 Welcome to today's lecture of NPTEL video course on Geotechnical Earthquake Engineering. Let us look at the slide here. For this video course Geotechnical Earthquake Engineering, a quick recap what we had learnt in our previous lecture. In the previous lecture, we were going through the module number 4 which is strong ground motion. So, in that, let us see in the slide, we were talking about various attenuation relationships available and earlier to that lecture, we discussed about development of the attenuation relationship worldwide mainly in California region where it started with the work of Campbell and then Bore et al and further we continued the attenuation relationship developed for India because we mentioned that it is not correct if somebody uses the attenuation relationship of an Indian site for future designer construction with attenuation relationship available for other countries. They must use the local attenuation relationship to take care of the local fault characteristics earthquake behavior, geology of the region, soil behavior and so many other local concern. So, that is why it is always advisable to use the local attenuation relationship for design so that correct estimation for the earthquake hazard etcetera further can be developed. In that connection, we have seen that not only a particular region of attenuation relationship is important, suppose we have given the example earlier for India if somebody wants to find out the seismic hazard or attenuation relationship or want to do any earthquake resistant design at a particular site say in Mumbai, they should not use the attenuation relationship available for Himalayan region or northeast of India. They should use the attenuation relationship available for the peninsular India. In peninsular India as we know, we can consider Mumbai city is lying in that region. So, that is why use of attenuation relationship for particular design problem and estimation of input data is very important and essential. So, what we had seen earlier in the previous lecture that various attenuation relationships which are available in the literature like that given by Singh et al in 1996, I mentioned this is the work done by professor R. P. Singh and his research group from IIT Kanpur. So, this is the attenuation relationship which can be used to estimate the peak horizontal acceleration and units are given over here because these are empirical relationships. So, that is why we have to very careful about what type of unit has proposed while developing that empirical relationship. And another relationship for the peak horizontal velocity also they proposed which is in centimeter per second and they considered the earthquake data for a period between 1986 to 1993 and this was developed only for the Himalayan region. So, these attenuation relationships for acceleration and velocity are valid only for Himalayan region and that too for the data considered within this period. And what was the range of the magnitude for which this equation is applicable, they use the body wave magnitude between 5.7 to 7.2, so that is the range of the magnitude for which these two proposed empirical relationships are valid. Also this in this equation the r is mentioned as the hypocentral distance in the kilometer unit. So, knowing all these constraints we should know what are the limitations of using this equation when somebody wants to further extend it or use it for the estimation of the attenuation relation for the strong ground seismic motion in the Himalayan region of India for PHA and PVA. Then we had seen in the previous lecture about the attenuation relationship proposed by Einger and Raghukant in 2004. As I said this is the work done by a professor Einger at IISC Bangalore with his PhD student at that time Dr. Raghukant who is now faculty at IIT Madras, they developed the attenuation relationship for peninsular India. So, if somebody wants to do any earthquake related design they have to use this equation not the Himalayan region attenuation relationship or northeast region etcetera or the California earthquake attenuation relationship for obvious reason. So, this is the basic form of the attenuation relationship which they have proposed as we know this is the basic form as proposed by Bore et al while developing the attenuation relationship for California region. So, in this equation Einger and Raghukant mentioned this value of y is nothing but the PGA peak ground acceleration in the G unit you will get and m is the moment magnitude and r is the hypocentral distance in kilometer unit and this C 1, C 2, C 3, C 4 all these are various coefficients which are different for different region within that peninsular India also. So, from their considered earthquake data they have proposed for Koena, Warner region these are the values of various coefficients whereas, for western central region these are the values and for southern region these are the values. So, suppose somebody is interested to do any earthquake analysis and design they have to use in Bangalore city then they should use these coefficients. So, like that we have to be region specific area specific based on the proposed various available attenuation relationships. Then we had also seen the work done by Sharma in 2000 where he proposed the attenuation relationship for peak vertical ground acceleration only for the Himalayan region. Before that he worked for the horizontal peak ground acceleration also this is he is a professor at IIT Roorkee in earthquake engineering department professor M L Sharma. So, this is the basic form of the equation he had proposed where A is nothing but the PGA in this equation it is nothing but the vertical component and M is the moment magnitude of earthquake and X refers to the hypocentral distance in kilometer unit. Now, how many database or data points he has used to derive or arrive at this empirical relationship 66 PGA values he has used in the vertical directions from the five recorded earthquake motions and also with respect to the vertical to horizontal acceleration ratio this is the equation proposed by him. We had also seen from our previous lecture that the for Calcutta region it is in eastern part of India for Kolkata city, Shively and Narayanan in 2012 they have developed or proposed the attenuation relationship for peak horizontal ground acceleration given by this expression where A is the peak horizontal acceleration, R is the closest distance in kilometer unit from the site to the zone of energy release that means nothing but the hypocentral distance, M is the magnitude and other parameters like few dummy parameters like F and P they have proposed to take care of the type of the fault and the what type of plate events it refers to. Other than this magnitude based attenuation relationships proposed for India we had also seen in our previous lecture that some researchers has proposed attenuation relationship for intensity based calculations also that is how the intensity of earthquake attenuate with respect to the distance that also was proposed. As we had seen earlier Martin and Zeliga in 2010 had proposed the intensity based attenuation relationship proposed in this form of equation where n is the cumulative number of observations per year and I is the intensity of earthquake which can be obtained. For various cities like Mumbai, Delhi, Bangalore, Kolkata, Chennai they had proposed various values of these coefficients A and B and what are the various numbers of return time period for the value of seismic intensity of 5, 6 and 7. So, they considered the earthquake data points of about 570 earthquake between the period of 1636 to 2009. I mentioned in my previous lecture that in olden days whatever earthquake data we have historical earthquake data those are mostly the intensity based because how the damages occurred at a particular location due to earthquake or how it was felt that time by the people who observed or who felt that earthquake. But later on in recent days these are mostly the magnitude based which of course we can correlate with respect to the intensity based scale. We had also seen the attenuation relationship proposed by Dunbar et al. This is the form of attenuation relationship for peak horizontal ground acceleration for South India only. So, this is for applicable only for South India that is the attenuation relationship for peak horizontal acceleration in terms of moment magnitude. Further, they had proposed this moment magnitude in terms of the intensity based calculation. So, in turn they have given another equation where one can get the attenuation relationship of acceleration with respect to the intensity based calculation. But what we have mentioned what about the major limitation of this equation that in this equation they did not propose any distance based that is how far is the earthquake energy release point that is the hypocentral distance which is another very important parameter which must come into any type of attenuation relationship because the attenuation relationship means that particular parameter which we want to analyze how it is decreasing with increase in the distance from the earthquake originated point or the hypocentral point. So, that distance is a very important parameter which is not reflecting in this equation. Then we had seen in our previous lecture as proposed by Dunbar et al. This is the attenuation relationship for pseudo spectral velocity. We have already derived what is pseudo spectral velocity and its complete derivation etcetera definition and they had proposed in terms of moment magnitude, hypocentral distance, focal depth and the time period based on the single degree of freedom oscillator system through which we actually arrive at any kind of spectral acceleration or spectral velocity response. So, about 261 accelerogram recorded on stiff soil and rock size for 6 earthquake events that was the total data points which were considered by them to arrive at this attenuation relationship for pseudo spectral velocity. Then we had also seen for India the attenuation relationship proposed by Nath et al as we can see in this slide in 2009 Nath et al proposed for peak horizontal ground acceleration attenuation relationship only for the Guwahati city. So, they considered the earthquake response of Guwahati city and used the stochastic approach to arrive at this attenuation relationship where m is the earthquake moment magnitude and r r u p is nothing but r rupture is the rupture distance in kilometer unit. Also further we had discussed in our previous lecture that the attenuation relationship proposed for the peak horizontal ground acceleration proposed by Einger and Raghukant in 2002 for only Bhuj city. So, this attenuation relationship is also very case specific that is why you can see this is for a only a particular earthquake that is for 2001 Bhuj earthquake they had proposed this attenuation relationship. Hence, in this equation if you look carefully only the hypocentral distance is a parameter or a function through which the attenuation relationship of PGA in the horizontal direction was proposed. There is no magnitude based calculation because it is for developed for a particular magnitude of earthquake. So, that is why the obviously there is no reason why that magnitude scale will come as a function in this proposed equation. Then we had also discussed the attenuation relationship proposed by Jane et al in 2000 for the peak horizontal earthquake acceleration. In this form of equation where various coefficients b 1, b 2, b 3, b 4 were proposed for basic four regions that is central Himalayan region of earthquake with thrust type of fault, some subduction zone of earthquake in northeast part of India with thrust type of fault, subduction earthquake in north India. This is for only one specific earthquake and Bihar-Nepal earthquake in Indo-Gangatid plain for strike slip type of fault that is also developed for only one specific earthquake. So, other two regions they were developed for three earthquake with so many numbers of recorded data points. So, accordingly you can see over here the coefficients b 1, b 2, b 3, b 4 they are different for central Himalayan region and for northeast region. And so for these two case A and case B they had proposed various coefficients b 1, b 2, b 3, b 4 and their values are of course different. But for C and D they have not proposed or shown any coefficients over here because those are case specific or only for a particular earthquake. So, obviously the variation is not there in terms of the earthquake magnitude is concerned. Then we had also seen how Sharma et al. in 2009 proposed the attenuation relationship but we cannot say that it is the attenuation relationship which can be used for only India because the data points which the or the events which they considered to develop this empirical attenuation relationship for acceleration were taken partly from India and partly from Iran actually more from Iran less from India. So, obviously this attenuation relationship cannot be said that it is completely the Indian attenuation relationship or should be used only for India. Also we had seen the attenuation relationship proposed by Mandal et al in 2009 Pranthik Mandal and the research group as mentioned over here. So, this is for the Bhuj region of earthquake between 2001 to 2006 so that is the period they considered all the magnitude of earthquake and their values were within this range 3.1 to 7.7 and these are the values of R J B means distance to the surface projection of the rupture point and this is the proposed equation of attenuation relationship for the earthquake acceleration PGA. Also we had seen the proposed attenuation relationship of Gupta in 2010 for P H A using or proposing by this empirical relationship for H is the focal depth and G is the geometric attenuation factor over here. So, this is the paper from which this information has been taken for Indobarmi subduction zone of earthquake intra slab earthquake events. Then we had also seen that attenuation relationship in terms of intensity that is already we obtain intensity based attenuation relationship two attenuation relationships one directly on intensity based there was no effect of the magnitude we mentioned on that because it is intensity based, but there was no effect of the hypocentral distance also mentioned in that intensity scale that is why we said that was the drawback. Now let us see what this Zeliga et al in 2010 had proposed the intensity based attenuation relationship this is the proposed equation where you can see it is a function of the magnitude as well as the distance. So, this is I will say a better equation compared to the previous intensity based attenuation relationship and they had considered all the earthquake failed since 1762 to up to 2009 and about 570 earthquake where about 100 of them were instrumented earthquake that is the recent earthquakes whereas, the remaining other earthquakes were non instrumented that is why those are mostly intensity based based on the amount of damages occurred and how it was failed from the set of 29 earthquakes were finally used for this development of the empirical relationship of the prediction where r is the hypocentral distance in kilometer unit m w is moment magnitude and a b c d are various constants which can be obtained from this chart as proposed by them for different regions for entire India they had proposed for Creighton region for Himalayan region and how many number of events they considered to develop that or to arrive at those coefficient that also is mentioned because I told you earlier that this is a ever evolving or changing process because after 2010 suppose somebody is interested to arrive at this attenuation relationship or want to modify it further up to today's date say 2013 they have to take up to 2013 whatever earthquake data is available for this region and then automatically if their number of events increases this co-efficient and their standard deviations etcetera will change. So, the same course those who will be watching after say 5 years or 10 years in this video mode of NPTEL they will understand that these equations are valid on those date only there is quite a good number of expected proposed attenuation relationships will probably come up by that time of another 5 years time or 10 years duration. So, this is a ever changing or ever evolving process whenever any major earthquake occurs if you incorporate those earthquake instrumented data points in your attenuation relationship or in the prediction of attenuation relationship these values of your co-efficient and corresponding standard deviation will keep on changing. So, hence your form of the equation will remain change, but this co-efficient etcetera needs to be updated with more record of earthquake data. So, with that in the previous lecture we completed our module number 4. So, in today's lecture we will start with our next module. Let us see in the slide we are starting today with our module number 5 which is wave propagation. So, in this module number 5 let us see what is wave propagation let us look at this slide. So, in this module 5 wave propagation. So, what is wave propagation it is excitation of a compliant medium is not instantly felt at other points within the medium. Because whenever any vibration any excitation occurs in any medium say it is a soil it is a rock or any other material in a medium. Obviously, that excitation or that vibration will not be immediately felt within the entire media at other points even in the water also. We have seen when you throw a stone in the water that waves etcetera starts forming. So, you are throwing the stone in the water it starts forming it takes little bit of time to propagate that wave. It is not that instantly when you are dropping that stone in the water immediately the waves will be felt at the entire media. It takes little bit of time for travelling those waves from your source point where you created that excitation or vibration from that point to another point in that media or other points. So, slowly it moves further in that media. So, that is what is mentioned in the slide over here. It takes time for the effects of the excitation or vibration to be felt at other distant points which is known to all of us. And the effects are felt in the form of waves that travel through the media. So, as we know these waves etcetera will form and slowly slowly they will travel and through that wave it this excitation or this vibration will be felt from one point where that disturbance has been created to any other point in that media. So, waves actually carry that excitation from source point to other points in that particular media. So, the manner in which these waves travel will control the effects they produce. So, how they carry this behavior or the excitation in a particular region or in the particular material that we will see in this wave propagation that is how this wave propagates or travels or carries this disturbance or excitation from one point to the another point when they are travelling through a particular media. Now, let us come to the basics of this wave that is when any excitation takes place at a particular point what happens finally the waves travels in this direction. So, this is the direction of travel. So, these are formation of the waves, waves gets formed the travel like this. So, this excitation finally from this point it reaches to another point it of course travels in various directions. We have shown here the basic of this travel of this excitation from one point to another point through this waves. So, what is the wave length in the distance unit? If we say to complete one cycle whatever is the distance that is nothing but the wave length. Generally, wave length we represent in this form of lambda or lambda c. Now, how the particles move? So, when waves travels, waves is travelling like this. So, finally the direction of travel of the wave is in this direction, but when the travel through a particular media the particles of that media also gets excited because these waves are travelling these are nothing but they are carrying the excitation through that media. So, while carrying that excitation what is happening? This particle starts jumping or moving like this. So, this is the movement of the particle. So, it can be either in this direction or it can excite in this direction also depending on what type of wave it is travelling through. So, to identify that particle motion is a different aspect than the wave propagation that is why it is shown separately over here, but wave propagation is the cause and particle excitation or particle motion is the result. So, when we are talking about the seismic waves that is when that excitation comes from and release of energy due to earthquake. So, obviously here also that excitation needs to be carried from one source point where the energy gets released to another point through these waves. So, what are these waves due to the earthquake? These are nothing but the seismic waves. So, when an earthquake occurs different types of seismic waves are produced. As we have seen in our previous lecture we have discussed thoroughly that what are the various types of earthquake waves and we have also seen how those wave mechanics is used to estimate the epicenter of the earthquake. So, when we are talking about wave propagation let me come back to the classification of seismic waves. There are two major classification two types of seismic waves body waves and surface waves. Now, within body waves again we have sub classification like p waves and s wave p wave is nothing but as you already know primary wave or compressional wave or p wave and s is secondary wave or shear wave. And s wave also can be further sub classified into two types of waves one is called as v waves another is called as h waves. What are these things? There is s waves in vertical direction and s waves in horizontal direction. So, those are the two components of s wave by which s wave can be further classified too. And surface waves when we are talking about the sub classification of surface waves there are two types of surface wave one is Rayleigh wave and another is Love wave. How they are movement and motions etcetera occurs? Let us look at this slide though we have discussed earlier in our module three for this course again let me go through that further. So, body waves like p wave and s wave s wave again is v waves and s h wave for p wave this figure a you can see over here. So, when it travels through a media what happens there will be compression and expansion successively in that media that is some portion of that media will be expanded like this some portion of the media will be compressed like this when the wave travels through that media. And this shows the picture of the undisturbed media and that creates a wavelength that is one cycle between the one compression to another expansion or another starting of a compression. So, that demarcates the wavelength in the case of p wave. So, compressional wave or p wave or primary wave which when it travels through the media what happens the particles also gets excited in the same direction of the motion of the wave whereas, for the s wave the picture b should be looked into you can see from picture b that when this is the undisturbed media this is the direction of the movement of the wave, but particle they excite in the perpendicular direction to that direction of movement of the wave in this case. So, that is why this kind of wavy form will get formed and in that we have this s v wave two components as I said one is s vertical component that is particle will move in this direction perpendicular to the direction of the movement of the wave and another is h that is in the horizontal direction of this one. This one will be horizontal movement of the particle compared to the direction of the propagation of the wave. So, you can see so far whatever earthquake excitations or accelerations we talked about horizontal acceleration and vertical acceleration majority of them will be caused by horizontal acceleration will be caused by this s h and the vertical will be caused by either this s v or the p wave that we will further discuss because that depends on the direction of propagation of your wave. So, this s h and s v they will create the perpendicular movement compared to the direction of the wave whereas in p wave it is in the same direction the particle movement as well the wave movement. Whereas, for surface wave like rally wave and love wave we have seen these are the behavior for a is for rally wave b is for love wave this type of rotary motion you can see over here for the surface wave this is the undisturbed media and this is the disturbed media when waves are passing through. So, this also creates a particle movement and particle movement is in this fashion compared to the direction of the propagation of the wave. Now, let us discuss the derivations and various types of waves when they travel through a particular media. So, first let us try with the simplest case of waves in unbounded media that is the media is non bounded in both the direction it is infinity it is traveling we can consider the media whatever the media whatever the material we are considering in both the direction it is extending to infinity. So, that is why unbounded there is no boundary. So, for that let us first talk about one dimensional wave propagation in this unbounded media. If we want to see what are the different types or categories of that one dimensional waves in unbounded media they can be classified into three categories of one dimensional wave. Those are called longitudinal wave, torsional wave and flexural wave. What are these waves how they are different that is when a wave is traveling through this one dimensional one dimensional means only in this direction only in this dimension of say x we are considering the travel of the wave or the direction of movement of the wave. Now, how the particles gets excited that decides that is what type of wave we are considering for longitudinal one dimensional wave. Wave movement direction is in this direction particle also gets excited or particle motion is also in that same direction. So, that is why based on the particle motion it is called longitudinal motion or longitudinal one dimensional wave propagation. Whereas, for the second case when we call it torsional when wave propagates in this direction in this x axis direction the wave propagates, but the particles get disturbed or moved in this direction in this fashion that is in the torsional way or in the perpendicular way to the axis of this x. So, that is called as torsional one dimensional wave propagation that is when the particle movement will be of this pattern. What is the flexural wave movement in one dimensional in this case also wave propagates in this direction, but the particle they move in this direction that is the direction of a bending. So, the difference as we know in longitudinal it excites in this direction particle in torsional it excites in this direction that is the torsional movement of the particles and in bending or in flexure the particle move in this direction. So, that is what it shows over here in this picture is the flexural one dimensional wave propagation when wave is moving in this direction, but the particle moves in this fashion. So, these are the three major categories of one dimensional wave propagation one can have. Now, let us come to the derivation of this one dimensional wave propagation. Let us start with the longitudinal wave in an infinite rod. So, let us draw here and derive it as we are considering one dimensional wave equation. So, let us say this is our x direction in this one dimension only we are considering our wave propagation and the boundary condition we can say is like this. So, that we make sure that it is in the one dimension only and let us say it is an infinite rod. So, longitudinal wave we are talking about wave in finite rod. So, how it behaves let us see and the material property what are the various material property we can consider like say rho is the density of that material is Young's modulus of that material say nu is the Poisson's ratio of the material and capital A is the cross sectional area of this rod. So, these are the various input parameters which we can consider. Now, let us take within this infinite rod a small infinite similarly small element of length dx. So, we are considering this shaded region we have taken an infinitesimal portion of the rod for consideration. So, in that we are now trying to find out how the variation of the stresses will be at this two ends of this infinitesimal small rod of dx length. So, what we can say here in this one end this is the other end you can see over here at this side let us say the stress can be expressed as sigma x. So, we are considering x at a distance say x naught at a time say t. So, we are considering that point that point we have taken over here. So, that point coordinate let us say it is x naught and at a particular time say t the stress is expressed in this format let us say that we are further reducing to sigma x naught at this phase. And at this end let us say the change in the stress in the x direction is this initial stress sigma x naught plus there is a change over this length of dx so this over this length of dx it has changed by del epsilon del sigma x naught by del x into this over this distance of dx. So, why this change of stress has occurred because we considered wave is passing through this media. So, this is the direction of the travel of the wave and for that the particles are getting excited in that direction because we have taken the longitudinal wave propagation. Because of that what will happen let us show here in through some other color say this point has come to this point this green one is the new position because particle has moved particle has displaced what made the particle to move that wave which is propagating through that. So, that excitation has made the particle to move by this distance that distance let us say is u. So, that u is nothing but let us say u at the distance at the coordinate x naught at a particular time say t and at this end this end also has moved by a certain distance let us say the new location is this one. So, particle has moved by this distance. So, how much that distance it has moved over a distance of dx its variation of this movement u will be u plus that del u del x over the distance of dx right. So, now if I use this one what I can further say the change in the behavior of this stress will be we will derive it now will be coming from let us see over here next derivation that is what is the change in the stresses at both the ends what we obtain here the difference of force is nothing but stress sigma x naught plus del sigma x del x over the distance of dx that stress variation if I multiply with the cross sectional area that will give me a force. So, force in the direction of the propagation of the wave and what is the force in the other direction that is what is the net force acting on that element is nothing but minus sigma x naught times that cross sectional area A. Let me put this back over here again you can see here. So, I have stress at this point in this direction and stress at this point in this direction. So, what will be the resultant force acting on this element this stress times cross sectional area will give me force in this direction and this stress times cross sectional area in this direction will give me force in this direction. So, the net force which is acting on this small element is nothing but this one. Now, this force is nothing but what this force is nothing but the inertia force which is acting on the system why this inertia forces acting because of the excitation because of the wave traveling through that media the particles get excited. Once the particle gets excited as we have already seen look at here it has moved by this distance at two ends of that infinitesimal small portion. So, this portion has moved by this amount this portion has moved by this amount. So, what is the acceleration of that we should know first. So, what is the acceleration? Acceleration is nothing but del 2 u by del t square that is the acceleration because u is the displacement as we have already taken. If you differentiate it twice with time you will get the acceleration and that particular direction. Why I am using del instead of d in this one dimensional case I can use d also, but when we take the multiple direction later on we will take the three dimension the common case or the generalized case we will see it is dependent on a particular direction which we are considering. So, that is why it is the del partial differential not the complete differential of d. So, that is why it is better always to use this del from the beginning itself. Now, this is the acceleration. Now, what is the mass of that element? Mass of that element is nothing but density is given to us cross sectional area is given to us and the distance which we are considering is known. So, area times distance is nothing but volume and times density will give you the mass. So, mass times acceleration is nothing but your inertia force that inertia force is nothing but which is causing that stress difference within that element. Now, is it clear why that stress difference between the two ends of the infinitesimal a l rod is occurring this stress difference between two ends. So, that is because of this inertia. So, now if I further simplify it what we can get from here if we simplify this one further you can see this sigma x naught times a and sigma x naught times a get cancelled from both the sides d x d x also gets cancelled. So, let me simplify it. So, it will be easier to understand sigma x naught times a plus del sigma x del x into d x into a minus sigma x naught times a equals to rho a d x times del 2 u by del t square. Now, in this case we can easily cancel this what else we can do the cancellation you can see on the both the sides there is d x d x a a why we can cancel because these are non zero parameters you have to always remember because d x is also non zero and cross sectional area is also non zero then only from both the sides we can cancel this two parameters. So, after canceling that what finally we are getting let us see we will plot it here we will further simplify that del sigma x del x equals to rho times del 2 u by del t square. So, that is the basic governing equation for the one dimensional wave propagation in infinitesimal media is the basic equation now we need to solve this. So, this is the formation of the equation that is the in this parameter this sigma x dependency this is the stress dependency and this is the displacement dependency. So, this is displacement dependency. So, how stress and displacement are connected when a wave is travelling through a media that we have we can see from here because from this simplification we got it over here you can see this equals to rho times del 2 u by del t square. Now, let us see further if we can simplify it again. So, let us look at this derivation once again now what are the other relationships are known to us we know that stress can be expressed as some modulus time strain. So, this is stress strain relationship this we are getting from stress strain relationship that is always we know stress is related to strain through some modulus also we can express this strain in terms of displacement. So, that will be in this case del u by del x because u is the displacement. So, this is the strain displacement relationship now this strain displacement relationship in this case when we are talking about one-dimensional wave propagation longitudinal wave propagation this modulus which I have mentioned over here m can be written as m as in this form let me write it down because this is already covered in basic solid mechanics course right. So, I am not going to cover that aspects this stress with strain relationship through modulus using any solid mechanics book or the NPTEL course on solid mechanics we can have this relationship when we are talking about the longitudinal wave propagation. So, this is called constrained modulus constrained modulus and mu is called the Poisson's ratio of the media and E is the Young's modulus of the media. So, constrained modulus can be expressed in this form which is connected through this stress strain relationship in this one-dimensional wave propagation. So, with this relation what further simplification we can do let us see the derivation again. So, we can write that what we already got the basic equation that del sigma x del x equals to rho times del 2 u by del t square that we had already derived now in this case we can simplify it further in this way or del 2 u del t square equals to 1 by rho del sigma x by del x. Now, what is del sigma x del x if we look at the previous relationship of stress strain and strain through displacement. So, let us look at here strain is related like this which is related to stress like this. So, I can put it like this that stress can be expressed as m times that constrained modulus times strain which is nothing but del u by del x or I can write del sigma x by del x is nothing but that constrained modulus times del square u by del x square I am differentiating it with x only. So, with that expression if I now put it in this equation what I will get on further simplification this equation will now take the form of del 2 u del t square equals to m by rho del square u by del x square. So, this is the further simplified version of that wave equation in one dimensional case for longitudinal wave. Now, this parameter m by rho is expressed as v p square we will discuss that is very soon this where this v p is known as primary or p wave velocity. So, we can write v p equals to root over m by rho where m is constrained modulus and rho is the density of the material. So, with that what we can write that del 2 u by del t square equals to v p square times del 2 u by del x square. So, that is the basic governing equation for longitudinal wave traveling in one dimension. So, with this we have come to the end of today's lecture we will continue further in our next lecture.