 So, earlier we had covered reverberations in rooms which were of regular ships, parallel walls, they could be 1D long rooms or 2 dimensional rooms or we had also talked about 3 dimensional rooms. A lot of rooms, however, big auditoriums, concert halls and lot of lecture theaters, they are not of regular ships. So, what we will cover today is how does sound behave if it gets propagated inside a large but irregular room. So, couple of things for large irregular rooms, one that because the walls are not perfectly parallel to each other. So, the reflections keep on happening for and you will have standing waves for virtually any frequency and at virtually any angle of incident because things are not very nicely parallel to each other. Second thing, at any given point sound moves in all directions, in parallel rooms if you have a forward going planar wave, then it will nicely behave and reflect and it will move in straight lines. So, the propagation will not necessarily be in all directions but in irregular rooms these reflections could happen in not because of the nature of reflections, the propagation of sound at any given point could be in all directions. Third thing, the number of modes in these rooms and the number of natural frequencies in these rooms is extremely large, it is even larger compared to some of the 3D room examples which we had seen. So, at virtually any given frequency if you move from one point to other point within a very small distance, you will see moving from one node to other node to other node. In last class we had seen because it is like a more or less a rectangular room, for low frequencies the distance between 2 nodes at least in this room which has parallel walls, it is a nice rectangular boxy room. At low frequencies the distance or the separation of nodes at low frequencies 125 hertz we had seen is not very small but in irregular rooms that may not necessarily be the case. So, if you combine all these 4, 5 effects which I had mentioned, then the sound field in such a room called a diffuse field, it is called a diffuse field. So, we will develop some relations how sound behaves in this kind of enclosures, so it is large irregular rooms. So, what I will do is I will introduce a couple of terminologies. So, the first concept I will introduce is developer relation between energy density in the room, energy density and the sound pressure level and this I have taken from some of the earlier class notes, it is just a mapping. So, I will define the term D average is P average square over rho naught c square and this is sound energy density is D average. So, the units of this guy are watts second, what second is energy or joules right and because it is density it is for every cubic meter. Couple of things to note P average is the average of RMS pressure in space and time, what does that mean? If I have a microphone when I am playing music or people are talking, the microphone is sensing some pressure level. So, I position a microphone at a point and I record data for let us say few minutes, 1 minute, 2 minute and then I take the average, the average of the different RMS values. So, for first one-tenth of a second I find the RMS, second one-tenth of a second I find the RMS and then I average it out. So, that is averaging the RMS pressure in time, in time. Now, I move my microphone to slightly different location and then I do the same exercise. So, then I am averaging it in space. So, this guy is average in both space and time. So, you measure for a little longer periods of time at couple of locations and then you get this P average and it is again the average of RMS pressure. It is not the peak pressure, it is not the minimum pressure, it is the average of RMS pressure and of course, rho naught is air density and C is velocity of sound in air. That is my sound energy density. The second concept I will introduce and this is a new concept, mean free path. Let us consider couple of walls in an enclosure. I have incident sound, it gets reflected by this surface. So, it comes and hits another surface, it gets again reflected and it gets again reflected and so on and so forth. The mean free path, I will define it as letter D and this is average distance between two reflecting surfaces. So, this distance could be D. This is reflecting surface one of absorption coefficient alpha 1. We will define and explain what these alphas are later, alpha 2, alpha 3. So, between two reflecting surfaces, average distance through which the sound travels is D and if I have to compute the value of D, it is 4 V over S. How did we get this relation? So, V is volume of room, S is total surface area, I mean reflecting area. This could be in meter cubes or feet cubes. This could be in meter square or feet square. So, the first time there was an acoustician, his name was Nudson. So, in 1932, he did some experiments. He computed experimentally the value of D. So, he designed 10, 12 different types of closed spaces. All these closed spaces were irregular in shape and they were very significantly different from each other. So, it is not that they were slightly different, they were very significantly different and what he found was that the value of D comes to 4 V over S. Later, based on some statistical principles, people also did some studies and from a theoretical standpoint validated the same expression. So, this expression D equals 4 V over S as an experimental basis and also a statistical basis. The value of D could be either in feet or in meters cube, but you have to be just consistent. The reason I am saying that is that because it is a relation based on statistics and experiments, we have to be careful about unit. So, we have talked about D average. The second concept we introduced is mean free path, which is D. Now, we will introduce another thing called sound energy absorption coefficient and this is alpha. Alpha is the energy absorption coefficient absorbed energy by a reflecting surface divided by incident energy and this I average it over all angles of incidence because the absorption coefficient will change as my angle of incidence is changing. So, whatever energy is being absorbed for different values of incident angles, I average them up and I get value of alpha. This is an experimentally. So, you find this value alpha for different materials through experiments. The other thing you may want to know is that this value of alpha is strongly dependent on frequency. It changes its frequency. I will give you two examples. So, the handout which you have today, I extracted these two pieces of data. For example, this wood panel, which is 2 to 4 inches thick, alpha equals 0.3 at 125 hertz and same thing because down to 0.10 at 4000 hertz. So, they changed significantly with frequency and then I will give you another counter example, carpet. It could be 5 over 8 inch thick, alpha is 0.2 at 125 hertz and in this case it is going up at 4000 hertz. So, whenever we talk about alpha, the natural question is that alpha at what frequency. There are some materials where alpha is more or less constant over the frequency range. On most of the materials, they have very strong frequency dependent behavior. So, alpha always has to be talked in context of a frequency number. In general, when people do not specify alpha, then you may assume with a little bit of latitude that it is maybe for 500 cycles per second. Third thing I wanted to mention is that this is a ratio of energies. It is not a ratio of pressures. It is a ratio of energies. So, if I have a irregular room, which has lots of reflecting surfaces, then for that room I can compute an average alpha and I can do that in such a way that I take s 1 is the first reflecting area, alpha 1 plus s 2, alpha 2. There are n surfaces, s n over total reflecting area in the room, where s equals s 1 plus s 2 plus s n. Question, what would be the alpha or reflection coefficient of a window, open window 1. So, if you have windows in your rooms, you should still account for those by prescribing an appropriate value. If it is closed, then you have to or if it is open, then you have to make it 1. So, alpha window equals if there is a room. So, this is for an empty room. If there is a room with people sitting on it, then again the handout, which you have, it provides values of alpha for individuals, whether they are sitting or standing or they are on a chair and so on and so forth. So, if there are people sitting in it, then we can say approximate average alpha would be s 1, alpha 1 plus s 2, alpha 2 is an alpha n plus alpha person times number of people sitting in the room or standing or whatever over s. So, this is with people. Typically alpha p exceeds 1, 1.5, sometimes it goes up to 7, 8. So, we have talked about three concepts, sound energy density, beam free path, sound energy absorption coefficient. A lot of times it is also called sound absorption coefficient. So, just be aware. So, using these three concepts, we will develop a property of a room, which is large and irregular and this property is called reverb time, reverberation time and we will designate it by a letter capital T. Couple of points, it is a metric, one of the metrics, metric is a measure large acoustic spaces. To figure out whether an auditorium sounds good or bad, this is one way of objectively calibrating yourself and saying that yes this room sounds good, this room does not sound good. Second thing, T is the time taken when SPL, that is the sound pressure level, which is fully developed decays by 60 dBs. What does this mean? You have an auditorium or a lecture theater L 7 and you start the sound system there or if people are talking, let that start happening and at a particular point you place a microphone and you will see that after a certain period of time, the sound pressure level will stabilize. Once it has stabilized, then you turn off the source and then sound pressure level will start decaying very fast. So, the time it takes to start from that P naught condition to P naught minus 60 dB, that time is called reverberation time of that room. What is 60 decibels in pressure? Two pressure values are off by 60 decibels, it means what is the ratio of pressure in absolute 1000. 60 dB corresponds to P T over P naught is 1000 and if I am computing the ratio of energies, ratio is 10 to the power of minus 6. Using this understanding, we will start now doing some math. So, again we will draw three surfaces. All these three surfaces are reflecting and they also do absorb some sound. So, let us say this absorption coefficient of this surface is alpha 1, this surface is alpha 2, this surface is alpha 3, such that the overall rooms absorption coefficient is alpha bar. That is my incident sound. It gets reflected, gets reflected again, gets reflected again. The intensity of incident sound is D naught. What will be the intensity of sound after it gets first reflection? D naught times alpha got absorbed. What is remaining? 1 minus alpha. This guy becomes T naught 1 minus alpha square. This is T naught 1 minus alpha cube. So, in a statistical sense, these are alpha, this is the average alpha for the room. In a more precise way, yes, you are right. It will be D naught times 1 minus alpha 1 times 1 minus alpha 2 and so on and so forth. But in a statistical sense, I am just using alpha average alpha bar. What we will do is construct a simple table. First column time, second column delta T, third column value of D. At T equals 0, delta T is 0 times 4 v over C s, T is 0. This is D 1 equals D 0. That is 4 v over C s. At first reflection, this is my reflection number. At first reflection, delta T is 1 times 4 v over C s. This is D naught times 1 minus alpha bar. This is D 1. Second reflection, delta T is 4 v over C s and this is D 2 equals D naught 1 minus alpha bar whole square. It did nth reflection. It is n times 4 v over C s. D n equals D naught 1 minus alpha bar n. Now, we know that we can also write n as total time elapsed, which is T over delta T and that is T times C s over 4 v. So, with that understanding, I can express at any given time, the sound energy density is D T equals D naught 1 minus alpha bar times C s over 4 v times T. I can again reframe it as exponent log 1 minus alpha bar bar. So, if I plot D T with respect to time, I get this kind of a relationship, an exponential decay relationship. So, this is in regular rooms, we had developed a relation for envelope of decay. In this case, the envelope of decay is also giving the exact value of pressures at different times of sound intensity at different times, because reflections are happening all the time and the overall field in the room is diffused. The other thing is we made this jump n equals T over delta T and this is because again the room is irregular. So, the values of delta T do not increase in you know quantum. They are continuously changing. So, this is also a continuous variable. In the earlier case, where we had parallel walls, this T was changing from you know this reflections were happening and T was progressing in one quantum, second quantum, third quantum and so on and so forth. Here, it is a continuous variable. So, that is my envelope of decay. We know that D is related to energy. So, it is directly proportional to P square, pressure square. So, I can also say that pressures will pressure in the room will also decay in a similar exponential factor such that it is P naught E log 1 minus alpha, but what will be the value of C s here C s 8 V T. So, instead of 4, I am putting an 8 here because the relationship between D and P is a square relationship. So, I level this expression as 1. We know that sound pressure level in decibels is 20 log in base 10 P R m s over P ref, where P ref is equal to 0. So, this is the value of equals 2 into 10 to the power of minus 5 Newton's per square meter. So, what is it that we are trying to do in through this entire exercise? We are trying to develop an expression for reverberation time. So, we do not have to forget that. So, if T equals T, which corresponds to reverberation time, then S P L at T equals 0 minus S P L at T equals 0 minus S P L at T. What is the difference? 60 decibels. So, I put this relation in this equation and what I get is 20 log P R m s over P ref at T equals 0 minus 20 log P R m s over P ref at T equals T equals 60 dB. I can rearrange this as 20 log P R m s at T equals 0 divided by P R m s at T equals 60. What is P R m s at T equals 0? P naught. What I get is 20 log P R m s at T equals 0. So, basically what I get is this thing. E L n 1 minus alpha bar and the whole thing is raised to the power of 8 V times T. Can everyone read this? I did some micro typing there, writing there. It could be read. So, just to make sure that you have read it correctly, 20 log 1 over exponent natural log 1 minus alpha in parenthesis, the whole thing raised to the power of C s over 8 V T. So, now, I do some mathematical manipulations. I get 20 log E minus L n 1 minus alpha C s over 8 V times T equals 60. Now, I take the log and what do I get 20? I take log of this guy. So, I get minus L n 1 minus alpha times log of E times C s over V T equals 60. So, I get 20 log of this is log is in log 10 base 10. So, log of times 0.434 that is log of E divided by 8. I am pulling these three together times minus L n 1 minus alpha times s. So, this entire thing becomes 1.086 times this entire expression times C T over V equals 60 and I call this expression as A prime. Is A prime a positive number or a negative number? The 1 minus alpha is less than 1. So, this A prime is positive. So, essentially what I get is reverb time equals, I can move everything on the right side 55 over V divided by A prime C and A prime is called absorption unit. The name is absorption unit. It is not a unit of the name is absorption unit and the unit of this absorption unit quantity is meter square in SI system and feed square, but more popularly it is also called Sabine S A B I N in British system. Sabine is feed square and A prime is called absorption. A prime is a property of the room because what is it? It is basically s times a function of alpha and essentially what it tells is how much sound is overall in an overall sense getting absorbed by the say that T is 50 times V over A prime C and we can rewrite because C we know the value of C. So, we can rewrite the same equation as T equals 0.161 V over A prime and this is a very famous equation in architectural acoustics. It is called Sabine's formula. It is named after a person obviously Sabine who in 1898 in Boston he measured coefficients this reverb time of this formula should work in both. So, you have to change it here. So, this is for SI system. Yes, you are right, but you can change it. This constant will change, but the formula is still will be still be called Sabine's. So, what Sabine did is hold on, what Sabine did is that he took an axis and he was computing with respect to he was essentially plotting this V over A ratio and what he found was that this is a hyperbolic relationship. So, from that he said that I can extract this value of T. So, he was doing that numerically. Now, he was probably not playing with this 0.161 parameter because that comes out naturally once you plot this entire hyperbola then that thing comes out. So, this is a very it looks a very simple relation. There are only three terms in it T, V and A and it looks very deceptive, but it has a lot of substance in it. So, one thing is that as V goes up, T goes, T will go up. Second, as alpha goes up, what happens to T? It goes towards 0. Why? Because you have in this A prime you have logarithm 1 minus alpha. So, this goes towards minus infinity. So, the third thing remember there is this relation is dependent on frequency. So, each room will have different river constants for different frequency because alpha depends on frequency. So, do not forget that. In this entire analysis, we have only assumed that sound is getting absorbed only on the reflecting surfaces. As it moves from one surface to other surface during that movement of sound from one surface to other surface, nothing is getting absorbed. But, in reality air also does attenuate the overall sound pressure level. So, this formula needs a little bit of calibration to account for absorption of air, but even without that it is fairly good relation. So, we had talked about how do you measure T? First thing is you excite a room with some sound source till SPL is stabilized where it takes a little bit of time for the sound pressure level to converge and become flat and then you switch it off the sound source. So, what you get is something like this. So, here if I am plotting P over P naught in decibels then when I am exciting it becomes stable after a while and then I switch this thing off and the sound pressure level decays something like this. So, if this is my reference which is 0 dB, I am recording data and this is minus 60 dB, then this value time is my river and this approach of measuring T is called switch off method. I want to make couple of qualifications in this entire discussion of the analysis. One is that this theory is good to the extent that sound field is what? Diffuse, it is well distributed throughout the room. So, that is one that we need to have a diffuse field. Second thing is that the room should not only be irregular, but it should not also have focusing elements. What do I mean by focusing elements? For instance, I have let us say light you know and then I can have a concave mirror which will focus light in a particular way. Similarly, I can have surfaces which focus sound at particular locations. So, I do not want to have any focusing elements in the room with that will kill the smoothness of SPL distribution in the room. They could even be a little bit curve, but they question overtly focus in one particular area or zone. Third thing is that the total absorption coefficient should be small or moderate. What does that mean? That if my alpha bar is fairly large, if alpha bar is fairly large then it does not take a lot of reflections for the sound to decay to a level which is not measurable. So, alpha has to be not too large. It could be fairly small also, but it does not have to be large. For instance, if there is a room and all the windows are open and it only has windows in practical sense, then this reverb time and all this discussion becomes meaning. So, one way to quantify that the rooms absorption characteristics are moderate or small is that V over log minus 1 minus alpha s is V over alpha s is greater than 1 meter. Some way to find it. If you go to lecture hall complex, you listen, you hear a lot of echoes especially in L 1 and some of the larger rooms. Whenever a professor is teaching, the sound is not clear and I think the biggest challenge there is the reverb time. It is too large. What I also wanted to show you is couple of slides which will be helpful just for record. This is the picture of Mr. Sabine. What you are seeing here on this plot is typical recommended reverb times for different applications. For instance, a classroom where most of the frequency content is in the bandwidth of speech. About 0.8 seconds is the optimum time. So, if you have a classroom where reverb time is significantly larger than this, then the speech will not necessarily be intelligible, won't be understood easily. Similarly, theaters where you have a mix of music and speech, they need a little larger reverb time. Then concert halls where you want to have a good amount of reverberation. The time is in excess of 2 seconds and so on and so forth. This is another chart. It shows in some quantifiable ways. For speech, 0.8 to 1.3 is a good reverb time. 2 seconds is certainly unacceptable. Even if I am in 1.4 to 2, it may be poor or fair. For pure music, I need a little higher reverb time. If the reverb time is too low, then sound does not excite the human emotion as much. This is the diffuse field. Yes, diffuse field is like a single one-way source. You can have, it could be omnidirectional. You could have, because the points are different, is that because you have irregular rooms. So, even regardless whether it is omnidirectional or unidirectional, at the end of the day, sound will be moving in all directions at all frequencies. In case, a window is open. After some reflections, it goes to the window. If a unidirectional source is there, then in that case, what will be the reverberation time? Again, because of the nature of irregularity of the room, by the time it reaches, unless you are directly projecting sound towards the window, that is a totally different case. By the time it reaches, by that time, the sound field should have become diffuse enough. So, there is no very hard objective cutoff that, but this is an approximate answer to you. What happens if the reverb time is too low? Let us say I am talking, we are talking in this room and if the reverb time is too low, it lets a 0 second. What does that mean physically? What that means is that the person who is speaking, he has to make more effort in making himself or herself heard. So, you also need some reverb time. 0 is not good because that requires more power for the person or for the sound system, but if it is too high, then if I say cat, by the time I am saying ta, c will also overlap with ca. So, it will become confusing. Another parameter which we can extract if we know the reverb time of a room is called bandwidth or queue of a room and we will talk about it. Let us say we have a spring mass damper system, f equals k x plus c x dot plus m x dot dot and if I plot the response of that kind of a system, I get something like this. Where this is my omega and this is my amplitude. This could be some transfer function which is a function of this point is what? Natural frequency. So, bandwidth of this kind of a system is defined as a range of frequencies between which the power goes down or the value of h goes down by a factor of 3 dB. So, this is 3 dB and this is let us say h naught, then this will be what? h naught over root 2. So, associated with this is a parameter called delta omega. Delta omega is called bandwidth. Omega over delta omega is called queue. It is an industry standard term. Even in vibrations, you hear this term queue. What is the queue of the system? So, if there is no damping in the system, what does queue become infinite? Queue becomes infinite when there is no damping in the system. So, queue is also kind of a reflection of how damped the system is at omega equals omega naught plus minus delta omega over 2. My energy is half. The direct variable h, which could be pressure or velocity or displacement, h is 1 over root 2. So, this is the bandwidth. Now, the question is that if I have a room and there is a diffuse field in it, what is queue of that room? So, we will try to use this concept and also the idea of T and try to figure out what is the value of queue of a room. So, there is an intermediate step to get to that answer. So, let us say you have a signal h naught e minus gamma t. The Laplace of it will be h s equals h naught over s plus alpha. So, we will plot this and we get something like this. This is my h naught. If this is h naught over root 2 alpha and this is alpha, then bandwidth is 2 alpha and queue, I am sorry, this should be gamma. This is also gamma. So, queue is what? Omega naught over 2 gamma. In this case, it is infinity because omega naught is 0. This is my omega naught. The point is that if I have an exponential decay form, then my bandwidth is 2 gamma. So, now we go to a diffuse room and in a diffuse room, my h t signal is essentially we have developed it. This is h naught e minus gamma t cosine omega t. What this means is that if I have a room and if I started playing a tone of omega naught in that room till it became stabilized and then it turned it off. Then the signal in the room will decay in such a way. It will still be cosine omega t, but it will decay over a period time in an exponential way e to the power of minus gamma t. That is what it means. This should be a little not that sharp. So, my center frequency is omega naught and this is h naught over 2 gamma root 2. This is h naught over 2 gamma. Essentially, my bandwidth of a room is 2 gamma and queue associated with the frequency which is getting excited in the room is. Now, in case of a room, we know that gamma is what? My h t signal minus l n 1 minus alpha bar times C s over a to v. Remember, it is 8 because we are not plotting in this case the energy. We are plotting the fundamental variable pressure or displacement or whatever. So, I can use this and these to figure out the overall queue of the room. We will do a very quick example. So, let us say my t equals 2 seconds. This is an example. Omega naught is 100 hertz. Then what is bandwidth and what is queue? So, we know that gamma is l n 1 minus alpha bar s C over 8 v. This is basically a prime C over 8 v. We also know that t is 55 v over 8 t a C a C a prime C. So, this gives me a prime C over v which I will later introduce here. A C over v is 55 over t. Now, I put this here and gamma is 55 over 8 t and that gives me gamma equals 55 over 8 into 2 equals. The number comes to 3.4 seconds. So, gamma is 3.4 seconds and delta w was 2 gamma is 6.8 seconds. Now, gamma is radians. Yes, you are right and that translates to 1.0942 degrees. So, q equals omega naught over delta w equals 100 over 1.094. That is approximately 90 hertz. What it physically means is that between 90 hertz and 110 hertz, the room will sound very still. If I play a tone, the decay of energy will be by a factor of 2 between 90 and 110 hertz. That is what it means. So, if this band is very thin, then the room will be very still in that band because if it will pick up, then it will be very still. If this band is a little bit wide, then it will be very smooth transition of energy. That is what it means. We will do three examples of reverb time, how it is used in industry applications. So, one is, we already talked about finding the constant of a room and using the constant to design a room accordingly. So, t helps us in designing acoustic spaces. In your assignment, there is one question where you will be expected to tune the characteristics of a particular room such that you acquire a t of, I think it is like 0.7 seconds. So, if you have a target value, then you can architect your room accordingly so that you hit your target value of t. So, that is one. Second one is, so in this context, again dependent on frequency, there is no reverb time which works for all frequency ranges. So, you have to design for a particular frequency or a set of frequencies. Second thing is, has to be irregular room, has to be fairly large enough and if it is not large, if it is a small quote unquote regular room, then the Sabine's approach, it does not work that well. So, in that case, there are some other approaches where they take ratios of reflected energy and direct energy and then they try to optimize that particular parameter. But in large irregular rooms where the sound field is fairly diffused, reverb time does a fairly good work in terms of optimizing the acoustics of them. The second thing is, you can use t to measure alpha of a particular material. How do you do that? So, way back in the course, we had talked about one dimensional long tube which has known reflective, no unknown reflective impedance and using the Kuhn's approach, we could figure out the reflectivity or reflection coefficient of this material and also the phase of the material by sweeping the microphone along the length of the tube. But that approach is good only if your sound is hitting in what direction? A normal direction. So, if I surface this and setting like that, then that is the way to develop, get reflective coefficient of the material. If I have to develop an understanding for different incident angles or in an overall average sense, what is the reflection coefficient of the material, then we can use t to find out the value of alpha. So, we will quickly cover that. Time is to measure alpha of unknown material. So, step one, we take a fairly diffused room which does not have a high absorption characteristic and then find the reverb time of that room without this material in it. So, find t without unknown material. So, t equals 0.161 v over a prime and if I do the math, I get minus l n 1 minus alpha r equals t over 0.161 v over 0.161 v s. s is the surface area. I know surface area of the room. So, I can get log alpha from this relation and I can also get alpha bar for the room without the material and that is basically 1 minus e minus t over 0.161 v s. So, I know alpha bar for this room from t, I can figure out the alpha bar for this. Then step two, I do the same measurement and I find alpha nu. So, in the same room, now I place this material. I know the surface area of the material and I find alpha nu. So, from alpha nu and from alpha, I can extract the value of alpha of the material. How do we do that? So, that is what exactly what we will do. So, alpha nu is basically alpha i s i plus s of the material. I is only for the room alone, s material alpha material and divided by s plus s mat. So, this is basically s i alpha i over s plus s mat plus s mat times alpha mat over s plus s mat. I do not know alpha mat. I know everything else. Also, I do not know this quantity as one single entity. I do not know this also. I know alpha i, but I do not know this entire thing by itself, because this is different than the alpha of the room. What I do is s i alpha i over s times s over s plus s mat plus s mat times alpha mat over s mat plus s mat plus s mat times alpha mat plus s plus s mat. What is this term? Alpha bar times s over s plus s mat. So, I know this s mat over s plus s mat times alpha mat. Now, I know everything. I know all these blocks and this is the only guy which I do not know. So, I can figure out alpha mat in this way. So, this is another application. The reverb time has usage in terms of designing architectural spaces, also characterizing materials, also developing targets for different types of rooms. Once again, it is dependent on frequency and the choice of what is the right reverb time for a particular acoustic space. It depends on how we are going to use that space. So, if I have a room where I will use it only for purposes of talking, then I have to shoot for one particular value of t. If I have to use it only for music, then I have to use may be another value of t. If I have to use for mixture of applications, then my target has to be different. So, that is all I wanted to cover today and then we will