CK-12 21st Century Physics: A Compilation of Contemporary and Emerging Technologies

by Andrew Jackson

  Attenuation

  All of this suggests that in order to image very small targets, the frequency of the ultrasound should be increased to something like MHz, which is certainly technically feasible. But this is where the other aspect of sound travel through a medium comes into play. Waves can lose their energy by scattering or by absorption. Together these two processes are called beam attenuation.

  Scattering results from parts of the beam deflecting from the straight path of travel. The most familiar example is the scattering of sunlight by our atmosphere. While the Sun is very bright, the rest of the sky is lit by the scattered light of the atmosphere. Scattering is often most important in mixtures and materials that are heterogeneous. The scattering of sunlight is aided by solid dust particles in the air. The most famous example of dust scattering sunlight was the spectacular sunsets that resulted from the famous Krakatoa volcano eruption in 1883.

  Scattering is not as important in ultrasound as absorption. The ultrasound beam is thermalized as the sonic energy is converted into tissue heating. This problem of absorption of the sound has a large impact on the amount of echo that returns to the probe.

  Scattering

  Where a portion of the wave deflects from the straight line path and is lost.

  Typically this will happen more often in materials that are heterogeneous (mixed) materials (not generally a problem here).

  Absorption

  Where the wave energy is converted into thermal heating of the material. Absorption is a major issue in ultrasound imaging.

  Absorption Coefficient

  The absorption process follows first order kinetics, which is familiar from radioactive decay. The intensity is measured as a function of the distance travelled in the tissue, i.e., . The change in intensity is proportional to the intensity and the distance , i.e., . Here is the absorption coefficient in units of inverse meters. It follows that the intensity decays exponentially with distance,

  where is the initial intensity at . The absorption coefficient governs how rapidly the sound is absorbed. The absorption coefficient is the product of two factors, the medium and the frequency of the ultrasonic beam. The frequency matters because the transfer of energy from the ultrasound beam to the tissue is more efficient if the frequency of the ultrasound matches the frequency of a (microscopic) process in the tissue.

  This is called resonance and is familiar from a child on a swing set. The parent exerts a small force each time the child is closest to him/her. If these forces are synchronized with the oscillations of the child, theses pushes "add up," resulting in a large amplitude [and thus energy] of the child.

  The competition between the need for the highest possible frequency to provide a good target image and the need for the lowest possible frequency to return a good echo is usually made around , which provides reliable imaging of objects that are in size.

  The absorption coefficient is the product of a coefficient that depends on the medium and the frequency of the ultrasonic beam,

  This means that as the frequency is increased (wavelength decreases), more of the signal is absorbed by the medium.

  Some Sample Amplitude Absorption Coefficients

  Tissue

  blood

  abdomen

  fat

  soft tissue

  muscle

  bone

  lung

  (Taken from Irving, H. B., Physics of the Human Body. Berlin: Springer Verlag; 2007:562.)

  Sample Problems

  Calculate the attenuation for a ultrasound that penetrates average soft tissue for a distance of and returns to the transponder. Consider the initial beam to be %.

  Repeat the calculation for a beam with twice the wavelength .

  Repeat the calculation for a beam with half the distance and twice the wavelength .

  Equipment

  The probe for the ultrasound is a transducer. This is a crystal of piezoelectric material (piezo is Greek for pressure or squeezing). The most common example of piezoelectricity is the high school demonstration with a crystal mounted under a lever. As the lever is rocked the crystal is squeezed and electric sparks shoot across a gap beside the crystal. A somewhat similar, but not exactly the same effect, is the triboluminescence of crushing Wint-O-Green Lifesavers.

  Try It!

  Take some fresh Wint-O-Green Lifesavers into a darkened room with a mirror. Put one in your mouth and crush it. You should be rewarded with a visible blue spark as the electrons are moved out of the sugars by crushing. The Wint-O-green flavor, methyl salicylate, (oil of wintergreen) acts to convert the normally ultraviolet light of this transition into visible light. When an electric field is applied to the crystal it contracts, as the field is reversed, the crystal expands. The contraction and expansion produce a pressure sound wave. The same process works in reverse so that when the echo comes back the pressure wave produces an electrical signal. The compression high pressure will cause the crystal to produce one electric field and the low-pressure rarefaction will produce the opposite electric field.

  Figure 8.3

  Transducers consist of a piezoelectric material. A varying electrical signal will cause the material to contract and expand, which produces a pressure sound wave.

  The same process can work in reverse: A sound wave hitting the piezoelectric material will give rise to a varying electrical signal.

  Another Problem

  The other transmission problem occurs when the ultrasound wave encounters a new medium with a different speed of sound. As the incident intensity of the beam encounters the new material it is either reflected or transmitted (refracted). The amount transmitted plus the amount reflected must be equal to the incident amount in order to conserve energy. The amount that is reflected compared to the amount that is transmitted depends on a property of the two materials called the impedance. The short answer is that if the impedances of the two materials match then the ultrasound is transmitted. If they don’t, the wave is reflected.

  The ultrasound wave represents a beam energy that must move into the new material if transmission is to occur. But, because the materials are different, the materials have different speeds of sound. An analogy is an airplane delivering packages to an airport that then transfers those packages to trucks for home delivery. If the air freight plane delivers packages (energy) faster than the trucks can haul them away, packages pile up. In that case, the packages are sent back (re-elected).

  Impedance, or complex resistance, can also be found in electricity, specifically in an LRC circuit. The inductor , resistor , and capacitor have a natural resonance where the electric current is the highest. Two circuits with matching impedances will resonate together.

  In the case of ultrasound, two processes can happen when the beam changes two media: Reflection and refraction.

  As the ultrasound pressure wave hits the boundary between the media, there can be no net pressure so exactly how much is reflected and how much is transmitted (refracted) depends on the impedances of tthe media.

  If the impedances match then ALL of the incident intensity if transmitted. For example, if the material is the same on both sides, then the beam is transmitted and not reflected.

  More Impedance

  The intensity, , of some pressure sound wave is defined as the power per unit area . But the power is the kinetic energy per unit time. Expressing the mass of the kinetic energy term as the product of the density, , and the volume, , yields the fourth equation. But the volume is simply the unit area times the distance that the wave travels at the speed of sound in time , i.e., .

  Cancellation of the and terms yields an equation with the impedance, , defined as the speed of sound in the material times the density of the material, .

  The impedance, , of human tissue is not that different from water, but is markedly different for air and bone. This is natural because of the differences in speeds and densities of the materials compared to water. This means an ultrasound beam will travel quite readily from water to t
issue to muscle and back but will reflect off of air (lungs) or bones. The amount of reflectance can be quite remarkable as the table shows. The amount that is transmitted plummets from % for water and soft tissue to % for air and soft tissue.

  Substance Density Speed Impedance

  air

  water

  fat

  muscle

  bone

  Transmission

  The actual intensity that is transmitted can be calculated by taking the ratio of the impedances times the incident intensity. The key point to see here is that the ultrasound "wants" some of the energy to be reflected in order to have an echo for imaging. But when all of the energy is reflected, nothing beyond that material can be directly imaged.

  The most common example of this impedance mismatch is the way that sounds travel very far across water surfaces like lakes. It is not uncommon to hear conversations that are occurring a mile away as if they were in the same room. Another example of this impedance mismatch between water and air is the way that sounds are muffled when underwater. Next time you are in a pool, have someone yell at you while you are underwater. The sound reflects but does not transmit into the water. At the same time, if someone clangs on a pool ladder in the water, the sound travels quite well to your underwater ears.

  The fraction transmitted is dependent on impedance matching.

  For water/air, .

  Almost all of the sound wave is reflected whether from air to water or water to air.

  Interface Reflect (%) Transmit (%)

  water/soft tissue

  fat/muscle

  bone/muscle

  soft tissue/bone

  bone/fat

  soft tissue/lung

  air/muscle

  air/water

  air/soft tissue

  Notice that just getting the ultrasound beam into the body is a problem as most of the energy is reflected at the air interface.

  A special gel is used to make good acoustic contact (match impedances) between the transducer and the body.

  Doppler Effect

  The Doppler effect is the change in the frequency as heard by a listener compared to the frequency emitted by the source. As the listener moves closer to the source, the listener encounters more waves in the same time. The listener’s frequency is higher than the source. The reverse is true for the listener who moves away from the source. An analogous change occurs when the listener is stationary and the source moves.

  The summary of the Doppler effect is that when the distance between the source and listener decreases, the listener hears a higher frequency. When the distance between the source and the listener increases, the listener hears a lower frequency.

  Moving Listener

  A similar argument for a listener moving away results in the same equation but with a minus sign.

  is the frequency that the listener hears.

  is the frequency of the source.

  is the velocity of the listener.

  Moving Source

  is the frequency that the listener hears.

  is the frequency of the source.

  is the velocity of the source.

  is the speed of sound.

  Moving Source and Listener

  Notice that if the source and the listener are moving in the same direction at the same velocity, the result is that the frequency is unchanged.

  As the listener and source close in on each other, the frequency will increase.

  As the listener and the source move away from each other, the frequency will decrease.

  Use relative motion to simplify problems by stopping the slower object.

  Doppler Demo

  Now Try This!

  Either use a tuning fork on a string or a constant frequency speaker (a piezoelectric buzzer with a 9-volt battery).

  Start the buzzer or strike the tuning fork.

  Twirl around your head and listen for the Doppler shifted sounds.

  This is more effective as the speed of the sound source increases.

  Did you hear it? That was the Doppler effect!

  What does this have to do with Ultrasound?

  Blood Flow

  Ultrasound can be used to diagnose the speed of blood flow by using the Doppler effect. The procedure is non-invasive because it doesn’t require inserting a probe into the blood vessel. If the blood is flowing at in the blood vessel then the Doppler effect calculation shows that the change in the Doppler frequency is about %, which would be incredibly difficult to measure. But the imaging system doesn’t measure the frequency directly; instead, it mixes the echo with the known original signal to produce a beat frequency (which is just the difference between the two frequencies). In this case the beat frequency is , which is very easy to accurately measure. This is a common practice in physics to measure an unknown signal precisely by mixing it with a known frequency to produce a beat frequency.

  Virginia Physics Standards of Learning

  This chapter fulfills sections PH.4, PH.9, and PH.10 of the Virginia Physics Curriculum.

  Chapter 9: Kinematics: Motion, Work, and Energy

  John Ochab. "Kinematics", 21st Century Physics Flexbook.

  Linear Motion and How to Describe It

  This section will teach you how to characterize one-dimensional motion by appreciating the use and construction of its representation using graphs.

  About the Chapter

  Understanding how things move is fundamental to our understanding of the physical universe. Critical to this understanding is the ability to portray motion in a manner that is clear, accurate, precise, efficient, and reproducible. “Linear Motion and How to Describe It” identifies the terms used to characterize motion and illustrates the graphical methods used to represent motion visually. The chapter is concerned with the “kinematics” of motion, without regard to the cause of the motion (i.e., no mention of forces).

  One-Dimensional Motion

  Learning Objectives

  to clarify the terms you use to characterize motion and to show their relationships

  to connect your physical ideas of motion to a graphical representation of motion

  to boost your ability to graph motion through tutorial exercises using a motion sensor

  Introduction to One-Dimensional Motion

  You are familiar with motion—a dog chasing a cat, a truck backing up, an apple falling from a tree, people walking in a park—these are just a few examples of motion of a mass or body. In this chapter you are going to study motion, a word synonymous with movement—the state of a body (an animate or inanimate mass) when not at rest. Of course, the straight-line (or “linear”) motions mentioned here are not the only types of motions that can occur. Rotational motion (such as the spinning of a bicycle wheel) and vibrational motion (such as the oscillations of a butterfly wing), or any combinations of linear, rotational, and vibrational motion are also possible. In this chapter, however, we are interested only in linear motion because it is the simplest type of motion and it provides a framework upon which more complicated types of motion of bodies can be characterized.

  Verbal Description of Linear Motion

  So how can you describe motion? Well, you can use verbal and/or written descriptions. Common terms used such as “speed,” “velocity,” “acceleration,” and “deceleration” come to mind, which are used, sometimes interchangeably, to describe the motion of a body. “Direction" is also a property of linear motion and can be dealt with simply by using algebraic signs, "" for one direction (e.g., East), "" for the opposite one (e.g., West) (analogous to a Cartesian coordinate system). Consider the following example:

  Example 1 An oil truck, sitting at rest at time seconds (s) is traveling at a speed of meters per second later.

  Question Is the description of the motion of the oil truck in Example 1 the most complete description of its motion? For one thing, you do not know how the oil truck got from rest to a speed of . What type of motion occurred during that time period? In which direction wa
s the oil truck traveling? What reference points were being used to locate the truck’s position at every point in its motion? What was the speed of the truck at a particular instant of time in the period?

  There is a saying “one picture is worth a thousand words.” Hence, is there a less ambiguous way, other than using just words, to describe the motion of the oil truck in the 6s? The answer is yes and that representation is a visual one. However, before we introduce visual representations of the motion, we need to standardize our definitions of the terms used to describe motion.

  Vocabulary

  distance, average speed, position, velocity, acceleration

  Distance

  Distance is the amount of length that a body has moved from one instant of time to another instant of time. It is always a positive number because it is just an amount, and says nothing about a direction. For example: .

  Average Speed

  Speed, in general, is a measure of how fast a body is moving without regard to the direction of motion. Average speed (symbol, ) is defined as the total distance a body travels per unit time interval. Because distance and time, , are positive quantities, speed is always a positive quantity. We can express mathematically in the following way:

  Example 2

  The car travels an average of miles in one hour: