the interior of the Earth to the biological tissue.
Denoting acoustic impedance as Z, in mathematical form we have:
Z = ρ.v
Where ρ is the density and v the speed of sound of the medium. This expression is valid for a plane wave moving in a fluid.
In SI International System units, the density is in kg / m3 and the velocity in m / s. Therefore, the units of acoustic impedance are kg / mtwo.s.
Likewise, the acoustic impedance is defined as the quotient between the pressure p and the velocity:
Z = p / v
Expressed in this way, Z is analogous to electrical resistance R = V / I, where pressure plays the role of voltage and speed that of current. Other SI units of Z would be Pa.s / m or N.s / m3, completely equivalent to those given previously.
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When you have two means of different impedances Z1 and Ztwo, part of a sound wave that hits the interface of both can be transmitted and another part can be reflected. This reflected wave, or echo, is the one that contains important information about the second medium..
The way in which the energy transported by the wave is distributed depends on the reflection coefficients R and the transmission coefficient T, two very useful quantities for studying the propagation of the sound wave. For the reflection coefficient it is the quotient:
R = Ir / Ior
Where ior is the intensity of the incident wave and Ir is the intensity of the reflected wave. Similarly, we have the transmission coefficient:
T = It / Ior
Now, it can be shown that the intensity of a plane wave is proportional to its amplitude A:
I = (1/2) Z.ωtwo .TOtwo
Where Z is the acoustic impedance of the medium and ω is the frequency of the wave. On the other hand, the quotient between the transmitted amplitude and the incident amplitude is:
TOt/TOor = 2Z1/ (Z1 +Ztwo)
Which allows the quotient It / Ior is expressed in terms of the amplitudes of the incident and transmitted waves as:
It / Ior = ZtwoTOttwo / Z1TOortwo
By means of these expressions, R and T are obtained in terms of the acoustic impedance Z.
The above quotient is precisely the transmission coefficient:
T = (Ztwo/ Z1) [2.Z1/ (Z1 +Ztwo)]two = 4Z1Ztwo / (Z1 +Ztwo)two
Since no losses are contemplated, it is true that the incident intensity is the sum of the transmitted intensity and the reflected intensity:
Ior = Ir + It → (Ir / Ior) + (It / Ior) = 1
This allows us to find an expression for the reflection coefficient in terms of the impedances of the two media:
R + T = 1 → R = 1 - T
Doing a bit of algebra to rearrange the terms, the reflection coefficient is:
R = 1 - 4Z1Ztwo / (Z1 +Ztwo)two = (Z1 - Ztwo)two/ (Z1 +Ztwo)two
And since the information related to the second medium is found in the reflected pulse, the reflection coefficient is of great interest..
Thus, when the two media have a large difference in impedance, the numerator of the previous expression becomes larger. Then the intensity of the reflected wave is high and contains good information about the medium..
As for the part of the wave transmitted to that second medium, it gradually fades and the energy dissipates as heat..
Transmission and reflection phenomena give rise to several very important applications, for example sonar developed during the Second World War and used to detect objects. By the way, some mammals like bats and dolphins have a built-in sonar system.
These properties are also widely used to study the interior of the Earth in seismic prospecting methods, in ultrasound medical imaging, bone density measurement, and imaging different structures for faults and defects..
Acoustic impedance is also an important parameter when evaluating the sound response of a musical instrument..
The ultrasound technique to image biological tissue makes use of high-frequency sound pulses. The echoes contain information about the organs and tissues they pass through, which a software is responsible for translating into an image.
An ultrasound pulse directed at the fat-muscle interface is incised. With the data provided, find:
a) The acoustic impedance of each tissue.
b) The percentage of ultrasound reflected at the interface between fat and muscle.
Grease
Muscle
The acoustic impedance of each tissue is found by substituting in the formula:
Z = ρ.v
In this way:
Zgrease = 952 kg / m3 x 1450 m / s = 1.38 x 106 kg / mtwo.s
Zmuscle = 1075 kg / m3 x 1590 m / s = 1.71 x 106 kg / mtwo.s
To find the percentage of intensity reflected at the interface of the two tissues, use is made of the reflection coefficient given by:
R = (Z1 - Ztwo)two/ (Z1 +Ztwo)two
Here Zgrease = Z1 and Zmuscle = Ztwo. The reflection coefficient is a positive quantity, which is guaranteed by the squares in the equation.
Substituting and evaluating:
R = (1.38 x 106 - 1.71 x 106 )two / (1.38 x 106 + 1.71 x 106 )two = 0.0114.
When multiplying by 100 we will have the reflected percentage: 1.14% of the incident intensity.
A sound wave has an intensity level of 100 decibels and normally falls on the surface of the water. Determine the intensity level of the transmitted wave and that of the reflected wave.
Data:
Water
Air
The intensity level in decibels of a sound wave, is denoted as L, is dimensionless and is given by the formula:
L = 10 log (I / 10-12)
Raising to 10 on both sides:
10 L / 10 = I / 10-12
Since L = 100, it results in:
I / 10-12 = 1010
Units of intensity are given in terms of power per unit area. In the International System they are Watt / mtwo. Therefore, the intensity of the incident wave is:
Ior = 1010 . 10-12 = 0.01 W / mtwo.
To find the intensity of the transmitted wave, the transmission coefficient is calculated, and then multiplied by the incident intensity.
The respective impedances are:
ZWater = 1000 kg / m3 x 1430 m / s = 1.43 x 106 kg / mtwo.s
Zair = 1.3 kg / m3 x 330 m / s = 429 kg / mtwo.s
Substituting and evaluating in:
T = 4Z1Ztwo / (Z1 +Ztwo)two = 4 × 1.43 x 106 x 429 / (1.43 x 106 + 429)two = 1.12 x 10-3
So, the intensity of the transmitted wave is:
It = 1.12 x 10-3 x 0.01 W / mtwo = 1.12 x 10-5 W / mtwo
Its intensity level in decibels is calculated by:
Lt = 10 log (It / 10-12) = 10 log (1.12 x 10-5 / 10-12) = 70.3 dB
For its part, the reflection coefficient is:
R = 1 - T = 0.99888
With this, the intensity of the reflected wave is:
Ir = 0.99888 x 0.01 W / mtwo = 9.99 x 10-3 W / mtwo
And its intensity level is:
Lt = 10 log (Ir / 10-12) = 10 log (9.99 x 10-3 / 10-12) = 100 dB
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