PROPAGATION OF SOUND IN FLUIDS March, 2008

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2 TABLE OF CONTENT CHAPTER ONE 1.0 INTRODUCTION 1 CHAPTER TWO 2.0 DEFINITION OF TERMS frequency Amplitude Intensity Liquids Gases Plasma 2 CHAPTER THREE 3.0 CHARACTERISTICS OF FLUIDS 3 CHAPTER FOUR 4.0 PROPERTIES OF FLUIDS Viscosity Flowing Gravity 4 CHAPTER FIVE 5.0 FLUID FLOW AND THE CONTINUITY EQUATION 4 CHAPTER SIX 6.0 FACTORS INFLUENCING PROPAGATION OF SOUND IN FLUIDS particles in fluids 5 2

3 6.2 particles in fluids 5 CHAPTER SEVEN 7.0 SPEED OF SOUND IN FLUIDS In a given ideal gas In non-ideal gases e.g van der waals gases Implications for atmospheric acoustics 7 CHAPTER EIGHT 8.0 BASIC CONCEPT OF SOUND PROPAGATION Speed in solids Speed in liquids In water in sea water WAVE-EQUATION FOR SOUND PROPAGATION IN FLUIDS REFERENCES 3

4 CHAPTER ONE 1.0 INTRODUCTION Fluid is substance that yields readily to any force that alters its shape; thus, it conforms to the configuration of a containing vessel. Fluids are either liquid or gas and, with solids, are the most important states of matter. The particles composing a liquid are not rigidly adherent to one another, but are held together more closely than those of a gas. Thus, the volume of a liquid remains unaltered in a sealed container and has a definite surface boundary. In contrast, a gas has no natural boundary and expands and diffuses into air, diminishing in density. It is sometimes difficult to distinguish solids and fluids because even rocks and glaciers flow under favorable conditions. 2.0 DEFINITION OF TERMS CHAPTER TWO 2.1 Frequency The frequency of a sound is the number of cycles, or oscillations, a sound wave completes in a given time. Frequency is measured in hertz, or cycles per second. 2.2 Amplitude Amplitude is the characteristic of sound waves that humans perceive as volume. The amplitude corresponds to the distance that air molecules move back and forth as a sound wave passes through them. As the amount of motion in the molecules is increased, they strike the ear drum with progressively greater force. This causes the ear to perceive a louder sound. 4

5 2.3 Intensity Sound intensities are measured in decibels (db). The intensity at the threshold of hearing is 0 db, the intensity of whispering is typically about 10 db, and the intensity of rustling leaves reaches almost 20 db. Sound intensities are arranged on a logarithmic scale, which means that an increase of 10 db corresponds to an increase in intensity by a factor of 10. Thus, rustling leaves are about 10 times louder than whispering. Intensity is experessed mathematically as: 2.4 Liquids A liquid in space will form the natural shape of a sphere. This is because the attraction between its atoms or molecules is greater than the forces from their kinetic energy moving outward. A sphere is a shape with the smallest surface area for a given volume of material. A liquid sphere or drop of liquid such as water that is falling toward the Earth through the atmosphere will be a slightly flattened sphere, due to the air resistance. 2.5 Gases The molecules in a gas have more energy than when the material is in the liquid state, such that they overcome the molecular forces. A gas in space or in the atmosphere will continually spread in a shapeless form. A gas that is heavier than air may gravitate toward the floor, where it then spreads out. 2.6 Plasmas Plasma is an ionized gas, usually at extremely high temperatures. That means some of its electrons have been stripped off. Plasmas have most of the same properties as gases. 5

6 CHAPTER THREE 3.0 CHARACTERISTICS OF FLUIDS The states of matter are solid, liquid, gas and plasma. A fluid is a subset of the states of matter, consisting of liquids, gases and plasmas. This is because they have common properties that are distinct from solids. A fluid does not have a specific shape as does a solid. Instead, fluids take the shape of their containers. They also will flow or pour when under the influence of a force such as gravity. A fundamental characteristic of any fluid at rest is that the force exerted on any particle within the fluid is the same in all directions. If the forces were unequal, the particle would move in the direction of the resultant force. It follows that the force per unit area, or the pressure exerted by the fluid against the walls of an arbitrarily shaped containing vessel, is perpendicular to the interior walls at every point. If the pressure were not perpendicular an unbalanced tangential force component would exist and the fluid would move along the wall. 4.0 PROPERTIES OF FLUIDS CHAPTER FOUR Fluids display such properties as: Not resisting deformation, or resisting it only lightly (viscosity), and The ability to flow (also described as the ability to take on the shape of the container). 6

7 4.1 VISCOSITY Viscosity is a measure of the resistance of a fluid to being deformed by either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction. Thus, water is "thin", having a lower viscosity, while vegetable oil is "thick" having a higher viscosity. All real fluids (except super fluids) have some resistance to stress, but a fluid which has no resistance to shear stress is known as an ideal fluid or inviscid fluid. The study of viscosity is known as rheology. 4.2 FLOWING The major feature of a fluid is that it flows when acted upon by some force. This makes a fluid different than a solid, which may be distorted by a force but will not start to flow. Typically, the force is that of gravity, but other forces can also apply. 4.3 GRAVITY Fluids under the influence of gravity will flow or can be poured. Liquid and gasses can be poured from one container to another. Since plasmas are typically very hot, they are seldom poured. Although, you cannot see carbon dioxide (CO 2 ), you can demonstrate pouring it from one jar to another. This is shown by using dry ice to fill a jar with CO 2 and then pouring it into a jar containing a burning candle. The candle flame will be snuffed out as the invisible CO 2 is poured into the jar. CHAPTER FIVE 5.0 FLUID FLOW AND THE CONTINUITY EQUATION Fluids, by definition can flow, but are essentially incompressible. This provides some very useful information about how fluids behave when they flow through a pipe, or a hose. Consider a hose whose diameter decreases along its length, as shown in the Figure below. The ``continuity equation'' is a direct consequence of 7

8 the rather trivial fact that what goes into the hose must come out. The volume of water flowing through the hose per unit time (i.e. the flow rate at the left must be equal to the flow rate at the right or in fact anywhere along the hose. Moreover, the flow rate at and point in the hose is equal to the area of the hose at that point times the speed with which the fluid is moving: Flow rate = (area) x (velocity) CHAPTER SIX 6.0 FACTORS INFLUENCING PROPAGATION OF SOUND IN FLUIDS A number of factors influence how far sound travels underwater and how long it lasts. 6.1 PARTICLES IN FLUIDS Particles in fluids can reflect, scatter, and absorb certain frequencies of sound just as certain wavelengths of light may be reflected, scattered, and absorbed by specific types of particles in the atmosphere. Seawater absorbs 30 times the amount of sound absorbed by distilled water, with specific chemicals (such as magnesium sulfate and boric acid) damping out certain frequencies of sound. Researchers also learned that low-frequency sounds, whose long wavelengths generally pass over tiny particles, tend to travel farther without loss through absorption or scattering. 6.2 EFFECTS OF SALINITY, TEMPERATURE AND PRESSURE Further work on the effects of salinity, temperature, and pressure on the speed of sound underwater has yielded fascinating insights into the structure of the ocean. Speaking generally, the ocean is divided into horizontal layers in which sound speed is influenced more greatly by temperature in the upper regions and by pressure in the lower depths. At the surface is a sun-warmed upper layer, the actual temperature and thickness of which varies with the season. At mid latitudes, this layer tends to be 8

9 isothermal, that is, the temperature tends to be uniform throughout the layer because the water is well mixed by the action of waves, winds, and convection currents; a sound signal moving down through this layer tends to travel at an almost constant speed. Next comes a transitional layer called the thermocline, in which temperature drops steadily with depth; as temperature falls, so does the speed of sound. However, at a point roughly 600 meters to 1 kilometer (0.4 to 0.6 miles) below the surface, further changes in temperature are slight (the water the rest of the way to the bottom is effectively isothermal). Now the dominant factor influencing the speed of sound is the increasing pressure, which causes sound to speed up. CHAPTER SEVEN 7.0 SPEED OF SOUND IN FLUIDS The speed of sound is variable and depends mainly on the temperature, and the properties of the substance through which the wave is traveling. For, example, in low molecular weight gases, e.g. such as helium, sound propagates faster compared to heavier gases, such as xenon. 7.1 IN A GIVEN IDEAL GAS. The sound speed depends only on its temperature. At a constant temperature, the ideal gas pressure has no effect on the speed of sound, because pressure and density are also proportional to pressure and they have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly. 7.2 IN NON-IDEAL GASES E.G VAN DER WAALS GASES The proportionality in this case is not exact and there is a slight dependence on the gas pressure even at a constant temperature. Humidity also has a small, but measurable effect on sound speed (increase of about 0.1%-0.6%). This is because some oxygen and nitrogen molecules are replaced by the lighter molecules of water. 9

10 7.2.1 IMPLICATIONS FOR ATMOSPHERIC ACOUSTICS In the Earth's atmosphere, the most important factor affecting the speed of sound is the temperature. Since temperature and thus the speed of sound normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source. The decrease of the sound speed with height is referred to as a negative sound speed gradient. However, in the stratosphere, the speed of sound increases with height due to heating within the ozone layer which produces a positive sound speed gradient CHAPTER EIGHT 8.0 BASIC CONCEPT OF SOUND PROPAGATION. The transmission of sound can be explained using a toy model consisting of an array of balls interconnected by springs. For a real material, the balls represent molecules and the springs represent the bonds between them. Sound passes through the model by compressing and expanding the springs, transmitting energy to neighboring balls, which transmit energy to their springs, and so on. The speed of sound through the model depends on the stiffness of the 20 springs (stiffer springs transmit energy more quickly). Effects like dispersion and reflection can also be understood using this model. In a real material, the stiffness of the springs is called the elastic modulus, and the mass corresponds to the density. All other things being equal, sound will travel more slowly in denser materials, and faster in stiffer ones. For instance, sound will travel faster in iron than uranium, and faster in hydrogen than nitrogen, due to the lower density of the first material of each set. At the same time, sound will travel faster in iron than hydrogen, because the internal bonds in a solid like iron are much stronger than the gaseous bonds between hydrogen molecules. In general, solids will have a higher speed of sound than liquids, and liquids will have a higher speed of sound than gases. 10

11 Some people have a faulty idea that the speed of sound increases with increasing density. This is usually illustrated by presenting data for three materials, such as air, water and steel. With only these three examples it indeed appears that speed is correlated to density, yet including only a few more examples which would show this assumption to be incorrect. The speed of sound increases with the stiffness of the material, and decreases with the density. If relativistic, effects are important, the speed of sound may be calculated from the Relativistic Euler Equations. In a non-dispersive medium, sound speed is independent of sound frequency, so the speeds of energy transport and sound propagation are the same. For audible sounds air is a non-dispersive medium. But air does contain a small amount of CO 2 which is a dispersive medium, and it introduces dispersion to air at ultrasonic frequencies (>20 28 khz). In a dispersive medium sound speed is a function of sound frequency. The spatial and temporal distribution of a propagating disturbance will continually change. Each frequency component propagates at its own phase velocity while the energy of the disturbance propagates at the group velocity. The same phenomenon occurs with light waves 8.1 SPEED IN SOLIDS In a solid, there is a non-zero stiffness both for volumetric and shear deformations. Hence, in a solid it is possible to generate sound waves with different velocities dependent on the deformation mode. In steel=20 the speed of sound is approximately 5,100 ms -1. In Berrylium, substance with the greatest known ratio of stiffness to density at room temperature and pressure, the speed of sound reaches 12,870 ms -1, which is the highest known for solids in standard conditions. In a solid with lateral dimensions much larger than the wavelength, the sound velocity is higher. It is found by replacing Young's modulus E the plane wave modulus M, which can be expressed in terms of the Young's modulus. 11

12 8.2 SPEED IN LIQUIDS In a fluid the only non-zero stiffness is to volumetric deformation (a fluid does not sustain shear forces). The speed of sound in liquids depends upon the temperature. This is useful in monitoring the temperature of oceans and other large bodies of water because pulses of low frequency sound can travel thousands of kilometers through the ocean and still be detected. The pulse traverse time can be measured with a network of stations to monitor changes in the temperature of the intervening water. When compared to changes predicted by climate models, this can give some indication of whether global warming from the greenhouse effect is occurring IN WATER The speed of sound in water is of interest to anyone using under water sound as a tool, whether in a laboratory, a lake or the ocean. Examples are sonar acoustics communication and acoustical oceanography. In fresh water, sound travels at about 1497 m/s at IN SEA WATER Sound speed as a function of depth at a position north of Hawaii in the Pacific Ocean derived from the 2005 World Ocean Atlas. The SOFAR channel is centered on the minimum in sound speed at ca. 750-m depth. In salt water that is free of air bubbles or suspended sediment, sound travels at about 1500 m/s. The speed of sound in seawater depends on pressure (hence depth), temperature (a change of 1~ 4 m/s), and salinity (a change of 1 ~ 1 m/s), and empirical equations have been derived to accurately calculate sound speed from these variables. Other factors affecting sound speed are minor. Equations for sound speed in sea water are accurate over a wide range of conditions, but are far more complicated. 12

13 Liquid Sound Speed in Liquids Temperature ( C) Speed in m/s Water Water Methyl alcohol Sea water 3.5% salinity Temperature measurement from sound speeds CHAPTER NINE 9.0 WAVE-EQUATION FOR SOUND PROPAGATION IN FLUIDS. Sound waves are traveling pressure waves in a fluid (gas or a liquid). A consequence of this is that there can be no sound propagation in vacuum. We verify this by showing that an otherwise noisy personal alarm device becomes completely silenced when placed in a vacuum. Corresponding to the pressure wave is a displacement wave s(x-vt) of atoms from where they would be if there had been no sound propagating. The relationship between the displacement wave and the deviation from average pressure is a characteristic property of the fluid: Note that ds/dx rather than s appears on the right hand side of the equation because ds/dx measures the deformation of the fluid from equilibrium: A constant s simply corresponds to an overall displacement of the fluid. The negative sign is there because a contraction ds/dx<0 gives an increase in pressure and thus a positive p. B is called the bulk modulus of the fluid and has dimensions of a pressure. It can be shown that where 13

14 is the ratio of the constant pressure to the constant volume specific heat. To derive the wave velocity in a fluid we write Newton's second law for a cylindrical slice of thickness and area A. The force on this slice is We also calculate Newton tells us to equate F and ma and this gives us the wave-equation From which we conclude that the speed of sound is Neglecting the we could immediately have written down this formula based on dimensional analysis or an analogy with the previously derived expression for the wave-velocity on a taut string. The formula was first derived by Newton himself. Putting in numbers we get for air at ambient pressure which is indistinguishable from the measured value. Note that we have derived the interesting result that at constant pressure the velocity of sound is greater in a light than a heavy gas. 14

15 REFERENCES Cole, R.H Underwater explosions. Princeton University Press, Princeton, N.J. Microsoft Encarta Reference Library Microsoft Corporation. All rights reserved. Wikipedia, the free encyclopedia 15