 So, in the last couple of lectures we have talked about sound propagation in regular rooms and also in irregular rooms. In case of irregular rooms, we developed the parameter reverb constant and while developing this parameter we had assumed that there is no damping in the air, but the walls themselves have some dampening effect and they do absorb some sound. The context of all this discussion, regular rooms and irregular rooms is that if there is some sound which is getting generated in the room, how does it propagate and it has direct applications in areas of in places, in acoustic spaces, lecture halls, auditoriums, theaters, places where do the recording given small rooms. I wanted to develop this reverb time further and also what we will see today is what does air do to this whole notion of reverb time. But before I start doing that, I wanted to generate some thinking that if you have a let us say a big room lecture hall where you sit and take lectures and there are 200, 300 students sitting in the room. What are different things as an engineer you would think which will help you design a good lecture hall? Can you cite some parameters? They do not have to be very well defined engineering parameters, but what will make as a listener a good room and it will distinguish it from a bad room. No time delays speakers face and the sound which is why should that matter if the speaker generates a sound and reaches you after in an upside case 5 minutes later. What does that do? Why is it bad or good? We will lose the synchronization. But it reaches first later reaches you after 5 minutes, the second later reaches you after 5 minutes plus some delta t. If you do not watch him then it is understood. So, there is no synchronization between his lip movement and understood. So, that is one that no significant delay between the sound source. I mean the time when the sound is coming out and when it reaches here, here because there is will be a disconnect between visual perception and auditory perception. So, that is all good. You wanted to say something? Sir, there should not be any notes and in the room. The sound should be uniform all over the room. Sound should be uniform all over the room and why is that good or bad or why is that good? I mean one person is sitting like this and if he changes his position, he should hear the same volume like same sound. Second is that anything else? There should not be any echo. There should not be any echoes. Why are echoes bad? We will hear double sound like. So, echoes is there. Anything else? Sir, the main importance should be on the vocal frequencies. Main importance should be on the vocal frequencies. The note for vocal frequencies should be less. In a lecture room? Yes, sir. But not necessarily in a place where you are playing music? Yes, sir. So, it depends on the application that how the room behaves should be tuned to whatever is the band in which sound is being generated. What about noise from outside? Noise conservation. You want to minimize that. Noise which is coming from outside has to be minimized. Echoes you already mentioned. So, we will also cover some of these other things which we talked about. There should be no resonance with the objects. No resonance. I have seen many a times these chairs and if it is a especially metallic objects, they start to vibrate and make sound in the bass room. In the room? If there is a loud bass, objects begin to chatter. So, there should not be any buzz, chatter, rattle, freaks, these kind of things. So, we will cover some of these. We will definitely cover echoes. We will cover some criteria so that we can figure out what is a good permit, not good acceptable noise level as it comes from outside. We will also cover a term called room constant. So, we have talked about reverb time and we will also introduce a term called room constant. So, that is all that is the spectrum things we will talk today. But we will start by including the effect of air damping on reverb time. We had seen that reverb time is 60 times v over 1.08 AC, a prime c and we had defined a prime c as minus s natural log of 1 minus alpha. So, what we will do is we will modify this particular equation by incorporating the damping of damping generated due to air. Let us say d naught is my initial energy going into the system and once sound travels a distance d which we had defined in earlier lectures as mean free, then just because of air this d naught it goes down and it becomes d naught e to the power of minus where m is fairly small compared to 1 and its dimension is 1 over length. So, dimension could be of m is 1 over meters or 1 over feet. So, using the logic we had used in our previous lecture, we can say that d which is at the energy density at a given point of time, time equals n delta t equals d naught 1 minus alpha bar n e minus n m d. This is after n reflections. This part the one in yellow is because of the reflections of the wall. This part is because of the role of the air. So, you have n reflections. So, it is n times m times d. We know that so we can develop this further. So, that is d naught 1 minus alpha bar and n is what t over delta t and then the other exponent part is minus again I will replace n by t over delta t. So, it is t times m d over delta t. Now, we know that d over delta t is what? d is the mean free part c velocity of sound. So, I will put that in here. So, I get d naught 1 minus alpha t over delta t e minus t m c. We also know that 1 over delta t is basically c s over 4 v. We had seen this earlier. So, I modify this further d 0 1 minus average alpha. So, I get c s t over 4 v exponent minus t m c. So, now, what I do is like I did for the earlier case I express this entire thing also in an exponential format. So, I get d equals d naught exponent. So, I get log 1 minus alpha bar s e t over 4 v minus t m c and if I rearrange my stuff what I get is exponent minus s l n 1 minus alpha bar plus 4 m v times c t over 4 v. I can make this whole thing as a prime here because now it includes the effects of damping also of the air. So, I get d naught e minus a dash air times t c over 4 v. So, this is essentially very similar to the earlier expression we had developed for a room with no damping due to air. So, I can use the same logic stream and I can say that reverb time if I have to include the effect of air is basically 60 times volume divided by 1.086 a dash air times velocity sound in air. So, couple of things one is that m this m parameter which is it actually very strongly depends on omega. So, this depends on omega and the relationship is that as frequencies go up m goes up. So, as omega goes up we have see we see through experimental data omega also m goes up. So, once again we have to be careful that when we find the value of a prime it is specific for a particular frequency or a narrow frequency band is important to understand. Second thing is that a prime air is approximately equal to a prime with for small values of for small rooms. If my v is a small then this term will pale in insignificance compared to the other term. So, for small rooms I may still get away with the without including the effect of air damping, but for very large rooms auditoriums halls it is prudent to use the damping effect. This completes the effect of air damping and what kind of an influence it has on reverb time. So, does the reverb time go up or go down as I include the effect of air damping it will go down. Because this term goes up and physically what it means is that what is the original fundamental definition of reverb time. Sir the sound decay is by the 60 decibel. Yeah which is a factor of a million. So, it will take lesser time to decay to that floor level if I include the effect of air damping which makes sense. So, this completes our discussion on the reverb time and now what we will do is we will move to a new concept and this is also used in architectural acoustics and it is essentially trying to develop a ratio of how much energy is received at the ear of the listener which is directly coming from the source and how much energy is coming from reflected you know due to reflections. What we will discuss is SPL due to direct and reverb fields I have a big I have my source and then I have a person some energy is coming directly to the individual direct and some other energy goes and it has a consequence of reflections it comes. So, this is my reverb the minimum amount of reflections the reverb energy is what direct field there is no reflection the minimum amount of reflections which a reverb energy will have which reverb energy will have will be one at least one will have see at least one reflection in very large auditoriums typically if you are far away from the source in general the reverb energy dominates compared to the direct field we will see that as we develop the mathematics of it and the SPL depends on several parameters it depends on volume it depends on how far you are from the distance of the source depends on power of the source and it depends on alpha and bunch of other parameters. So, first we will do is direct SPL direct field so, I have a source this is my ear or microphone. Sir reverb energy how can it dominate sir there will always be some. We will see that and this is the distance between the source and the ear I ear which is the intensity at my ear and let us say the source is emitting W watts of energy. So, I ear is what based on some of the earlier concepts we have developed intensity is. Sir p power. Power per unit area. So, it is watts over 4 pi r square and we know that we can also write this as p direct the whole thing squared I have to average this divided by 2 rho naught and I can write this in a short form over 2 rho naught. So, again note that p d is average r m s pressure from this relation and from this relation what I get is p d square is essentially 2 rho naught c times whatever is the strength of the source divided by 4 pi r square r p d equals the square root of rho naught c W over 2 pi r square. So, this is my first relation and we will apply the standard disclaimer that this everything of this nature is valid to the extent my r is what. Very little less than lambda over 2 pi. Yes it is fairly large than one sixth of the wavelength of the frequencies we are considering. So, this is my direct. So, now what I will do is I will develop something for the reverb field. We do it for a reverb field. So, for the reverb field we have to think how the sound develops in the room. If the room has absolutely no dampening surfaces and air the sound does not leak out of the room then as my speaker is pumping energy into the system what will happen to the reverb energy over a period of time. It will just keep on going on increasing infinitely. If there is at least one dampening surface which absorbs some energy or if there is air which is also absorbing. Then the reverb field the way it will behave is that it will keep on growing, but the rate of growth will start over a period of time. It will start slowing and it will become flatten it will flatten out and at a certain after a certain point of time whatever is being pumped in to the speaker is getting absorbed by the walls and there is some sort of an equilibrium. So, what we are trying to do in this analysis is that we are trying to find that what is that equilibrium state. So, it is not the transient state what we are trying to figure out is what is that equilibrium state at which the reverb field does not grow over a period. So, in this analysis what we will essentially do is we will compute whatever is going in input power and whatever is going out which is the output thing and once they are equal then I can that is my condition for input power equals what what is the input energy going in the system or power going in to the it is whatever is whatever is pumped by the speaker. No, but it has to see at least one reflection that is going in. So, what is that value what is being pumped by the system W. W times 1 minus. W times 1. So, this is power flow into the room now what we will do is whatever is going out. So, power going out equals energy going out divided by energy going out divided by time I will number them 1 2 this is 2 power going out is essentially power lost per reflection. So, that is energy lost per reflection and then what is the time time is delta t. There is time between two reflections. So, this is energy lost per reflection over delta t and we know that delta t equals what d over c and it is also 4 v over s c. So, now, d r we define a term d r it is reverb energy density reverb energy density and that is essentially t r upper case r implying it is reverb pressure r m s the whole thing square divided by rho naught c square is from earlier notes and e r is reverb energy in the room. So, what is reverb energy in the room in terms of d r d r is reverb energy density. So, reverb energy in the room is this is energy density d r is energy density unit of energy density is joules per cubic meter. So, total reverb energy in the room is what times volume. So, it is d r times volume. So, that gives me v e r dot r m s the whole thing square over rho naught c square 3 and total energy lost each reflection will be how much in terms of e r? e r times alpha energy lost each reflection is here. So, total energy lost per reflection e r times alpha which is same as d r times v times alpha 4. So, whatever is the flow going in we know w 1 minus alpha and that equals whatever is going out. So, now we equate them we have to do one more step this is the total energy lost per reflection. So, total power which is decaying per reflection which gets attenuated per reflection or it gets deducted per reflection total power will be how much? No no no in terms of this we know now we know what is energy lost per reflection is d r times v times alpha. So, in terms of this power divided by delta t. So, power lost I will not say lost power decay per reflection equals v r times v times alpha over delta t and this equals whatever is going in which is w times 1 minus alpha. So, w times 1 minus alpha equals d r v alpha over delta t and I substitute the value of delta t as c s over 4 v. So, what I get is d r v alpha times c s over 4 v and now what I do is this alpha bar and my if I make use further this particular relation d r equals p r r m s square divided by rho c square and I plug in there what I get is v times p r r m s square divided by rho naught c square alpha bar times c s over 4 v. So, I do the math and I simplify this and essentially what I get is reverb r m s pressure which is average in time and space the square of it is basically whatever the speaker is pumping in times 1 minus alpha over s alpha bar times 4 rho naught c and now I introduce a new term called room constant r and r is s alpha bar divided by 1 minus alpha bar and the unit of r is meter square or feed square. So, e r r m s is basically 4 w rho naught c over r. So, that is my reverb pressure I will read out these two relations my reverb pressure is 4 w rho naught c over room constant and my direct pressure is what parts over 4 pi square times rho naught c. So, what you see is that your reverb pressure does not change in the room and your direct pressure very rapidly it decays with radius. So, to answer your earlier question if you are reasonably far away from the source within the room the reverb pressure will dominate whatever you are and we will put some numbers on that and my total pressure and I can add the energies basically this is p r square is directly proportional to reverb energy p d square is directly proportional to the direct energy and I can add the energies I cannot add the pressures themselves, but I can add the energies. So, p t square is basically w rho naught c times 4 over r plus 1 over 4 pi r square. So, this is the relation for my total pressure. Sir, this distance is minimum when we consider this straight line. Sir, the part travel will be minimum. Sir, in that case a direct pressure will decay and when the when we consider the reflected then the distance travelled is more, it will decay more then. Sir, then how the. Yes, but the diffused pressure the reverb pressure it does not change from point to point it will decay more when you are in relative terms see all you are doing is you have to compare you have to compare your direct pressure in a square way 1 over r square way decays as you move out your diffused pressure is not changing within the space a diffused pressure is not changing within the whole room. So, if you are fairly far away from the source then a direct pressure will be fairly small while the diffuse pressure will still remains. This relation is basically addition of two energies that is one thing I want to say and this works only to the extent that both these sources are what correlated or uncorrelated we have talked about this earlier the source of the music is uncorrelated and my sound pressure level basically 10 log 10 times p t r m s over t ref basically it is a square of this and square of this. So, now I plot my S P L with respect to r and the curve is going to look like somewhat like this is my r r is the distance between the listener and the source and this is my sound pressure level and what I get is something like this an asymptotic behavior where what is this value this corresponds to p what p d this is my total p t. So, p t converges to p d as I move away from the from the source. So, this is not p d I said that by mistake this is river field another question that I know that room constant is S times alpha bar over 1 minus alpha bar if there are no walls in the room. So, that it is just I am having a speaker in a field and I am just running it then what is r what is the value of r infinity for if I am too close to the room not to the source then my direct field is dominating basically that is what this relation is saying and as I move further there will be a threshold where this term equals this term and once I have crossed that particular threshold then the 4 over r term starts dominating compared to 1 over 4 pi r square that particular value is where I have this equivalence between 4 over r and this the other term 1 over 4 pi r square that we call it r naught. So, r naught is basically r square average size auditorium r naught is approximately equal to 20 feet about 6 meters 4 w r c and various p direct. So, r naught is about 20 feet for reasonably sized auditorium if I make my auditorium bigger then this r naught also goes up because r goes up and for small rooms this is around 2 feet less than a meter. So, even in a room like this most of the sound which you are hearing is essentially reverb energy not directly. Consider a scenario where you have a recording equipment and there is a bunch of people who are generating creating some music their instruments and all that and if you are too far from the music then the microphone will essentially capture the reverb field. If it is too close to the musical instruments then it will capture predominantly the direct field. Now, the limitation of capturing the direct field only is that each of these instruments they have their polar patterns based on their geometry and all that. So, if you are too close then based on each instrument's polar pattern the microwave will pick selectively some frequencies more than other frequencies. So, that will not necessarily be a faithful reproduction of the sound. If you are far then you do not capture that, but essentially what you are doing is you are picking up peaks and nulls and you are picking up a lot of information contained in the modes and the natural frequencies of the room. So, again that this starts the faithfulness of the music production. And also sir the absorption of the air absorption will be there. Air absorption, but that you can amplify, but yes air absorption and it depends on the frequency. It depends on the frequency, but lot of times these rooms are not that big. Recording studios are not that large, but yes I mean if you want to do this recording and sitting in the auditorium and where the performance happening in on the stage and then air effects will also become dominant will become prominent. So, that will also curtail some of the higher frequencies. So, it is very difficult to reproduce live music whatever you hear in a room when live music is being played faithfully capturing that in a on a CD or a DVD or an or a cassette. It is very difficult and there are technical reasons behind that. It is not that the companies do not know what they are doing, but it is difficult. So, lot of times they try to put their recording instrument at this value or not. So, it is like some sort of a compromise. Another thing is that we know that room constant is 4 W rho naught c over r no I am sorry c r m s. So, this is my room constant. So, as room constant goes up it as it goes up my t goes down and. So, if I have a larger room my reverb energy will go down in that room. Now, we know that r is s alpha over 1 minus alpha bar. So, what that means is that as alpha goes up what happens to a it goes up or down sorry first is depends alpha is always less than 1 r goes up. So, as alpha goes up which means the absorption coefficient of the room is going up r will go up and p will go down more sound is getting absorbed. So, less reverb energy similarly as s which is the surface area of the room goes up r goes up and p goes down I just wanted to curve. So, first thing I wanted to show in this overhead is that what you are seeing here is the dependence of alpha on humidity which we have not explicitly talked about in earlier lecture also. But if I increase the relative humidity in a room then my alpha it actually decays from 10 to 20 percent it goes up as I move from 10 percent relative humidity 20 percent, but after that it goes down significantly. So, humidity is a very important parameter as we are trying to figure out what is the value of alpha and that has an impact on the time constant of the room. And what this is showing is that so as alpha changes how does room constant change and this is also plotting the value of for larger rooms as alpha goes up we expect room constant has to go up and as alpha I am sorry as v goes up for a given value of alpha room constant is going almost linearly and as I reduce my alpha those linear relationship they just move in parallel. The other thing is so we will again start again from this relation or actually we will start from this relation and we will go further. So, S t l is 10 log 10 and we had seen this rho w rho naught c times 1 over 4 pi r square plus 4 over r this whole thing divided by e ref square. So, now I can break it and that is essentially 10 log 10 of w plus 10 log 10 of rho naught c plus 10 log 10 of this bracketed term and if I also take the log due to this p ref I get a value of 94 p ref is 2 times 10 to the power of minus 5. So, I take that log and then multiplied by 10 I get 90. Now, we know that like S p l we had also it is shown earlier that p w l decibels in power is basically 10 log whatever the power is being measured divided by reference power and that reference power is 10 to the power of minus 13. So, if I plug this back here minus 9 minus 5 that is power we usually take minus 10 to the power of minus 9 more. I think this is fairly standard we will check that 10.9 is for intensity 10 to the power of minus 9 is for intensity. So, the story will not that is a number. So, if I plug this back in here basically what I get is S p l equals p w l plus 10 log this number I mean this bracketed term and then I get another some constant term and that constant will change whether I am going to SI units or I am going to feed pounds system, but essentially it is p w l plus 10 log this whole bracketed term and this C is 0.5 dB for British system. And why I am talking about British system is some of the curves which you will see have charts which are based on British system, but for SI system this is what is that minus 36 anyway. So, for British system this is what you have. So, if I plot the relative S p l then essentially what does relative S p l mean that I normalize it with respect to w itself then relative S p l is essentially dependent on 10 times log of this whole bracketed term plus this constant 0.5 dB. My reference yeah relative to this w. So, what I wanted to show you was this particular chart. So, what you are seeing here are relative sound pressure level in decibels for which we had developed the relation just now and here you are increasing r as you move away from the source and then what you have are different curves. So, for r equals infinity which is open here you have a straight line. Now, let us consider this and essentially what it shows is that let us say minus 20 dB line and let us say I am only 2 or 3 feet away from the source. If I am just 2 or 3 feet away from the source then the relative sound pressure level is fairly close to whatever I have for a room which has no walls because this point on the curve is fairly close to this r equals infinity line. But as I move away depending on what is the value of room constant my sound pressure level starts becoming constant and it is significantly different from r equals infinity line. So, just wanted to show how r plays an important role in these calculations that is what this chart I will show you. For instance for this 1000 r equals 1000 line when I am at 20 feet then I am here where it is the relative sound pressure level is fairly constant. But if I for the same curve on r equals 1000 when I become get closer and closer and let us say I am only 2 feet away then this point is very close this point is very close to an open room response room with no walls. So, the next thing I just wanted to talk about is echoes and there is some sort of a standard definition for an echo what is an echo and what is not an echo when intuitively we all know that whenever sound gets reflected it is an echo. This is source which emits sound it gets reflected comes back to the listeners here. If the delta t between the original sound and the reflected sound when it hits the ear is less than 15th of a second then the perception of echo is not that great. If it exceeds 1 15th of a second then you say oh there is an echo there is some sort of a quasi engineering definition. So, it is like 1 15th of a second if it is delta t is greater than 1 15th of a second then you say that there is an echo. Now, in fairly large auditoriums not even large auditoriums even in moderate size rooms you hear this echo in auditoriums where the reverb time is in several seconds echoes will be there and they are always there, but the impact these echoes they have on the our understanding and perception of sound it not only depends on whether the ear is hearing the echo or not, but also it also depends on how strong is that echo in terms of SPL. So, if the difference of original sound and the echoes SPL is significantly large then even though that echo is there it does not have a very strong adverse impact on our assessment of the large. So, in that context I wanted to show you another chart. So, what you are seeing here is echo delay in milliseconds 1 15th of a second is about 67 milliseconds 67 70 milliseconds. So, on the horizontal axis you have echo delay in milliseconds and on the vertical axis it is echo intensity in decibels and this is relative to the original sound level. So, if echo intensity is equal to original sound level then what is the dB level difference between the two 0 dB. So, 0 dB line will be just on the top of the curve on the graph. So, and then you have different curves here. So, we will take one point let us say a delay of 100 milliseconds minus 10 dB of a difference. So, let us say this is minus 10 dB of a difference. So, that falls on the 10 percent curve what that means is that out of every 100 people who listen to an echo where the delta t or delay time is 100 millisecond and the echo intensity is 10 decibels less than the original sound 10 percent of the people will get confused 90 percent will be fine actually it is the other way 10 percent will be fine 90 percent will get 10 percent will be 10 percent will get confused and 90 percent will be fine. Now, if I increase the intensity level then I have a 90 percent curve and the decibel level is over 0. So, I do not know how they constricted above 0, but essentially as you increase the decibel level of the echo more and more people get confused. So, this also gives you some approximate idea how to ensure that whatever echoes you are hearing in the room they are below a certain threshold. So, that most of the people do not have a poor view of the sound in the room and then the last one is noise background. So, the noise background could be for instance in this room there is some fan have running in this podium there is some noise coming from this fan source it could be due to there may be power generator in the room or outside the room it could be an air conditioner it could be AC ducts sending in air it could be people walking outside the auditorium could be traffic there are so many sources of noise. And there are again some fairly good guidelines what kind of a noise level is acceptable because you cannot eliminate the noise absolutely and what is not acceptable. So, to do that if you go by the Berenek book what it says is that you have two step the step one identify what is your need. So, identify this parameter SC and SC is a speech communication you know speech communication number and there is a chart we will show that chart. For a broadcast studio where you have a person speaking or a musical instrument playing and that is being broadcast over a wide range that number is very low because you need very little amount of noise coming in from outside for a factory that SC number is fairly high. So, for instance again for broadcast studio that SC criteria you have to the SC number is 15 to 20 then you go to music rooms it goes up to 25 then you go to a conference room where people also chat while they are listening and participating still remains 25 then court rooms, libraries, small private restaurants, restaurants is 45 noisy factories are very noisy. So, it could be anywhere between 40 to 65. So, that is a step one figure out what is this SC criteria curve and then in step two then what you do is you refer to another table and you identify max SPL levels for each frequency or for different frequency bands you figure out what are the frequency levels which are permitted and this curve helps you do that. So, suppose you are in that broadcast studio you pick up this curve SC equals 20 and you say oh my range of frequencies which I am going to use in this room are 300 to 10000 hertz. So, my SPL noise coming out from the outside or unwanted sources has to be below this dark line. So, it is very straight forward, but these are like some sort of guidelines. Now, there are also dashed lines on this curve and the only time you use those dashed lines is when all or most of your noise is in this 300 to 4800 band. If your noise content is also in this particular band below 300 hertz then you stay with the dark line, but you can use this dashed line if most of your noise is above this 300 basically what it means is that since your most of your noise is coming in that higher frequency range you can have lower noise floors for less of you. So, that is some qualitative measures which you can use to design sounds. So, we have talked about reverb time we have talked about direct and reverb energy densities and how it relates to all characteristics. We have talked about room constant we have tried address the role of echo extraneous noises and in the next lecture we will try to cover some applications or whatever we will not definitely not introduce new topics we will try to cover some applications and you had mentioned. . . . . . . . . . . . . . . . .