 Let us start our today's lecture of this NPTEL video course on Geotechnical Earthquake Engineering. On this course, we are now going through our module 5, which is wave propagation. Let us quickly recap what we have learnt in our previous lecture. In the previous lecture, we have seen for one-dimensional wave propagation in an infinite rod. For the case of longitudinal wave, what is the particle velocity? How we can estimate the particle velocity from the displacement function and using the strain displacement relationship as well as the stress versus strain relationship? We arrived at this relationship and the v p is nothing but the primary wave velocity or p wave velocity or longitudinal wave velocity which is expressed as root over m by rho. So, this is the relationship as we have seen earlier as I was mentioning v p is nothing but root over m by rho, m is nothing but the strain modulus rho is the density of the material and this is the relationship between the particle velocity with respect to stresses and this parameter the multiplication of density with respect to the velocity of the wave of this p wave that function rho v p is known as specific impedance. So, that is the specific impedance of the material. Then in the previous lecture, we have derived what will be the governing equation of motion for a torsional wave when it travels through an infinite rod in one dimension. So, this is the derivation we have seen the difference of torque at both the ends of this infinitesimal element of the rod with the different magnitude of rotational displacement theta at this end and at this end. Then after considering the equilibrium we had derived at this relationship of torque versus rotation in this form where J is the polar moment of inertia after that we had applied the relationship between torque and rotational displacement through the shear modulus and polar moment of inertia in this form. And on further simplification what we had arrived finally is this one where this parameter again we have defined as square of shear wave velocity. So, shear wave velocity is nothing but root over G by rho where G is the shear modulus of the material and rho is the density of the material. And hence the governing equation of motion in terms of the acceleration this is the rotational acceleration and this is the second order differential of the rotational displacement with respect to the x axis system or space coordinate system related through the shear wave velocity of the media for this torsional wave. So, general form of governing equation of motion for any wave traveling in this one dimension can be represented by this equation. So, for longitudinal wave it will be v p for torsional wave it will be v s and corresponding displacement right for longitudinal wave it is u for torsional wave it is theta that is the rotational component of the displacement. And we have seen for this type of governing equation of motion we have the general form of solution expressed in this format. Now, this general form of solution again we can get a particular solution depending on the loading conditions. So, these functions depends on what type of load is applied to the system. So, we have also seen for harmonic loading this is the response what we have obtained where we have defined this parameter k as a wave number which is nothing but ratio of the circular frequency to the velocity of wave in that media. We had also defined another parameter which is known as wavelength lambda which is related to the wave number in this format. And we have seen the similarity between the relationship of this wavelength lambda with respect to wave number k. And in the time coordinate system we have seen the relationship between time period with respect to the natural circular frequency. After that what we have seen in the previous lecture we can see the slide here. We have considered various boundary conditions like this where center line where displacement will be 0. Then at the fixed end where the displacement has to be 0. Also a boundary effect where both says stress and displacement has to be 0. Then at the free end where stress has to be 0. Then we have seen the boundary effect that is material boundary when there is a layered body. How this incident wave then transmitted wave and reflected wave they are related to each other through this example that is for one dimensional case. When wave is travelling in this one dimension in a layered body like this we have seen this is material one this is material two. We have defined the x coordinate system in such a way that this is the positive x direction and this boundary itself is the x equals to 0. And incident wave comes from material one. Some of that goes in the material two as transmitted wave and some of the wave comes back in the same media as reflected wave. Then we have seen how this stresses of incident transmitted and reflected wave can be represented in a harmonic fashion. Also how the displacement functions can be represented through this form. Then applying the known stress strain relationship through this modulus that is when we are talking about longitudinal wave we have to take constrained modulus like this. So, we have obtained what are the relationship between amplitudes of the stresses and displacement. So, this is the relationship of amplitude only stress amplitude with respect to the displacement amplitude. Also we had seen in the previous lecture that at the interface of two material this boundary condition has to be satisfied that is the displacement compatibility and equilibrium of stress. So, these two conditions has to be maintained. Hence this incident wave displacement at x equals to 0 plus the reflected wave displacement at x equals to 0 should be equals to the transmitted wave displacement at x equals to 0. So, that is from the displacement compatibility whereas from equilibrium of stress condition we will get stress due to that incident wave at x equals to 0 plus stress due to the reflected wave at x equals to 0 should be equals to the stress due to the transmitted wave at x equals to 0. Now, let us continue this in today's lecture further. So, what we can write from the previous known relationship that if we equate the amplitudes the displacement amplitude of incident wave plus displacement amplitude of reflected wave should be equals to the displacement amplitude of transmitted wave in the similar form. So, this is from the displacement compatibility relationship and from the equilibrium of stress relationship we get the amplitude of incident wave plus amplitude of reflected wave should be equals to the amplitude of this transmitted wave. Now, at the interface using the known relationship of this one what we can write further by putting the values of this a i and a r with a t we will get this relationship. So, on further rearrangement if we do the further rearrangement of this relationship of displacement amplitude to the reflected and incident wave with respect to the transmitted wave what we will get suppose this reflected wave amplitude we are interested to know because in most of the cases what will be known the incident wave properties will be known from a media we will know what is the stress and displacement of incident wave we have to now find out what are the stresses and displacement in the reflected and transmitted wave that is what we are more concerned about in practice. So, this is the correlation if you put this simple relationship in this equation then on further simplification you will get like this and knowing this a i and a r to determine as so a t also you can determine with respect to a i in this form because a r we have represented in this format. So, hence you can represent your transmitted wave amplitude also in terms of this incident wave amplitude displacement amplitude. Now, let us look at this parameter very carefully this rho times v that is density times the velocity what is that we have already mentioned that is nothing, but in the previous lecture we discussed it is specific impedance this is nothing, but specific impedance of material 2 and this is the specific impedance of material 1. So, that ratio which is called as impedance ratio. So, what is impedance ratio? So, the definition of impedance ratio is nothing, but it is the ratio of specific impedance of two materials and when we are taking that ratio always in the numerator there will be the material which is in the transmitted zone that is the incident wave where from it is the material is the wave is coming that will be in the denominator. So, that is why you can see this rho 1 v 1 for material 1 is in the denominator that is the incident wave region that is the specific impedance of the material where the incident wave is coming from and rho 2 v 2 is the specific impedance of material 2 where the transmitted wave goes in. So, that is the definition of impedance ratio this is the very important parameter because this defines through the material parameter of two materials this gives us the idea how much displacement amplitude of transmitted wave and how much stress amplitude of transmitted wave will occur for a particular incident wave. Why this is important as we I am mentioning because suppose when earthquake occurs at a large depth. So, basically from base rock level or bed rock level that wave will start travelling where that earthquake focus is existing through this form of seismic waves. Now, when this wave travels and finally, it comes close to the ground surface we should know that after travelling through various layers of material that is from rock to stiff soil to soft soil typically what will be the changes in the displacement amplitude and stress amplitude of the stresses because that is most important we need to find out. So, there this use of impedance ratio is very important which we will be easily able to obtain from knowing the density and that particular velocity of the wave suppose we if we are talking about shear wave velocity we have to take shear wave velocity of two material. If we are talking about the primary wave velocity we have to take corresponding primary wave velocity in the two medium like that. So, we have to find out the impedance ratio of the two material, but remember the impedance ratio for primary wave or shear wave etcetera will remain same for between two material between two given material why it will not vary between primary and secondary because if we have remembered the relationship between primary wave and secondary wave that is three is through a constant relationship between the ratio. So, it is a ratio of two similar parameters. So, that is why it gives us the similar value of impedance ratio whether it is a p wave or s wave. So, now what we can see from this slide the displacement amplitude of the reflected and transmitted waves they can be represented through this impedance ratio in this simplest format. So, these two equations are very very important we should be able to find out what is the displacement amplitude of the reflected wave with respect to the incident wave amplitude displacement amplitude over this impedance ratio and what is the displacement amplitude of the transmitted wave related to this displacement amplitude of the incident wave through this impedance ratio that is why impedance ratio is so important. And how to estimate this incident wave displacement amplitude if you know the loading condition or the stresses which are getting developed for your incident wave. So, this is the stress amplitude this is how the displacement is related to the stress amplitude in this format that we have already seen the derivation in the previous lecture. Also for the reflected wave displacement amplitude is related to the reflected wave stress amplitude in this form and the transmitted wave displacement amplitude is related to the transmitted wave stress amplitude in this form through their modulus. Now, stresses they can be expressed in the similar fashion like through this impedance ratio like this that is the reflected wave stress can be obtained using this relationship for a given stress amplitude of incident wave also the transmitted wave stress amplitude can be obtained from the incident wave stress amplitude through this impedance ratio. So, what does it mean if suppose we know the stress amplitude of the incident wave of any particular earthquake and if we know the material property of two different layers of media we can easily estimate all other parameters can you see. So, only parameter we should know is this sigma i. So, if sigma i is known you can compute from sigma i a i once you know a i you will get a r a t once you know sigma i you will get sigma r sigma t clear. So, let us look at this table this table is taken from Kramer's book table 5.1 what are the influences of this impedance ratio value that alpha z value on this displacement and stress amplitude of the reflected and transmitted waves. If you see the impedance ratio equals to 0 what does it mean impedance ratio equals to 0 let us go back to the definition impedance ratio 0 means this is 0 am I right. So, in that case incident wave displacement amplitude and reflected wave displacement amplitude they remain same transmitted wave displacement amplitude is double of this that you can easily get by putting alpha z equals to 0 in this equation and incident wave alpha i will be reflected wave minus alpha i and transmitted wave stress amplitude will be 0. What is this situation can you guess it is nothing but it is nothing but a free boundary where the stress at the boundary condition has to be 0 that is what it shows suppose if we are talking about shear wave velocity when this becomes 0 when we comes close to the ground surface because it cannot travel in your air media am I right. So, in that case you will get this rho 2 V 2 equals to 0. So, that is nothing but when you come to a boundary and end of a material there you will get displacement amplitude gets doubled but stress amplitude there will be 0 that is what if you remember in our previous lecture we have discussed this we have mentioned that that boundary effect of free end what happens at the free end stress becomes 0 but displacement is doubled can you see this. So, that is what now we have seen through this mathematical expression also which is valid. Now, let us check another case impedance ratio equals to 1 what does it mean impedance ratio equals to 1 that means this material and this material are same that is there is no in fact any material boundary the wave is traveling through the same material. So, what is expected same incident wave only will continue. So, let us look at the displacement amplitude of incident wave A i reflected is 0 which is quite obvious nothing should gets reflected entire thing should continue as a transmitted. So, that is the same wave will continue. So, A i will continue and let us look at the stress conditions incident wave stress amplitude is sigma i there is nothing called reflected and entire thing gets transmitted that is the same wave continues. So, sigma i clear. So, these are the cross check whether these derivations of equations are correct or not through this common known conditions and what is the impedance ratio of infinity what does it mean when it will become infinity. That means if this one becomes 0 suppose there is a wave safety wave which travels in the air also goes to another media. So, we have incident wave displacement amplitude reflected wave displacement amplitude is negative of that that is in opposite direction negative sign indicates it goes in the opposite direction always remember all the negative sign indicates the opposite direction transmitted nothing gets transmitted whereas, for stress amplitude what happens the incident wave and reflected wave stress amplitudes are same for transmitted wave stress amplitude it gets doubled. Now, what are the common practical form of impedance ratio for our geotechnical earthquake engineering if we go back to this definition of impedance ratio we will see typically when we come from layered soil. So, let us say here say we are coming from stiff soil rock soft soil this is the typical profile. Now from rock your waves are coming incident wave. So, some of that will get transmitted some of that will get reflected am I right. Now, depending on the values of this rho and v of this material and this material rho of rock v of rock and rho of soil v of soil which one will have more density of course, the rock compared to soil. Now, which one will have more velocity of wave of course, again rock compared to soil. So, the which material will have more specific impedance of course, rock will have more specific impedance than the soil. So, when we are talking about this alpha z between these two layers we are considering alpha z between these two layers which is rho of soil v of soil by rho of rock v of rock because generally this waves are coming from here. So, this is our material 1 here and this is material 2. So, that is what what we have seen in this slide the definition of impedance ratio is specific impedance of material 2 divided by specific impedance of material 1. Now, which parameter is more this parameter is much higher than this. So, now rho s v s is much lower than rho r v r that is the typical case for our geotechnical earthquake engineering which practically we will get at site. So, at site if we get this situation obviously our alpha z should be all practical cases will be less than 1. So, that is the practical range of alpha z it should be between 0 to 1 equality of 0 also we have seen equality of 1 also we have seen. So, most practical site condition in terms of our geotechnical earthquake engineering is alpha z should be between 0 to 1. So, if alpha z is at 0 to 1 let us see what happens let us look at this table once again. So, alpha z value is between 0 to 1. So, this are the common ranges as it is given 0.25 0.5 it can be different values based on your known soil parameters or rock parameters that can be any value between 0 to 1 what happens in this cases let us say alpha z is 0.5 for that 0.5 what happens when displacement amplitude incident wave is a i reflected is a i by 3 just putting that alpha z equals to 0.5 you will get this you will get this also right similarly you will get this and this that is what we are doing here. So, alpha z of 0.5 will give us reflected wave displacement amplitude much lower than what is the displacement wave amplitude for the incident wave. But let us look at the transmitted wave it is 4 a i by 3 what does it mean that means the displacement amplitude of transmitted wave increased from that what was there for the incident wave let us look at the stresses stress incident wave is this one reflected wave is this minus that is opposite direction and transmitted wave stresses two-third of the incident wave. So, stresses is not increasing it reducing for most of our practical geotechnical earthquake engineering problem. But where the big difficulty arises difficulty arises with respect to this displacement that is the reason an earthquake wave which is not imperceptible or not that much damaging at large depth in rocks media when it reaches through various soft soil there will be manifold increase of this displacement amplitude of that wave that creates more displacement when it reaches or goes through a stiffer to a softer media that is the reason as I have already mentioned in the introductory lecture Mexico City earthquake of 1985 is an example of this phenomenon of soil amplification that amplification mostly occurs in the case of when you have a very soft soil. So, from bedrock to if the wave travels through various soil layers and finally, it goes through a soft soil media you will see displacement amplitude of the transmitted wave in the soft media gets magnified gets increased much more than what that incident wave was clear. So, we have to be very careful about the values. So, which we can easily estimate in the beginning itself knowing the material property of various layers. So, that is why knowing the material property is very important to estimate the impedance ratio which automatically will give us the what will be the displacement amplitude of any transmitted wave. Now, let us come to the three dimensional wave propagation so for three dimensional wave propagation let us first take a note of the stress notations. So, this is a three dimensional element. So, far we have discussed about only one dimensional wave propagation. Now, we are considering wave is travelling in all the three directions. So, this is x coordinate system y coordinate system and z coordinate system we have the normal stresses in x direction sigma x x in y direction sigma y y in z direction sigma z z and the shear stresses like sigma x y this indicates as we know the first parameter indicates the plane the second one indicates in which direction it works. So, sigma x means it is acting on x plane in y direction this is sigma x z in x plane again, but in z direction these two are the shear stresses. Again on this phase we can have sigma y x sigma z y so these are this is actually sigma y z it should be that is on y plane in z direction now considering the stress notations of equivalence that is shear stresses of sigma x y should be equals to sigma y x y because to maintain the equilibrium of the body otherwise if they are different there will be a formation of a couple. So, to maintain the equilibrium the other phase shear stress has to be the same and in opposite direction like sigma x y in the opposite direction should act over here and this sigma y x should be equals to sigma x y to maintain the equilibrium of the cube or of the soil element in three dimension. Similarly, sigma x z should be equals to sigma z x and sigma y z should be equals to sigma z y. So, that is why this shear stresses over here sigma z y in y direction sigma z x in x direction this is sigma actually z y it will be y z because in z direction. So, direction will come last, but as we know sigma y z equals to z y that is why it is written as sigma z y similarly this is sigma y x in the x direction which is equals to the sigma x y. So, knowing this stress condition let us start deriving that is for three dimensional wave propagation what will be for three dimensional wave propagation what will be the basic equation of motion that we are planning to find out. So, now let us take the stresses on the face let us say this is our x axis this is our y axis and this is our x axis. And this is our z axis let us take only in one direction first then it will be easy to understand in the similar other directions as well. So, now in the x direction we have at this face. So, this is basically the origin as you can see. So, on this face let us say the normal stress is sigma x x and what are the dimensions of various sides of this small infinitesimal element. Let us say in x direction it is d x in y direction let us say it is d y and in z direction let us say it is d z. So, this to this is d x this to this is d y and this to this is d z. These are the dimensions of the material and let us say the density of the material is known as rho corresponding displacements also let us define in x direction let us say displacement is denoted as u in y direction let us say displacement is denoted as v and in z direction displacement is denoted as w u v w. These are the corresponding displacement in the direction of x in the direction of y in the direction of z fine. Now, if stress when wave is travelling through this infinitesimal element what is happening? This is the stress normal stress at one face the other face on this side the stress will be sigma x x plus there will be change of the dimension of sigma x x over del x into this distance d x that is the increment in the change or stress condition on the other side of the element. Why it happens? Because wave travels through this media. So, there is an inertia acting on the system that inertia will create the stress difference between two faces like we have seen in the one dimensional wave propagation equation also the similar thing we are observing over here. So, let us look at the derivation over here let us not look at the slide here first we will derive then we will go to the slide. So, now we are considering the shear stresses the shear stress in this direction is this is sigma x y and in this direction we have sigma x z. Now, as I said let us first concentrate only direction of wave in the x direction of wave in the x direction only. So, we will be interested to know all the stresses acting in x direction. Now, this is not in x direction this is not in x direction. So, we need not to consider these at this moment because we are only considering the x direction stresses. So, what are the x direction stresses we will see here x direction stresses will be on this space there will be this direction this is sigma it is in which plane it is in z plane in x direction. So, it will be z x if we can write the same notation as we have mentioned x z also we can write am I right. So, it is basically sigma z in x direction so z x as we know sigma z x equals to sigma x z that is why we could write this as sigma x z remember that what will be the change over a travel of this distance of d y there will be on this face in this direction opposite direction this should be sigma x z plus del sigma x z changing over the distance of d y. Del of into it is changing over the length of y it should be y it has change through this distance of y. So, what it should be sigma x z plus del sigma x z that is varying with respect to the distance of y. So, this should be not del z this will be del y over the distance of d y because see here it is sigma x z when it goes to on when it goes to on this face it moves by a distance of d y. So, it is change will be over the length of this d y so del y times d y am I right. Now, what will be the another one in x direction in x direction if I want to find out one more stress what that should be on this plane on this plane I have taken then x y another x y should come from I have sigma x y over here. So, other face will be on this front face and back face. So, this back face will have this direction the back face back side in the x direction. So, what it should be that is on which plane that is x y plane am I right. So, that is sigma I can write x y it is essentially sigma y x because sigma y x equals to sigma x y that is the reason why I could write it as sigma x y it is on the back side. So, when it travels through a distance of d z on this front side on this front side this face it will be this one what it should be it should be sigma x y which is changing del sigma x y over the distance of it travels in the d z directions over del z times d z am I right. Now, if I consider the equilibrium of this forces that is on the outer face or after the travel in the x y z directions the increase in the stresses whether it is normal stress or shear stress in x direction. So, all x direction forces now if we consider what we will see let us write it down further we can write down that sigma x x plus del sigma x x del x into d x this is working on which plane this force is acting or this stress is acting on the plane with cross sectional area of how much this is d z this is d y multiplied by d y d z that is the force that is the force that is the force positive force in this direction opposite force in the other direction is sigma x x times same d y d z. So, minus sigma x x d y d z clear to that is the resultant force in positive x direction due to the normal stress difference am I right. Next I have to write another one which is sigma x y plus del sigma x y plus del sigma x del sigma x y del z into d z this shear stress let us look at here this stress is acting on which plane the cross sectional area is d x times d y am I right look at this plane front plane this shear stress is acting on this front plane. So, if I multiply that shear stress with respect to this area cross sectional area d x times this vertical d y we will get the net force in the x direction. So, d x d y in the other direction what is the force which is acting this dotted one in black color on the other side of the face on this side on that side. So, that will be minus sigma x y times d x d y fine what else we have another force let us look at here. We have in this we have taken actually the direction will be different in the positive direction it is increasing. So, it should be the incremental portion in should be always in positive direction and this should be in this negative direction. So, that is the direction cosines also we can check why it should be in this direction because this and this should be in this formant and this the other side should be in this. So, that they are in equivalence. So, with this direction what we can write now this is my positive direction in x this is the shear stress incremental part that is acting on which plane what is the cross sectional area over here that is d x times d z. So, sigma x z plus del sigma x z del y times d y times d x d z minus what is this one acting is this one in the other direction sigma x z. So, sigma x z times d x d z now this net force which we are getting this should be equals to it should be equals to the inertia in x direction. What should be the inertia mass times acceleration now what is acceleration in x direction that is if you differentiate displacement twice with respect to time displacement in x direction is u we have already mentioned. So, del 2 u del t square is the acceleration and what is the mass mass is density into the volume what is the volume of that element d x d y and d z. So, inertia has two balance that difference in stresses at two phases of the element in x direction. So, this is the derivation for in x direction only. So, what we can write this gets cancelled this gets cancelled this gets cancelled. So, on simplification of this equation we can further write that del sigma x x del y times d z times d z times del x into d x d y d z plus del sigma x y del z into d x d y d z plus del sigma x z d del sigma x y del z into d x d y d z plus del sigma x z d del y into d x d y d z that is equals to rho d x d y d z times del 2 u by del t square. So, this term I can cancel from both the sides because this is the volume of the element which is non-zero has to be non-zero. Therefore, what we can write del sigma x x del x plus del sigma x x del x plus del sigma x z del y plus del sigma x y del z equals to rho del square u by del t square. So, this is our basic governing equation of motion for three-dimension wave propagating in x direction. So, this is in x direction only. Similarly, if I want to find out what will be the equation in y and z direction in the similar format if we want to derive the form of the equation will be let me write it down for a y and z directions. So, in y direction it should be sigma del sigma y y del y because this is the normal stress component plus del sigma x y del x plus del sigma y z. So, del z del z should be equals to rho times del square v by del t square and we will get here in z direction in the similar form del sigma z z del z that is the normal stress plus del sigma z z del z that is the normal stress plus del sigma x z del x plus del sigma y z del y equals to rho del square w by del t square. So, these are three-dimension governing equation of motion which we will get after solving this three-dimensional wave equation in the solid which is shown over here. You can see in this slide three-dimensional elastic solid which we have derived now. So, this is the final derived equation as I have mentioned just now. So, now using the strain notations the strain notations what we can use the normal strain in x direction in y direction in z direction those are nothing but epsilon x x is del u del x epsilon y y is del v del y epsilon z z is del w by del z these are normal strain. What are the shear strain? These are the shear strain epsilon x y. So, del v del x plus del u del y epsilon y z del w del y plus del v del z epsilon z x del u del z plus del w del x and what are the torsional strain? Torsional strain in x y and z directions can be represented by these equations. So, now using the stress strain relationship how we can use the stress strain relationship for this three-dimensional wave equation. This normal stress in x direction sigma x x can be represented through strain through some modulus. So, c 1 1 epsilon x x plus some modulus c 1 2 epsilon y y plus c 1 3 epsilon z z plus c 1 4 epsilon x y this modulus is related to the shear because it is relating to the shear strain these are relating to the normal strain. So, basically what you can see other two shear strain all these modulus coefficients that is stress versus strain can be easily written in the form of a matrix that is what it shows right. So, what is that form of the matrix in the simplest form if we want to rewrite it. The stress strain relationship is something like this sigma x x sigma y y sigma z z sigma x y sigma y z sigma z x these are the six shear strain. These three normal stresses and three shear stresses they are related through some modulus which is nothing but a modulus matrix through the strain. What are those strain epsilon x x epsilon y y epsilon z z epsilon x y epsilon y z epsilon z x. So, these strains are first three normal strain and these three are shear strain they are related through this matrix what we were showing over here c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 c 2 3 c 2 4 c 2 3 c 2 3 c 2 4 c 2 4 c 2 5 c 2 6 so on this one will be c 6 1 this one will be c 6 2. Similarly, it goes over here up to c 6 6 like that. So, this matrix is nothing but modulus matrix. So, this stress strain relationship can be expressed through this modulus matrix. So, with this we have come to the end of today's lecture we will continue further in the next lecture.