Transmission from one fluid to another: Transmission from one fluid to another: oblique incidenceoblique incidenceApplying the condition of continuity of pressure at Applying the condition of continuity of pressure at the boundary z=0 yieldsthe boundary z=0 yieldsApplying the condition of continuity of normal Applying the condition of continuity of normal velocityvelocity(3-7-11)(3-7-10)Law of reflection:Law of reflection:the angle of incidence is equal to the the angle of incidence is equal to the angle of reflection, and Snells lawangle of reflection, and Snells law : :Snells law :Snells law :From the equation (3-7-10), we obtain :From the equation (3-7-10), we obtain :From the equation (3-7-11), we obtain :From the equation (3-7-11), we obtain :whenwhenGives the angle of incidence for which there is no Gives the angle of incidence for which there is no reflection and therefore complete transmission. reflection and therefore complete transmission. This angle known as the This angle known as the angle of intromission angle of intromission : :It will be exist only if It will be exist only if The totally reflected The totally reflected From the reflection coefficient equation, we seeFrom the reflection coefficient equation, we seeThe angle of transmission is real and less The angle of transmission is real and less than the angle of incident.than the angle of incident.A transmitted beam exits in the second fluid. A transmitted beam exits in the second fluid. (1)The angle of transmission is again real The angle of transmission is again real but greater than the angle of incidencebut greater than the angle of incidence(2) when(2) whenWhere is the critical angle defined by Where is the critical angle defined by (3) When In this case , the transmitted wave assume a In this case , the transmitted wave assume a very peculiar form, the reflection coefficient is very peculiar form, the reflection coefficient is a complexa complexConsiderConsider560 12a18000000900 Angle of incidenceIRIIRIReflection of Reflection of coefficientcoefficientTransmission through a layer: Transmission through a layer: Normal incidenceNormal incidenceAssume that a layer of uniform thickness L Assume that a layer of uniform thickness L lies between two dissimilar fluids and that a lies between two dissimilar fluids and that a plane wave is normally incident on its plane wave is normally incident on its boundary, as suggested in following Fig. boundary, as suggested in following Fig. 0lLet the characteristic impedances of the Let the characteristic impedances of the fluids be Zfluids be Z1 1, Z, Z2 2 and Z and Z3 3 respectively respectivelyFig. Fig. Reflection and transmission of plane waves Fig. Reflection and transmission of plane waves normally incident on a layernormally incident on a layerWhen an incident signal in fluid I arrives at the When an incident signal in fluid I arrives at the boundary between fluids I and II, some of the boundary between fluids I and II, some of the energy is reflected and some is transmitted energy is reflected and some is transmitted into the second fluid. The portion of the wave into the second fluid. The portion of the wave transmitted will proceed through fluid II to transmitted will proceed through fluid II to interact with the boundary between fluids II interact with the boundary between fluids II and III, where again some of the energy is and III, where again some of the energy is reflected and some transmitted. The reflected reflected and some transmitted. The reflected wave proceeds back to the boundary between wave proceeds back to the boundary between fluids I and II, and the whole process is fluids I and II, and the whole process is repeated.repeated.From the wave equation and the equation of From the wave equation and the equation of motion, we obtain( neglect the time factor):motion, we obtain( neglect the time factor):In fluid IIn fluid IIn fluid IIIn fluid IIIn fluid IIIIn fluid IIIAt x=0, two boundary conditionsAt x=0, two boundary conditionsAt x =At x =l l, two boundary conditions, two boundary conditions(3-7-28)(3-7-28)Define the Define the reflection coefficientreflection coefficient of the layer :of the layer :Similarly, define the refraction and intensity Similarly, define the refraction and intensity transmission coefficients of the layer :transmission coefficients of the layer :From the equation of (3-7-28)From the equation of (3-7-28) Where Z Where Z21 21 is the is the input impedanceinput impedance of the of the layer. The boundary between fluid I and fluid layer. The boundary between fluid I and fluid II corresponds to an impedance. Then the II corresponds to an impedance. Then the form of the pressure reflection coefficient form of the pressure reflection coefficient Note : ZNote : Z21 21 has both real and has both real and imaginary componentsimaginary componentsFrom (3-7-28)From (3-7-28)Now the Now the sound intensity transmission coefficientsound intensity transmission coefficient is isThere exist a number of special forms of the There exist a number of special forms of the transmission coefficient which are of particular transmission coefficient which are of particular interest. In the following cases we can further interest. In the following cases we can further simplify the sound intensity transmission coefficient. simplify the sound intensity transmission coefficient. From the relationship of sound energy :From the relationship of sound energy :When medium 3 is the same as medium 1, When medium 3 is the same as medium 1, we have Zwe have Z3 3=Z=Z1 1, and , and If, in addition,If, in addition, (1) If the length of middle (1) If the length of middle mediummedium(2) If (2) If (3) If (3) If T=1T=101/201/103/201/51/4T0.9770.9870.9981.00 Maximum transmission of plane acoustic Maximum transmission of plane acoustic waves from water into steel is required. What waves from water into steel is required. What should be the optimum characteristic should be the optimum characteristic impedance of the material to be placed impedance of the material to be placed between the water and the steel? If the between the water and the steel? If the thickness of the layer of material to be used is thickness of the layer of material to be used is 0.02m and the frequency of sound transmitted 0.02m and the frequency of sound transmitted is 1000Hz, find the speed of sound in the is 1000Hz, find the speed of sound in the material and the density of the material.material and the density of the material.Textbook P 277 3-15Textbook P 277 3-15The sound intensity transmission The sound intensity transmission coefficient for transmission through coefficient for transmission through three media at normal incidence is three media at normal incidence is given by :given by :Where ZWhere Z1 1,Z,Z2 2,Z,Z3 3 are the are the characteristic impedancescharacteristic impedances of the media, kof the media, k2 2 is the wave number of medium 2, is the wave number of medium 2, and l is the thickness of medium 2.and l is the thickness of medium 2.For maximum transmission of For maximum transmission of acoustic intensity,acoustic intensity,If IfThen Then and the transmission coefficient becomes and the transmission coefficient becomes Where ZWhere Z1 1=1.48*10=1.48*106 6 MKS rayls is the characteristic MKS rayls is the characteristic impedance of water, and Zimpedance of water, and Z3 3=47*10=47*106 6 MKS rayls is MKS rayls is the the characteristic impedancecharacteristic impedance of steel of steelHence, ZHence, Z2 2=8.35*10=8.35*106 6 MKS rayls MKS raylsTherefore 100% transmission of sound Therefore 100% transmission of sound occurs only for bands of frequencies occurs only for bands of frequencies centered about the particular frequencies centered about the particular frequencies for whichfor whichOr cOr c2 2=4=4lf lf=4(0.02)1000=800m/sec, and=4(0.02)1000=800m/sec, and p=p=Z Z2 2/c/c2 2=8350000/800=10500kg/m=8350000/800=10500kg/m3 3HomeworkHomeworkTextbook P277 3-17, 3-18Textbook P277 3-17, 3-18