 In the last lecture, we had discussed different aspects of radial propagation of sound waves through some illustrations and through some actual illustration problems. What we plan to do today is continue that journey a little bit further forward and then once we are done with that then we will start talking about a new concept known as directivity and this particular concept of directivity it becomes really important especially in context of radial wave propagation. So, what we plan to do right now is that we will do a problem for illustration purposes and in that context we will consider a sound source which is propagating sound in a radial wave and we assume that while this radial propagation is happening there are no reflecting surfaces around this sound source. So, the only wave which is getting propagated through this sound source is the forward traveling wave and there is no reflection or no backward traveling waves in this picture. So, we consider a sphere and the radius of that sphere is r and that r is very small. So, sphere of radius r not and this thing this particular sphere is emitting. So, this is my r and it is emitting sound waves and what we know is that on the surface of the sphere the pressure is p naught cosine omega t. So, at this surface my pressure is p naught cosine omega t. So, pressure at surface equals p naught cosine of omega t and what we are interested in finding out for this particular problem is that what is the value of power flow which is happening from this sphere outwards and this power will have will be complex in nature. So, it will be complex power flow will have a real component and it will also have a have an imaginary component. So, to do this we start looking at the relation for pressure. So, we know that pressure for a spherical source depends on radius and it also depends on time. So, p is a function of r and t and that is nothing but real of this complex amplitude p plus divided by r times e j r omega t times e j oh excuse me e j I have to correct this e j omega t times e j omega r over c and because this is a forward travelling wave this is a negative sign here. So, that is my relation for pressure now we know that at r equals r naught this value p is equal to p naught cos omega t. So, we know given that p at r naught and at time t equals p naught cosine of omega t. So, I will call this equation one this is equation two. Now, what we do not know right now is the value of p plus. So, p plus is a constant and it will it may have an imaginary component as well as a real component and we do not know what this number looks like. So, let us assume. So, we assume we assume that without losing any sense of generality that p plus is having some magnitude and then it also has a phase element to it and that we express as e j times phi. So, now we plug this relation in equation one. So, what we get is p of r t equals real of magnitude of p plus divided by r e j phi times e j omega t times e minus j omega r over c I call this equation three and now I further process this equation. So, what I get is we I know that at r equals r naught at r equals r naught this is my expression for p. So, I plug that value. So, I plug the value of r as r naught and on the left hand side I put this value. So, what I get is that for r equals r naught p of r t is equal to p naught cosine of omega t and that is nothing but real component of magnitude portion of p plus divided by r naught times e j e j times phi times e j omega r over c times e j omega t. Now, I realize that this term is real. So, I can take it out of the parenthesis and I then split these complex terms into real portion and imaginary portion. So, what I get is p naught cosine of omega t equals magnitude of p plus which is this divided by r naught times real of. So, this is equal to I will rearrange some of these terms here. So, what I get is e minus j omega r naught over c times e to the power of j and then in parenthesis I have. So, I combine this term and I combine this term. So, I get phi plus omega t moving further what I get is amplitude of p plus divided by r naught times real of. Now, e j e to the power of minus j omega r naught c can be expressed in trigonometric terms as a cosine part and it also has a sinusoidal part. So, what I get is cosine of omega r naught over c minus j times sin of omega r naught over c and this whole thing is in parenthesis and then I have to add to that excuse me multiply that I have to multiply this with another set of two terms. So, and that is cosine of phi plus omega t plus j times sin of phi plus omega t. Now, if I take the real portion of this what I get is amplitude of or magnitude of p plus times and I am now taking only the real components. So, I get cosine of omega r naught over c times cosine of phi plus omega t and then plus sin of omega r naught over c times sin of phi plus omega t this is what I get and that equals. So, this is basically a function which looks similar to that of cosine of a plus b equals cosine a times cosine b plus sin a times sin b. So, what I get here is p plus divided by r naught times this entire thing can be rewritten as cosine of omega t plus phi minus omega r naught over c. So, that is my expression number 4 and on the left hand side I have p naught cosine of omega t. So, this is my expression number 4. So, I will rewrite it here on the next sheet. So, p naught cosine of omega t equals magnitude of p plus divided by r naught times cosine of omega t plus phi minus omega r naught over c. Now, this expression and this is again expression 4 this expression can hold true in general only if the following conditions are met. So, the first condition is that the amplitude or actually the magnitude of the right left hand side should equal magnitude of the left hand side. So, the first condition for this equation to hold good is that p naught should be equal to p plus over r naught which implies that magnitude of p plus should equal p naught times r naught. The second condition which has to hold true is that omega t this term should equal this term. So, omega t should equal omega t minus omega r naught over c which implies that phi should equal omega r naught over c. So, these are my equations I label them as 5 a and 5 b and once I plug these equations in my expression for p plus what I get is that p plus equals p naught r naught times e j omega r naught over c. So, that is my equation 6. Now, with this understanding I rewrite my expression for p. So, we know that p of which is a function of r and t equals real component of magnitude of p plus which is p naught r naught divided by radius r times its phase component which is j omega r naught over c times e j omega t plus omega omega t plus omega r over c and I can rewrite this entire thing as real of p naught r naught over r times e to the power of j times omega t plus omega r naught over c minus omega r over c. So, this is my expression for pressure likewise I can write the relation for velocity particle velocity is u of r and t and this is equal to p of r and t excuse me. So, I will write the expression for complex velocity. So, complex velocity is a function of r and t and that is equal to complex pressure divided by impedance. So, this is p naught r naught over z times e j omega and then t plus r naught minus r divided by c and the relation for complex pressure is p of r and t equals p naught r naught over. So, there should be an r here this times e j omega t plus r naught minus r divided by c. Now, I can re express I know that in my previous example what is the value of complex impedance for radially propagating waves and we know that value of z value of z equals 1 over j omega rho naught r plus 1 over rho naught c the whole thing inverse. So, I put this equation this expression in expression for u. So, what I get my expression for complex velocity is this. So, complex velocity equals p naught r naught over r times e j omega t plus r naught minus r divided by c times 1 over j omega rho naught r plus 1 over rho naught c. So, these are my relations for complex pressure and complex particle velocity. Now, our original intent was to find what is the value of complex power. So, complex power equals complex pressure times complex velocity and that can be also written as complex pressure times p star over z star and when I do this and I calculate it what I get here is magnitude of complex pressure is squared divided by z star. Now, we know that 1 over z equals 1 over j omega rho naught r plus 1 over rho naught c. So, which means that 1 over z star can be written as 1 over rho naught c minus 1 over j omega rho naught r. So, once I so now that I know what is z star and I also know that the magnitude of complex pressure is p naught my complex power equals p naught square times 1 over z star which is 1 over rho naught c minus 1 over j omega rho naught r and once I simplify this what I get is p naught square over rho naught times 1 over c minus 1 over j omega r naught. So, that is my expression for complex power. Now, there are couple of observations we can make when we look at this relation for complex power. One is first observation I can make is that this is of course, this is a real component which is p square divided by rho naught times 1 over c and then there is a there is an imaginary component which is p square over rho times minus 1 over j omega r naught. Now, if we want to make this imaginary component small. So, if imaginary component is to be small then 1 over c should be very large compared to omega r naught which means that omega r naught should be extremely large compared to c which means r naught should be extremely large compared to c over omega and c over omega equals c over 2 pi f equals and c over f is wavelength. So, lambda over 2 pi. So, the condition for imaginary component to be very small is if r naught which is the radius of the sphere which is emitting these sound waves is very large compared to lambda over 2 pi or roughly one sixth of the wavelength of sound waves which are being emitted by the radial source. So, that is essence that is the essence of this example that if I have a sound source which is emitting sound waves in a radial way and if it is a spherical source then it will be emitting complex power and the imaginary component of that complex power will be very small if the wavelength or the one sixth of the wavelength of sound waves which are being emitted are small compared to the radius of the sphere. So, this closes my this illustration problem and now I will move on to the next topic which is related to this concept of directivity. Now, we have talked about directivity earlier also, but not in so much of an so much in an explicit sense. So, earlier when we talked about directivity it was in context of interference of waves to radially propagating waves which are getting emitted from two point sound sources which are separated by distance and we saw that when these waves at point which is far away meet they interfere sometimes constructively sometimes destructively and we have some sort of a polar pattern which is not necessarily symmetric with respect to theta. So, we have what we did see earlier was in not such an explicit sense that when there are two sources point sources and when they emit sound waves the overall sound pattern is not necessarily radially symmetric and there is some directionality associated with this type of a phenomena. So, what we plan to do in remaining part of today's lecture and may be also in the subsequent lecture is explore this idea of directivity further and we will start with this discussion on directivity by developing some directivity patterns. So, that is what we will do and then once we are done with directivity patterns then we will look at some other ways or metrics of measuring directivity through terms such as directivity index and some other parameters. So, what the first step what we plan to do is we will start developing directivity pattern for some different set of sources. So, let us define directivity pattern. So, directivity pattern is a graphical representation it is a graphical representation and of sound sources emission as a function of direction specified plane. So, this is important for a specified plane and at a specified frequency. So, this is important to understand that these graphical representations are for specific frequencies. So, you may have one directivity pattern for a set of sound sources which may be significant at a given frequency let us say at 100 hertz and this graphical pattern directivity pattern may be significantly different if we alter the frequency and we make it 1000 hertz or something like that. So, it is for a specified frequency and then the other thing is that you have these directivity patterns plotted on specific planes. So, once you change the plane then the pattern may or may not remain necessarily same. So, we will start by illustrating the directivity pattern of a simple spherical source. So, directivity pattern a spherical source. So, we know that the pressure emitted by a sound source a point sound source and if this is a spherical a radially propagating wave with no reflections then the pressure function is real of complex pressure constant divided by r times e to the power of minus j omega r over c times e j omega t. Now, we see that in this relation p is not a function of theta p does not depend on theta. What that directly means is that if I have a sound source let say it is a point sound source and it is emitting radial sound waves then the directivity pattern for such a sound source will be circular in nature. So, this is the sound source is located exactly at the center of the circle and that is my directivity pattern. So, let us say that the SPL. So, in this case this is my 0 degree this is 90 degrees 180 degree and that is 270 degrees. So, here we have spherical and because it is spherical is spherically symmetric and because we are looking at the pattern on a plane. So, on that plane this is spherical symmetric translate to circular pattern it translates to a circular pattern. Now, it just happens that for a spherical source because everything is circularly symmetric the directivity pattern does not change when we change the frequency and also the directivity pattern does not change when we change the plane of observation. So, for this particular source a spherical source the directivity pattern does not change with respect to changes in frequency and also with respect to changes on in the plane of observation. So, we now move further and we look at a similar pattern for two simple sources. We are looking at directivity pattern for two simple sources and we are going to refer in this context some of the work which was done earlier in context of interference of two sound waves. So, let us say we have two sources source 1 and source 2 and they are separated by some distance and the midpoint of these sources is let us say this point. So, this is the midpoint which is not a source it is just a point of reference. The distance between this midpoint and s 1 is d over 2 and the distance between midpoint and s 2 is again d over 2. Now, I have a point far away. So, at this point the pressure is p and this pressure depends on three parameters r theta and t. So, I will define t in a moment. So, let us say from this midpoint I construct a line which reaches this point the angle of this and let us say this vector is r. So, the angle of this vector r with respect to the horizontal line is theta. So, the pressure at this far point far away point will depend on how far what is the value of r what is the value of theta and at what time are we observing. So, right away we see that whatever the pressure is going to be observed at this point it depends on theta. So, it is not a spherically or circularly symmetric as we saw was the case for a simple source which was emitting spherically symmetric or radially symmetric sound waves. So, I construct two more lines in this case one is between s 1 and point of observation and let us say that vector is r 1 and then the other line is r 2 which connects the point of observation and s 2 and then I drop a perpendicular line from point s 1 to this radius r that line and that is that this particular line in green is perpendicular to vector r. So, what we see here is. So, what we will do is we will again revisit some of the relations we developed earlier and then we will start talking more about directivity patterns. So, earlier what we had assumed was that the volume velocity of sound source 1 was same as volume velocity of sound source 2 and this is V V. So, the magnitude of this volume velocity is same second thing we had assumed earlier was. So, these are assumptions second thing which we had assumed earlier was that the phase of volume velocity 1 minus phase of volume velocity 2 is some constant phi and the third thing which we had assumed was radius r is extremely large compared to D where D is the overall distance between the two sound sources. So, with this understanding we now develop the relationship for pressure, but before we do that we would like to know how is r and r 1 how are they connected. So, what we know is that if r is extremely large compared to D then r 1 is approximately equal to r minus D over 2 sin theta and r 2 is approximately equal to r plus D over 2 sin theta. So, these are the two things. So, with this set of assumptions and these two approximations earlier in our previous class we were able to develop an expression for pressure such that pressure at point this point P which is r distance away and theta angle away from midpoint. So, that pressure equals 2 times volume velocity over 4 pi r j omega rho naught E j omega t minus r over c times cosine of pi D over lambda sin theta minus phi over 2. So, that is the pressure relationship which we had developed earlier.