- Transmission and reflection of the sound wave
- Transmission and reflection coefficients
- Applications and exercises
- - Exercise resolved 1
- Solution to
- Solution b
- - Exercise solved 2
- Solution
- References
The acoustic impedance or specific acoustic impedance is the resistance that material means have to the passage of sound waves. It is constant for a certain medium, which goes from a rocky layer inside the Earth to the biological tissue.
Denoting acoustic impedance as Z, in mathematical form we have:
Z = ρ.v
Figure 1. When a sound wave hits the boundary of two different media, one part is reflected and the other is transmitted. Source: Wikimedia Commons. Cristobal aeorum / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)
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, density is in kg / m 3 and velocity is in m / s. Therefore, the units of acoustic impedance are kg / m 2.s.
Similarly, acoustic impedance is defined as the quotient between pressure p and 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 velocity that of current. Other units of Z in SI would be Pa.s / m or Ns / m 3, completely equivalent to those given previously.
Transmission and reflection of the sound wave
When you have two means of different impedances Z 1 and Z 2, 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.
Figure 2. Incident pulse, transmitted pulse and reflected pulse. Source: Wikimedia Commons.
The way in which the energy transported by the wave is distributed depends on the reflection coefficients R and the transmission coefficient T, two quantities that are very useful for studying the propagation of the sound wave. For the reflection coefficient it is the quotient:
R = I r / I o
Where I o is the intensity of the incident wave and I r is the intensity of the reflected wave. Similarly we have the transmission coefficient:
T = I t / I o
Now, it can be shown that the intensity of a plane wave is proportional to its amplitude A:
I = (1/2) Z.ω 2.A 2
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:
A t / A o = 2Z 1 / (Z 1 + Z 2)
This allows the quotient I t / I o to be expressed in terms of the amplitudes of the incident and transmitted waves as:
I t / I o = Z 2 A t 2 / Z 1 A o 2
By means of these expressions R and T are obtained in terms of the acoustic impedance Z.
Transmission and reflection coefficients
The above quotient is precisely the transmission coefficient:
T = (Z 2 / Z 1) 2 = 4Z 1 Z 2 / (Z 1 + Z 2) 2
Since no losses are contemplated, it is true that the incident intensity is the sum of the transmitted intensity and the reflected intensity:
I o = I r + I t → (I r / I o) + (I t / I o) = 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 - 4Z 1 Z 2 / (Z 1 + Z 2) 2 = (Z 1 - Z 2) 2 / (Z 1 + Z 2) 2
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.
Applications and exercises
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.
- Exercise resolved 1
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
- Density: 952 kg / m 3
- Sound speed: 1450 m / s
Muscle
- Density: 1075 kg / m 3
- Sound speed: 1590 m / s
Solution to
The acoustic impedance of each tissue is found by substituting in the formula:
Z = ρ.v
In this way:
Z fat = 952 kg / m 3 x 1450 m / s = 1.38 x 10 6 kg / m 2.s
Z muscle = 1075 kg / m 3 x 1590 m / s = 1.71 x 10 6 kg / m 2.s
Solution b
To find the percentage of intensity reflected at the interface of the two tissues, the reflection coefficient given by:
R = (Z 1 - Z 2) 2 / (Z 1 + Z 2) 2
Here Z fat = Z 1 and Z muscle = Z 2. The reflection coefficient is a positive quantity, which is guaranteed by the squares in the equation.
Substituting and evaluating:
R = (1.38 x 10 6 - 1.71 x 10 6) 2 / (1.38 x 10 6 + 1.71 x 10 6) 2 = 0.0114.
When multiplying by 100 we will have the reflected percentage: 1.14% of the incident intensity.
- Exercise solved 2
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
- Density: 1000 kg / m 3
- Sound speed: 1430 m / s
Air
- Density: 1.3 kg / m 3
- Sound speed: 330 m / s
Solution
The intensity level in decibels of a sound wave, 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 = 10 10
The units of intensity are given in terms of power per unit area. In the International System they are Watt / m 2. Therefore, the intensity of the incident wave is:
I o = 10 10. 10 -12 = 0.01 W / m 2.
To find the intensity of the transmitted wave, the transmission coefficient is calculated, and then multiplied by the incident intensity.
The respective impedances are:
Z water = 1000 kg / m 3 x 1430 m / s = 1.43 x 10 6 kg / m 2.s
Z air = 1.3 kg / m 3 x 330 m / s = 429 kg / m 2.s
Substituting and evaluating in:
T = 4Z 1 Z 2 / (Z 1 + Z 2) 2 = 4 × 1.43 x 10 6 x 429 / (1.43 x 10 6 + 429) 2 = 1.12 x 10 -3
So, the intensity of the transmitted wave is:
I t = 1.12 x 10 -3 x 0.01 W / m 2 = 1.12 x 10 -5 W / m 2
Its intensity level in decibels is calculated by:
L t = 10 log (I t / 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:
I r = 0.99888 x 0.01 W / m 2 = 9.99 x 10 -3 W / m 2
And its intensity level is:
L t = 10 log (I r / 10 -12) = 10 log (9.99 x 10 -3 / 10 -12) = 100 dB
References
- Andriessen, M. 2003. HSC Physics Course. Jacaranda.
- Baranek, L. 1969. Acoustics. Second edition. Editorial Hispano Americana.
- Kinsler, L. 2000. Fundamentals of Acoustics. Wiley and Sons.
- Lowrie, W. 2007. Fundamentals of Geophysics. 2nd. Edition. Cambridge University Press.
- Wikipedia. Acoustic Impedance. Recovered from: en.wikipedia.org.