2. Acoustic Wave Equation

Transcription

1 2. Acoustic Wave Equation 2.1. Acoustic Wave Equation Acoustic wave equation is a second order partial differential equation. The acoustic wave equation for one dimensional plane wave in Cartesian coordinates is (, ) = 1 (, ) Where is sound pressure, is dimension, is time and is propagation speed of sound. The above one dimensional plane wave is a special case of sound wave, the general sound wave equation in Cartesian coordinates is + + (,,, ) = 1 1 (,,, ) The above general sound wave equation is only valid in Cartesian coordinates. It is not valid in cylindrical coordinates and spherical coordinates. In reality, a point sound source radiate spherical wave and is usually formulated in spherical coordinates. To prepare for transferring the acoustic wave equation to spherical coordinates, the acoustic wave equation is formulated in vector format which is independent of choice of coordinate and can be transfer to many different coordinates. The acoustic wave equation in vector form is (, ) = 1 (, ) Where the symbol is gradient operator and is called Laplacian operator. The physical meaning of the gradient is the same in any coordinate system. However, the actual formulation of the gradient depends on the choice of coordinate system. The gradient in Cartesian coordinates is = + + And the Laplacian operator in Cartesian coordinates is = + + The Laplacian operator in Spherical coordinates is = sin sin + 2 sin Where = sin cos, = sin sin and = cos, as shown in Reference page. The acoustic wave equation in spherical coordinates will be explained and solved in a latter section. Example A one dimensional sound pressure in Cartesian coordinates is given as

2 (, ) = ( cos() () ) a) Show that the assumed sound pressure (, ) can be expressed in the following complex exponential form as: [Form 3] (, ) = () () + () () + () () + () () 1 4 b) Show that the sound pressure (, ) expressed in complex exponential form (from part a) can also be expressed in trigonometric form as, Example Solution (, ) = cos() cos() a) Write the assumed sound pressure in the complex exponential form: (, ) = cos() () Remove the symbol by adding its complex conjugate and divide the whole thing by 2 (, ) = cos() () + cos() () 1 2 Move the common factor cos() out to the front of the parentheses (, ) = cos() () + () 1 2 Change cos() to its complex exponential form by the Euler s formula Multiply each terms to get (, ) = () + () () + () 1 4 (, ) = () () + () () + () () + () () 1 4 b) Show that the sound pressure (, ) expressed in complex exponential form can also be expressed in trigonometric form: Rearrange these four complex exponential terms into two pairs of complex conjugate terms as p(, ) = () + () () + () 1 4 Change complex exponential form to trigonometric form as p(, ) = ( + ) + ( ) 1 2 = ()cos() Rearrange the trigonometric cosine functions to get p(, ) = ( + ) + ( ) 1 2 = (()() sin() sin() + ()() + sin() sin()) 1 2 Simplify the above equation will give p(, ) = cos() cos() 2

3 Example A complex exponential function is give as (, ) = () 1 () 4 a) Show that it satisfies the acoustic wave equation (, ) = 1 (, ) b) Shows that when =±, the sound pressure p can satisfy the acoustic wave equation Example Solution a) Calculate using the above complex exponential form: = () () 1 4 = () () 1 4 Then = = () () 1 4 = () () 1 4 Calculate using the above complex exponential form = = () () 1 4 = () () 1 4 b) Substitute the calculated and into the acoustic wave equation: = 1 () () 1 4 = 1 () () 1 4 = 1 =± It shows that when =±, the sound pressure p can satisfy the acoustic wave equation. 3

4 2.2. Three Physics Principles Behind the Acoustic Wave Equation Acoustic wave equation is derived from three equations based on three fundamental Physics Principles: I. Euler s force equation (based on Newton s Law of Motion) It is based on Newton s 2 nd Law of Motion: = It provides a relationship between pressure difference and flow velocity of air particle. II. Equation of Continuity It is based on conservation of mass It provides a relationship between particle flow velocity and density difference III. Equation of State based on Kinetic theory of gases and Conservation of Energy It is based on the kinetic theory of gases. And, it is based on Conservation of Energy. It provides a relationship between pressure difference and density difference of air particle Velocity (flow) I. Euler s Force Equation (Based on Newton s Law of Motion) II.Equation of Continuity (Based on Conservation of Mass) Pressure (difference) Density (difference) III. Equation of State (Based on Kinetic theory and Conservation of Energy) Each of these Physics principle will be explained here. I. Euler s Force Equation Euler s force equation shows the relationship between sound pressure (pressure difference) and molecular flow velocity. Euler s force equation in Cartesian coordinates is + + (,,, ) = (,,, ) Where, is sound pressure (pressure difference),,, are coordinates, is time, 4

5 is the averaged molecular density, is the flow velocity of air molecules. Euler s force equation in vector format is independent of coordinates systems. Euler s force equation in vector format is (,,, ) = (,,, ) Where the gradient operator in Cartesian coordinate is = + + For one dimensional plane wave, Euler s force equation in Cartesian coordinate can be simplified as = The Euler s force equation can be derived from Newton s 2 nd law of motion. To simplify the derivation of the Euler s force equation, we will use the one dimensional plane wave in Cartesian coordinate as example. Molecular flow velocity: + According to Newton s Law of Motion, the force applied on the above cubic will cause an acceleration on this cubic with three dimension of,,. The free body diagram of the cubic is shown here. 5

6 + Absolute + Difference Averaged Absolute Force Mass Acceleration =Averaged=Constant=1 ATM The governing equation of the cubic can be expressed as + +() = It can be simplified as = When the above equation be compared to Newton s law of motion, the left hand side of the equation is the force ( ) due to pressure difference, and the right hand side of the equation is the mass ( ) multiplies the acceleration ( ). Simplify the above by eliminate will get one dimensional Euler s force equation in Cartesian coordinates as = 6

7 II. Equation of Continuity Equation of continuity shows the relationship between molecular flow velocity and molecular density. Equation of continuity in Cartesian coordinates is = + + Where, is the flow velocity of air molecules, variable is coordinates, is time, is the averaged molecular density and is the instantaneous molecular density. Equation of continuity in vector format is independent of coordinates systems. Equation of continuity in vector format is = Where the gradient operator in Cartesian coordinate is = + + For one dimensional plane wave, equation of continuity in Cartesian coordinate can be simplified as = The equation of continuity can be derived from the law of conservation of mass. To simplify the derivation of the equation of continuity, we will use the one dimensional plane wave in Cartesian coordinate as example. Mass flow rate : = Mass passing through an unit area per unit time + : The above figure shows the mass come in and out of a unit cubic. The mass flow 7

8 rate,, is defined as mass passing through an unit area per unit time. So the amount of mass passing through an unit area, say, is equal to. According to the law of conservation of mass, the increase of mass in the unit cubic volume is equal to the Mass Flow In minus Mass Flow Out as shown in the figure below. + Increase = Mass in - Mass out of mass Mass flow rate = Mass passing through an unit area per unit time : Equation of continuity can be concluded as = + We will compare the above equation to the law of conservation of mass. The left hand side of the equation is the increase of mass in this unit cubic ( ) after a finite time ( ). The right hand side of the equation is the mass flow in ( ) minus the mass flow out ( + ) The above equation can be simplified to = And = The mass flow is replaced with to get = This is the one dimensional equation of continuity in Cartesian coordinate. 8

9 III. Equation of State The third equation needed to derive the acoustic wave equation is called the equation of state. The equation of state describes the relationship between air pressure change and mass density change as Where, is the averaged air pressure, is the instantaneous pressure, is the averaged air mass density, is the instantaneous mass density, is ratios of specific heat = 1 Pressure Density Pressure Density Equation of State = : Absolute density : Averaged density : Absolute : Averaged pressure The equation of state can be derived from the law of conservation of energy and the Kinetic theory of gases. The energy of an enclosed system is the summation of translational and rotational energy of gases as = Assume the rotational kinetic energy is linear related to the translational kinetic energy as 1 2 = ( 1) 1 2 Where is equal or greater than 1. When the particle does not have rotation, =1. When the particle has rotation, >1. 9

10 Kinetic Energy Considering Rotation of Air Particle Kinetic energy ( ) = + = = = Colliding Pressure = = Use this linear relationship between rotational energy and translational energy, the total energy of the system will be = = ( 1) 1 2 = 2 According to the kinetic theory, the pressure is directly related to the translational energy and is independent of the rotational energy. The pressure due to the colliding of single molecular is = 1 3 The derivation of the colliding pressure is based on the fundamental of kinetic energy as shown in the figure blow. Colliding Pressure (Kinetic theory of gases) = = = = = = = = = = Based on the kinetic energy and colliding pressure, the equation of state can be derived using the law of conservation of energy. According to the conservation of energy, in a closed system, the increase of energy is equal to the work done to the system. Increase of Energy: = ( ) 10

11 Work Done== ( ) The total energy includes both translation energy and rotational energy and is equal to as shown before. And the work done by a constant external pressure due to a system of decreased volume is, therefore = ( ) Work Done () = = Work Done = Increase of Kinetic Energy Kinetic energy ( ) = Even though the pressure is not effected by the rotation energy, the pressure is also directly related to the energy (only the translational energy portion, not the total energy portion) in the form of = 1 3 So the change of translation energy will be + = 1 3 Combine the change of total energy (translational and rotational) and the change of colliding energy (translational energy only) will give the following equation 2 =3( + ) and 3 = (3 +2) and (3 +2) = 3 Because the mass of the system remain constant, that is =, therefore + =0 or =. = (3 +2) 3 In the above equation is the pressure difference between the absolute pressure 11

12 and the averaged constant pressure ( = 1 ATM). Similarly, In the above equation is the density difference between the absolute density and the averaged constant density. Therefore, we can replace with and with to get (3 +2) = 3 Finally, ratios between the normalized pressure difference and the normalized density difference () is called the ratios of specific heat and is defined as (3 +2) 3 Replace () with the ratios of specific heat gives = The ratios () of specific heat is related to the ratios (α) between total energy and translation energy. The ratios () for air can be measured and is = (for air) This concludes the derivation of equation of state. Derivation of Equation of State = = + Conservation of Energy Work Done = Increase of Energy ==. + = Kinetic energy ( ) = = = + = = + Colliding Pressure = = = + 12

13 The bulk modulus is defined by the equation Based on the derived relationship between pressure and volume = and the definition of the ratio of specific heat (3 +2) 3 The adiabatic bulk modulus becomes (3 +2) = = 3 13

14 2.3. Derivation of Acoustic Wave Equation The acoustic wave equation is based on the three Physics principles as introduced in the previous section. The sound pressure can be related to the air density using the first principle: I. Euler s force equation: = The air density can be related to the flow velocity using the second principle: II. Equation of Continuity: = Finally, the air density can be related back to the sound pressure using the third principle: III. Equation of State: = Velocity (flow) I. Euler s Force Equation (Based on Newton s Law of Motion) II.Equation of Continuity (Based on Conservation of Mass) Pressure (difference) Density (difference) From the Euler s force equation, we have the relationship between pressure difference and flow velocity. The first step is to replace the flow velocity with the air mass density so that the relationship between pressure difference and mass density can be formed. This is accomplished by combining (I) Euler s force equation and (II) equation of continuity and eliminate the flow velocity as shown in the figure below. Now we have direct relationship between pressure difference and the air mass density. III. Equation of State (Based on Kinetic theory and Conservation of Energy) The second step is to replace the air mass density with the pressure difference. So that the acoustic wave equation have only one variable pressure difference. This is accomplished by combining (III) equation of state and the relationship between the pressure difference and the air mass density from the previous step. The third step is to replace the pressure difference with sound pressure (defined) as = This will give us 1 = This is the one dimensional acoustic wave equation in Cartesian coordinates. 14

15 I. = II. = Derivation of the Acoustic Wave Equation = III. = = = = = By introducing the speed of sound as = With the speed of sound, the acoustic wave can be formatted in the general wave form as = 1 For air, the speed of sound can be calculated with the following measured numbers The speed of sound is = (For air) = [] (1 ATM) = 1.30[ ] = 1.30 = This concludes the derivation of the acoustic wave equation. 15