Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. ENDOSCOPY/LAPAROSCOPY TABLE OF CONTENTS 1 Overview................................................................................................................................ 2 2 Introduction........................................................................................................................... 2 3 Optics Basics for Endoscopy.............................................................................................. 3 3.1 Refraction ................................................................................................................................... 3 3.2 Total Internal Reflection............................................................................................................. 4 3.3 Rays, Lenses, and Images......................................................................................................... 5 3.4 Light: Photons and the EM Spectrum....................................................................................... 6 3.5 Blackbody Radiation .................................................................................................................. 7 4 Imaging in Endoscopy ......................................................................................................... 8 4.1 Numerical Aperture and Dispersion.......................................................................................... 8 4.2 Image Formation and Resolution ............................................................................................ 10 5 Good Vibrations – The Harmonic Scalpel ........................................................................ 11 5.1 Oscillatory Motion .................................................................................................................... 11 5.2 Standing Waves........................................................................................................................ 13 5.3 Resonance ................................................................................................................................ 14 5.4 The Harmonic Scalpel .............................................................................................................. 15 6 Sample Questions .............................................................................................................. 16 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 1 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 1 OVERVIEW Endoscopic surgeries have become widespread with the availability of optical technology, including fiber optics and lens systems. The endoscopic assemblage consists of a lens system, a light source, and a collection system that projects an image from inside the body onto the surgeon’s eye or a camera. High resolution, high magnification images available in real time allow for minimally invasive, safer surgeries. The harmonic scalpel is a device used in endoscopic surgeries that has the ability to both cut tissue and coagulate blood, reducing bleeding during surgery. Vibrational motion is foundational to its functionality: a piezoelectric diode converts electrical energy into mechanical wave energy that travels along a metal rod, ending in a cutting/coagulating tip. Along with vibration, heat created by the motion plays a crucial part in the operation of the scalpel. The section “Optics Basics for Endoscopy” will cover the physics concepts of refraction, converging and diverging lenses, ray diagrams, the thin-lens equation, magnification, total internal reflection and a brief look at light as electromagnetic radiation and blackbodies. The section “Imaging in Endoscopy” will examine numerical aperture, modal dispersion, image formation with fiber optics and the resolution of images. The physics concepts surveyed in “Good Vibrations – The Harmonic Scalpel” include periodic motion, simple harmonic motion, energy in oscillations, resonance and standing waves. 2 INTRODUCTION Endoscopy is both a diagnostic and therapeutic tool and a safe way to investigate body cavities and organs. An endoscope is a thin, flexible telescope that produces real-time images with high resolution and magnification. The endoscope tube can be inserted through an opening in the body, such as the mouth, anus, or through a small incision in the skin. Reflected light rays from the endoscopes fiber optics are collected by a CCD camera where electrical signals are produced. Those signals are sent to a monitor for real-time imaging. In addition to the fiber optics and lens system, the endoscope also has an instrument channel which surgeons can utilize very small surgical scissors, suction device, and forceps. Several branches of medicine and surgery utilize this tool. Figure 1: An example of a typical endoscope. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 2 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 2: The internal arrangement of endoscope components. 3 OPTICS BASICS FOR ENDOSCOPY 3.1 REFRACTION The speed of light in a vacuum is a constant, c = 2.998×108m/s. However, when light passes into a different medium, the material properties of that medium determine how fast light can travel through it. The index of refraction, n, is a material-dependent quantity that relates the speed of light in a vacuum to the speed of light v in a particular medium: n= c v (1) For example, the index of refraction for air is 1.00029, corresponding to a small decrease in speed compared to vacuum. For water, n = 1.33; for glass, n = 1.5 (different types of glass have different values of n). Refraction occurs when a ray of light travels from a medium with one index of refraction n1 to a medium with a different index of refraction n2. If the light ray strikes the second medium at an angle, the ray will bend upon entering the medium: closer to the normal (the line perpendicular to the surface) if n2 is greater than n1, and away from the normal if n2 is less than n1. This is shown in Figure 3. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 3 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 3: Refraction of Light Refraction is expressed mathematically with Snell’s Law that relates the angle of incidence θ1 to the angle of refraction θ2: n sinθ = n sinθ (2) Here θ1 is the angle of incidence and θ2 is the angle of refraction in the second medium. 3.2 TOTAL INTERNAL REFLECTION Total internal reflection of light within a medium is key to transporting light by optical fibers, which are used to carry information. The information may be in digital form or light rays comprising an image. An example of the latter is endoscopy, where light is transported into and out of the body with optical fibers. Analyzing total internal reflection begins with Snell’s Law: n sinθ = n sinθ (3) When light travels from one medium to another, some of it is reflected back into the first medium and some is transmitted into the second medium. The larger the angle of incidence becomes, the larger the angle of refraction. At some point, the angle of refraction will become 90o. When this happens, the angle of incidence is called the critical angle (θC): n sinθ = n sin90 = n sinθ = (4) For angles larger than the critical angle, light is no longer transmitted to the second medium but is entirely reflected back into the first medium. Figure 4 shows an angular progression through the critical angle, from the incident ray traveling perpendicularly through the interface to the ray being refracted by 90 at the critical angle to total internal reflection beyond the critical angle. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 4 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 4: Angular progression through the critical angle and total internal reflection 3.3 RAYS, LENSES, AND IMAGES The path of light changes when it travels from one medium to another because of refraction. This property allows lenses to be used to guide light, whether through a microscope, through the body in optical fibers or through an eye. A converging lens, shown on the left of Figure 5, brings parallel rays of light to a point, called the focal point. A diverging lens causes parallel rays of light to diverge or move apart from one another. Light leaving such a lens will appear to originate from the lens’s focal point, as shown on the right of Figure 5. Note that by convention, light rays are drawn propagating from left to right. Figure 5: Parallel rays of light entering converging (left) and diverging (right) lenses When rays of light coming from an object reach one of these lenses, a ray diagram can be composed to show where the image of the object will appear. This is demonstrated in Figure 6. F is the focal point of the lens;ay is the focal length, the distance between the midpoint of the lens and the focal point; do is the object distance, the distance between the midpoint of the lens and the location of the object; di is the image distance, the distance between the midpoint of the lens and the location of the image; ho is the height of the object and hi is the height of the image. Figure 6: Ray diagrams for light leaving an object and traveling through a converging (left) and diverging (right) lens. Rays 1, 2 and 3 are known as principal rays. Note that they emanate from the same point on the object. Ray 1 leaves the object parallel to the optical axis, the line perpendicular to the lens that bisects it. Upon reaching the lens, the ray is traced to the focal point as shown. Ray 2 travels through the center of the lens and continues in the Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 5 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. same direction. Ray 3 is drawn through the focal point on the same side of the lens for converging lenses, or toward the focal point on the opposite side of the lens for diverging lenses. An image is either upright, as in the in Figure 6b, or inverted, as in Figure 6a. The image is real if it is on the opposite side of the lens as the object, as in Figure 6a, or it is imaginary if it is on the same side of the lens as the object, as shown in Figure 6b. Using the geometry of lenses, it is possible to derive an equation relating the image location to the focal length of the lens and the location of the object. An equation for magnification, the amount by which the image is enlarged or reduced compared to the original object, can be similarly derived. The thin lens equation, so named because an assumption is made that the thickness of the lens is small compared to the overall radius of curvature for the lens, is: 1 1 1 + = d d f (5) where do is the object distance, di is the image distance, and f is the focal length of the lens. The magnification, M, is given by where hi is image height, and ho is object height. M= h d =− h d (6) In order to perform calculations with these equations, sign and other conventions must be strictly observed: ∑ ∑ ∑ ∑ ∑ ∑ ∑ Light is drawn propagating from left to right. The focal length is positive for a converging lens and negative for a diverging lens. An upright image is positive (has a positive value for magnification), an inverted image is negative (has a negative value for magnification). An object to the left of a lens has a positive object distance (real object). An object to the right of a lens has a negative object distance (virtual object - yes they do exist!). An image to the left of a lens, or on the same side as an object, has a negative image distance (virtual image). An image to the right of a lens, or on the opposite side as an object, has a positive image distance (real image). 3.4 LIGHT: PHOTONS AND THE EM SPECTRUM The theories of electricity and magnetism can be used to show that a changing electric field produces a changing magnetic field, and a changing magnetic field produces a changing electric field. The physicist James Clerk Maxwell showed that together, these oscillating fields can carry energy away from the source producing the Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 6 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. oscillations in the form of an electromagnetic (EM) wave. EM waves are transverse, and unlike mechanical waves, they can travel through a vacuum where their speed is a constant 3×108 m/s. It has been shown that the frequency f of a wave multiplied by its wavelength λ yields the speed of the wave. For EM waves whose vacuum speed is a constant c, this is expressed with c = λf (7) This equation is valid for the EM spectrum, or all EM radiation. Maxwell revealed light to be an EM wave. The wavelength of visible light ranges from 400 nm (violet) to 700 nm (red) (infrared ranges from 700 nm to 0.3 mm, and ultraviolet ranges from 400 nm to 3 nm). The range of wavelengths in the EM spectrum is divided into bands, as shown in Figure 7. Figure 7: Electromagnetic Spectrum A half a century after the discovery of light as an EM wave, Einstein, using Max Planck’s quantum hypothesis, proposed that light could also be treated as particle-like with quantized energy expressed by E = hf (8) E = nhf (9) where E is the energy of a photon or particle of light, h is Planck’s constant equal to 6.63×10-34 J s, and f is the frequency of the EM wave. The energy of a single photon, or particle of light, is given by this equation. The reason Max Planck had theorized quantized energy was to explain blackbody radiation, which requires quantization. Since light energy is quantized, it can only come in multiples of hf; that is, for any integer n, 3.5 BLACKBODY RADIATION Blackbody radiation is the phenomenon behind why some lighting is better for certain applications than others, whether it is infrared light therapy applied to tissues or using a high-intensity xenon light as a light source in endoscopy. An ideal blackbody is an object that absorbs all light (or EM) radiation that hits it (very reflective Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 7 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. objects are not blackbodies). Thermodynamics shows that if an object absorbs radiation very well, it also radiates energy very well. If an object is a good enough absorber of radiation, it can be approximated as a blackbody. The sun can be treated this way, as can certain sources of light such as incandescent bulbs. For such an object, the frequency or wavelength and intensity of the EM radiation it gives off can be measured. When plotted on a graph of intensity vs wavelength, known as the blackbody spectrum, the resulting curve has a peak at a wavelength (λpeak) that depends on the temperature T of the object, regardless of what the object is made. Figure 8 shows this for three objects of temperatures 6000 K, 5000 K, and 3000 K. Mathematically, the peak is found by Wien’s displacement law: λ × T = 2.898ª10 mK (10) Figure 8: Blackbody radiation curves, known as the blackbody spectrum, for three different temperatures. 4 IMAGING IN ENDOSCOPY 4.1 NUMERICAL APERTURE AND DISPERSION In endoscopic surgery, fiber optic bundles are used to carry light into the body. Endoscopes use fiber optic bundles to transmit that light back out of the body to form an image of an internal feature. A typical optical fiber is made up of a core material surrounded by a different material called the cladding, as shown below. The effectiveness of the optical fibers depends on a number of factors. One of the properties to take into account is the numerical aperture, which is a measure of how much light can enter the optical fiber. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 8 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 9: Acceptance angle and total internal reflection. θmax is half of the full acceptance angle, θr is the refracted angle, and θc is the critical angle at the core-cladding interface. The formula for numerical aperture is: NA = nsinθ = n −n (11) where n is the index of refraction outside the fiber (very often this will be air), ncore is the index of refraction of the core material and ncladding is the index of refraction of the cladding material. θmax is shown in Figure 9 and is the maximum angle from the axis at which light can remain confined within the core of the fiber for total internal reflection. This angle is the acceptance angle and is half the angle that makes up the full acceptance cone. Any light hitting the optical fiber within the acceptance angle will travel along the fiber. Outside of this angle, light will be refracted but will not be totally internally reflected. Another optical phenomenon that should be addressed is dispersion of a pulsed light signal. Dispersion is an event in which the phase velocity of a wave depends on its frequency. There are different types of dispersion that can occur in optical fibers, but the end result of all of them is that the component light rays entering a fiber simultaneously do not exit it at the same time. Modal dispersion is one type that can cause significant problems with signal transmission. It is a distorting mechanism that occurs in multimode fibers where the signal is spread in time because the optical signal is not the same for all modes. Light rays spreading out from their source result in modal dispersion within an optical fiber. These rays will be totally internally reflected at different angles to the core-cladding interface and will therefore travel different distances down the optical fiber. Figure 10 shows two cases of modal dispersion. The greater the dispersion, the longer time it takes for the entire original light pulse to exit the fiber and the more time must pass before the next pulse of light can be received. The left side of Figure 10 depicts the case when the critical angle at the core-cladding interface is small (refer back to Figure 9 for the location of the critical angle). On the right of Figure 10, the critical angle is larger and there is less dispersion because some of the light is refracted out of the optical fiber. The remaining rays experience fewer reflections as they travel along the core. For these rays, there is less spread or broadening of the light pulse because there is less variation between the paths different light rays take (compare the paths of light in the left figure with the paths of the remaining rays in the core of the right figure). Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 9 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Chromatic dispersion also occurs in optical fibers and is depicted in Figure 11. When light waves enter a medium their speed decreases. However, this decrease in speed also depends on the frequency of light; shorter wavelengths of light slow more than longer wavelengths. The discrepancy adds up to a difference in the distance traveled over long lengths of fiber optic cable (since the index of refraction is that travel more slowly also have a higher index of refraction n). = , shorter wavelengths of light Figure 10: Modal dispersion Figure 11: Chromatic dispersion in a fiber optic cable leads to the longer wavelength of light traveling farther. 4.2 IMAGE FORMATION AND RESOLUTION Once a fiber optic bundle carries light into the body and illuminates a feature, the light reflecting off the object must be transported coherently back in another fiber optic bundle to form an image for the viewer. Whatever light enters the fiber bundle will exit the opposite end the same way; that is, if an unfocused image enters the bundle, the image will still be unfocused when it exits. The focused image of the object must be projected onto the end of the bundle if this is what the viewer will observe. This can be accomplished using a lens system. For the sake of simplicity in the problems below, a single lens system as shown in Figure 12 will be analyzed. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 10 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 12: Optics to project an image into the fiber bundle A single fiber acts as an individual pixel to the eye of the viewer. To create an image, this spot of light is added to many other spots of light as each optical fiber transmits light reflecting from part of the object. The spatial resolution of the image depends on the geometry of the fiber optic bundle: the diameter of the fibers, fiber spacing, the number of fibers in the bundle, and so on. The images in Figure 13 show a cross section of a fiber bundle through which the image of a circle is transmitted. The left figure displays the circle image projected onto the bundle; the right figure shows how the image would appear to a viewer on the opposite end of the bundle. Notice that where the image is not fully projected onto an optical fiber, the resulting intensity of the light leaving the fiber is lower. Figure 13: Resolution of a fiber bundle 5 GOOD VIBRATIONS – THE HARMONIC SCALPEL 5.1 OSCILLATORY MOTION Periodic motion is the regular cycling of an event. Clocks, metronomes, and planetary orbits are a few examples. The number of repetitions or cycles per unit of time is the frequency; in SI units, frequency is given in cycles/second or Hertz. The inverse of frequency is the period, or how much time is required for one cycle. In other words, for period T and frequency f: Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 11 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. T= 1 f (12) For a certain type of periodic motion, one where a restorative (returning to equilibrium) force depends linearly on the distance from equilibrium, oscillations are described by simple harmonic motion. A mass on a spring obeying Hooke’s Law, F = −kx (13) x = Acos(ωt) (14) 2π T (15) is an example. The repetitive, single-frequency oscillation that results can be described with a sinusoidal function (sine or cosine function). Position as a function of time is expressed with where x is the position of the mass, A is the amplitude or maximum distance from the equilibrium position, t is time, and ω is the angular frequency. Simply put, every linear position of an oscillating particle or mass corresponds to a rotational position on a circle. If a mass on a spring has oscillated through a quarter of its cycle, it has moved through 90 of a circle; if it has moved through half its cycle, it has moved through 180 of a circle, and so on. The equivalence of frequency f and angular frequency ω is given as: ω = 2πf = where ω is measured in radians/second. An equation for the acceleration a of the mass can also be derived; it is a = −Aω cos(ωt) (16) F = ma = −kx (17) The equations for position and acceleration are used to find the period of a mass on a spring. Beginning with Newton’s law and Hooke’s law, ma = m[−Aω cos(ωt)] = −k[Acos(ωt)] (18) After canceling out similar factors, it can be seen that ω = or ω= Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. (19) 12 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. From this and the equations for frequency the final equation for the period of a mass or particle on a spring is given as T = 2π m k (20) A simple harmonic system can be useful as an approximation for some more complicated systems. Employing energy conservation to solving the system may simplify a complex problem. One example of a simple system is a horizontal mass on a spring with no loss of energy to friction. The total energy E of such a system is given by the sum of the potential energy as derived from the spring force and the kinetic energy: 1 1 E = mv + kx 2 2 (21) 1 = kA 2 (22) Here the first term represents kinetic energy, m is the mass in the system, v is the speed at some given time, x is the position at that time, and k is the spring constant. When the mass is at the turning points in the periodic cycle, x is equal to the amplitude A and the speed of the mass at that moment is 0. At that point all of the energy is potential energy (U), so the total energy available in the system can be written as: E=U 5.2 STANDING WAVES Unlike a wave that only travels from one end of a string to another or a water wave that moves continually away from a source, when an oscillator creates a wave that is reflected back on itself, superposition adds the displacement of the waves at each point. A standing wave is created when the original wave and the reflected wave interfere constructively. This is depicted in Figure 14. The top figure shows the displacement of particles for a standing wave on a string while the bottom figure shows the displacement of particles for a sound wave in a pipe with two open ends. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 13 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Figure 14: Standing wave on a string (top); standing wave in a pipe (bottom). In this particular vibrational mode, a full wavelength is present in both the string and the pipe. In this diagram, “N” stands for node, which is where the particles ideally experience no displacement. “A” stands for antinode, where particles experience maximum displacement. A standing wave on a string is a transverse wave that has nodes at either end. Pipe instruments carry longitudinal sound waves; if the pipe is open at one end, the closed end is a node and the open end is an antinode, while two open ends cause two antinodes. Harmonics occur when more than the minimum number of nodes and antinodes are present; they are determined by the boundary conditions of the system. The frequency and wavelength for vibrational modes on a string are expressed as: f =n λ = (N-A-N) (23) where n is for the nth harmonic; it can take on the values n = 1,2,3,.... The frequency and wavelength for vibrational modes in a system carrying longitudinal waves with a node at one end and an antinode at the other (clarinet, human ear canal) are f =n λ = (N-A) (24) where the harmonics can take on the values n = 1,3,5,.... Finally, for vibrational modes in systems carrying longitudinal waves having antinodes at both ends (flute, harmonic scalpel) the frequency and wavelength are f =n λ = (A-N-A) (25) where again n takes on integer values, n = 1,2,3,.... In the case of N-A-N and A-N-A, the fundamental node or first harmonic (n = 1) corresponds to the mode where only a half wavelength appears on the string (N-A-N) or in the column (A-N-A). For a column with a node at one end and an antinode at the other, the first harmonic is only a quarter of a wavelength long (N-A). Figure 14 shows the second harmonic for a string (top) and a column with antinodes on each end (bottom). 5.3 RESONANCE A system begins to vibrate when energy is added to the system via a disturbance. Oscillating objects, whether they are strings, pendula, piezoelectric crystals or buildings swaying in an earthquake, each have a natural frequency at which this added energy creates vibration. Energy may be lost from the system by friction or other nonconservative forces; in this case the oscillation is damped. The oscillations will at some point cease, although how quickly depends on the degree of damping. Driven oscillations, however, occur in systems where an external force is applied that constantly adds energy. An oscillation may be driven and damped at the same time. The driving force drives the system at a frequency independent from the natural frequency of the system. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 14 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. Although the following equations will not be used extensively here, they are given to provide some insights into the phenomenon of resonance. Summing the forces on a driven, damped oscillator (spring force, damping force, driving force) yields an equation for the amplitude A of vibration; this is: A= F 1 m (ω − ω ) + ω γ (26) / ), where Fo is the driving force (a constant), m is the oscillating mass, ω0 is the natural frequency (given by is the drive frequency, and γ is a damping constant. The mechanical energy of the system is found by summing the kinetic energy and the potential energy; doing so yields the equation: 1 E = mA [(ω + ω ) + (ω − ω ) cos(2ωt − 2δ)] 4 (27) where E is the energy, t is time and δ is a phase constant. According to this equation, the mechanical energy of the system oscillates due to the cos function. Resonance occurs when the frequency of the driving force and the natural frequency are the same (ω → ω0). A couple of key characteristics about resonance can be observed from Equations 26 and 27. It can be seen from Equation 26 that the amplitude of oscillation becomes increasingly large as resonance is approached. When ω and ω0 are the same, the maximum amount of power is absorbed by the oscillator. Additionally, Equation 27 shows that at resonance, the mechanical energy of the system is constant and equal to: 1 1 E = mA (ω + ω ) = mA ω 4 2 (28) (if not at resonance, the left side of the equation is the average energy for an oscillation although it may not be the value for any one cycle; it is still constant). The driving force adds energy to the system to sustain the amplitude of oscillation while the damping force perpetually removes it; both forces are nonconservative. For the average energy over an oscillation to remain constant, the amount of energy added to the system by the driving force must equal the energy removed by the damping force. 5.4 THE HARMONIC SCALPEL The physics of oscillations, simple harmonic motion, standing waves and resonance are the foundation for the functionality of the harmonic scalpel. The scalpel has an extended cylindrical-shaped shaft that vibrates in a longitudinal direction (along the direction of the shaft). The longitudinal wave in the harmonic scalpel is a standing wave. In terms of standing wave models, the harmonic scalpel is most similar to a pipe with two open ends, where the piezoelectric crystal at one end and the blade at the other end move to maximum displacement. Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 15 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. A reduction in shaft diameter at a few points along the instrument results in the mass of the tip being less than the mass at the handle end. By the principle of conservation of energy, the same amount of energy used to vibrate one end must be transferred to the other end. Since the tip end has less mass, the amplitude of vibration will be greater there than the amplitude near the handle end. This is also called an acoustic horn. Although it is a rigid body, oscillations in the harmonic scalpel can be analyzed to some extent by considering an analogous system of a mass on a spring (certain systems can be approximated in this fashion; for example, atomic interactions are sometimes modeled with atoms as point masses with spring-like restorative forces between them). Energy equations for oscillatory motion can be applied, although they represent an approximation of the actual system. The piezoelectric crystal at the handle end of the scalpel converts AC power to mechanical energy, thus creating the vibrational motion. Driving the crystal at its natural frequency with AC causes resonance in the crystal and maximizes the vibrational energy. Although this maximum depends on the natural frequency of the crystal, the greater the driving force (AC power) at that frequency, the greater the amplitude of vibration along the harmonic scalpel shaft. 6 SAMPLE QUESTIONS Problem: Assume the index of refraction for air is approximately 1. If light enters a material at an angle of 23◦ and exits at an angle of refraction of 16◦, what is the index of refraction for the material? Solution: Using Snell’s Law and solving for n2: = 1(sin(23 )) = 1.42 sin(16 ) Problem: An object is 28 cm from a lens. The image created by the lens is twice as large as the object. The image is not inverted. What is the focal length of the lens? Solution: We can solve for image distance and then use the thin lens equation: =− + (28)(2) = −56 = − + 1 = 55.99 0.05357 = = 0.01786 Copyright: Portland State University, 1825 SW Broadway, Portland, OR 97030. 16 A Look at the Physics of the Human Ear Physics for Engineers and Scientists II Donya Kaveh and Jorge Grijalva April 2021 The human ear is one of the most important organs of the body—one that is responsible for one of the five human senses: hearing. The chief physiological objective of the ear is to receive and transmit sound waves and remove background noise. Additionally, the ear is responsible both for the amplification of sound waves and the analysis of frequency and intensity. [1] The auditory system of human body is divided into three parts. [1] These consist of mechanical, sensory and auditory system. The mechanical system consists of ear that captures and amplifies sound waves. [1] The sensory system converts the sound waves which are basically mechanical pulses to electrical signals that are sent to the auditory nerve. Finally, the auditory system consists of brain that is responsible for decoding and interpretation of electrical signals that were received the auditory nerve. [1] The human ear consists of three parts: inner ear, middle ear and outer ear as shown in Figure 1. [2] The inner ear contains a spiral shaped cochlea and perilymphatic fluid while the middle ear consists of three bones hammar, anvil and stirrup that connect the outer and inner ear. The outer ear consists of the pinna which captures sound waves and the ear drum which transmits sound vibrations to the middle ear. [2] Figure 1. Three Parts of the Human Ear. [2] Sound is measured in decibels (db SPL) where SPL stands for “Sound Pressure Level.” SPL can be defined using the following logarithmic scale: [4] Sound stimulus (in db SPL) = 20log10 P Pref Where Pref is defined as the sound pressure of air which is taken to be 20 µPa. [4] Inner ear The inner ear contains a spiral shaped system consisting of three tubes shown in Figure 2. There are two outer and one inner tubes. [3] The two outer tubes are contained in the vestibular and tympanic chambers that in turn are present at the tip of cochlea while the inner tube is contained in cochlea duct. [3] Both the inner and outer tubes are separated by a basilar membrane consisting of about 15,000 hair cells. Perilymphatic fluid present in the inner ear are responsible for the transmission of sound vibrations to tiny hairs within hair cells comparable to cellular flagella. [3] The excitation of tiny hairs in inner ear causes conversion of vibrations to electrical signals. These signals are then transported to brain by the auditory nerve. The impulse rate of neurons within the auditory nerve depends on two factors: frequency, f and sound intensity, I. [3] Figure 2. Schematic of (a) cochlea (b) a section of cochlea. [3] A hair cell is shown in Figure 3 below. Every human hair cell contains hair-like structures known as stereocilia whose job it is to bend and vibrate when sound waves are received. [3] This vibration serves to stimulate hair cells and cause calculated excitations of neurons within the auditory nerve. Both neuron firing and impulse rate depend on two major factors: intensity, I and frequency, f. [3] Figure 3. Components of hair cell. [3] Middle Ear The middle ear cavity consists of three ear bones: hammar, anvil and stirrup. These ear bones are structurally interconnected to each other whereby a vibration of the eardrum causes sound vibrations to be transmitted to the inner ear via these bones as a medium. [2] This occurs when one end of the hammar is connected to the eardrum and its other end touches the second bone or anvil. [2] The free end of the anvil is connected to the third bone, stirrup. The free end of the stirrup is connected to the membrane of the oval window that resides within the inner ear. [2] The middle ear cavity also contains the Eustachian tube in its lower part and connects to the throat so as to regulate inner and outer air pressure. This causes differences in air pressures to be low so as to prevent the rupture of the ear drum. [1] Figure 4. Middle ear. [1] The middle ear is responsible for the amplification of the entire structural system. It consist of the oscilles muscle which acts as a lever to amplify vibrations originating in the eardrum membrane. [1] The pressure changes within the ear drum cause a force to be induced at the eardrum, Fm that causes a torque, τm to be induced at incus. τm induces a force F0 and pressure P0 at oval window that occurs at A0 as shown in Figure 5. [1] Figure 5. Ampli cation of system by eardrum and oval window. The torques are equal at both eardrum and oval window of middle ear. τm = τ0 τm = Fm x Lm = F0 x L0 = τ0 Fm x L m = F0 x L0 As, Fm = Pm x Am Pm x A m x L m = P0 x A 0 x L 0 Rearranging, fi P0 Am Lm = x Pm A0 L0 [1] The pressure ratio can change significantly with changes in length and area ratios of eardrum and oval window as shown below whereby an area ratio of 13 and length ratio of 1.3 can cause pressure ratio to be about 19.5 P0 = 19.5 Pm Am Lm = 13 and = 1.3 A0 L0 This causes a significant increase of intensity by 26 db as shown below (20)log10 P0 = (20)log1019.5 = 26 db Pm Outer ear Outer ear consists of pinna that collects sound wave and directs to the ear canal which is about 2-3 cm long. The end of ear canal contains eardrum which vibrates when sound wave hit them. This causes sound waves to transmit from ear drum to the middle ear. [1] The important part of outer ear is ear canal that is about 2-3 cm long. The outer ear is open on one side and closed on the other by eardrum. This represents a tube closed on one side as shown in Figure 6 and therefore sound waves coming inside the ear canal can cause resonance at certain frequencies. These frequencies can be found by the formula below: [1] fn = nv 4L (n =1, 3, 5…) Figure 6. Tube closed on one side. [1] Where v represents speed of sound wave that is 330 m/s, L represents length of ear canal which is about 2.5 cm, n represents an odd integer while fn is frequency evaluated at a particular n value. For example resonant frequencies at n=1 and n=3 are: f1 = (1)(330) 4(0.025) f3 = (3)(330) 4(0.025) f1 = 3300 Hz f1 = 9900 Hz This leads to increased sensitivity of ear at high frequency ranges. [1] Eardrum One of the most important components of the outer ear is the eardrum. This structure has an area of 65 mm2 and a thickness of 0.5 mm. The chief function of the eardrum is to absorb and transfer pressure changes caused by sound waves from the ear canal to the oval window. Moreover, the eardrum connects the outer and middle ear through the ear canal and oval window. Significant changes in the pressure in this area can lead Tympanic membrane perforation, is solved by Eustachian tube that connects between middle ear and mouth cavity and reduces large pressure when needed. [1] There are two possible effects when sound waves hit an eardrum: transmission and reflection as shown by Figure 7. The sensitivity of hearing is maximized when reflection is minimum and transmission is maximum. [1] Figure 7. Incident, transmitted and re ected sound waves. I represents the intensity. Sound waves travel at different speeds in different mediums because of impedance which is defined by [1] Z = ρ.v fl Where Z represents impedance, ρ represents density of material and v is speed of sound. Impedance is important in determining the level of transmission of sound wave through a medium. The transmission and reflection ratios can be defined by [1] 2 I ref (Z2 − Z1) = Iin (Z2 + Z1)2 (2Z2)2 Itran Z1 = x Iin Z2 (Z2 + Z1)2 This shows that transmission and reflection ratios depend on impedance. For example sound wave travels in air and water differently due to different reflection and transmission ratios as given below as an example: [1] For air, For water, Zair= 430 kg/m2.s Z water= 1.64 x 106 kg/m2.s Zmuscle= 1.48 x 106 kg/m2.s Zmuscle= 1.48 x 106 kg/m2.s I ref = 0.9988 I in I ref = 0.0026 I in I t ran = 0.0012 I in I t ran = 0.9974 I in The above calculations show that for air there is a large difference between impedances and therefore there is impedance mismatch. This causes transmission ratio to be low and reflection ratio of sound waves to be high. For water, there is a small difference between impedances and therefore there is impedance match. This causes transmission ratio to be high and reflection ratio to be low. Therefore impedance match will guarantee good transmission of signal. [1] References [1] https://www3.nd.edu/~nsl/Lectures/mphysics/Medical%20Physics/Part%20I. %20Physics%20of%20the%20Body/Chapter%204.%20Acoustics%20of%20the%20Body/ 4.3%20Physics%20of%20the%20ear/Physics%20of%20the%20ear.pdf [2] https://www.aplustopper.com/human-ear-structure-and-function/ [3]https://courses.physics.illinois.edu/phys406/sp2017/Lecture_Notes/ P406POM_Lecture_Notes/P406POM_Lect5.pdf [4] https://www.open.edu/openlearn/ocw/pluginfile.php/654899/mod_resource/content/1/ sd329_1_reader1.pdf
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