Practice: Write the mathematical representation of the wave graphed in the following two figures.
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Concept #1: The Mathematical Description Of A Wave
Hey guys in this video we're going to talk about the actual specific mathematical description of a wave instead of just talking about attributes to a wave like the period or the wavelength of the speed we're going to describe a wave fully with a single function, let's get to it. Since waves are oscillatory we have to use oscillating functions to describe them these are our trig functions our sines and our cosines. Alright we're not going to worry about the other trig functions because they aren't truly oscillating functions the displacement that the wave begins with or the initial placement is going to determine the type of trig function whether it's going to be a sine or a cosine let's take this first wave on the left it begins at the origin and then it increases decreases etc. This we know is a sine wave so this wave will be described by a sine function now the wave on the right the wave right above me begins initially at the maximum displacement at the amplitude. Then it decreases and increases and decreases etc. This is a cosine wave wave and so the mathematical function describing that wave is going to be a cosine. Now both of these graphs show a displacement versus time, these are both oscillations in time now simple harmonic motion is described by oscillations in time and you would have very similar functions sines or cosines to describe them but waves propagate in space so we can not only describe their oscillations in time we have to also describe their oscillations in space now if a wave happens to be a sine wave in time it's also a sine wave in position and the same applies for cosines. Now waves like I said are more properly described in terms of oscillation for both space and time so let's do that now for a sine wave we would say it has some amplitude times sine of K, X, minus omega T, now I will tell you what K and omega are in a second but this has both space dependence or position dependence and time dependence exactly like we want or for a cosine wave we had A cosine K, X, minus omega T. So the question is what's K? and what's Omega? K is something new that you guys haven't seen before called the wave number where its 2 Pi divided by the wave length Alright omega is something you guys have seen many times before it's simply the angular frequency or 2 Pi times the linear frequency.
Let's do a quick example a wave is represented by the following function. What is the amplitude? the period? the wave length? and the speed of the wave? So we want to gain all that information from the single equation that describes this waves oscillations in both space and time so first this equation isn't quite of the form that we have seen before we want to take this coefficient right here and we want to multiply it in words because we want to have our equation of the form Y equals A cosine K, X, minus omega T. So we need to multiply this number inside so we can find readily what K is and what omega is. So this is going to be Y equals 0.05 cosine of well X is coefficient right now is 1 so it just gets the 2 Pi over 10 which is 0.628 centimetres inverse. The co sorry the coefficient of T is 7 so if you do 7 times 2 Pi over 10. Which becomes 439.8 inverse seconds now really quickly. Notice that the units of 7 are meters and the units of our number all on the outside are centimetres those need to be the same unit to cancel once you create them the same unit either both of them centimetres or both of them meters then you will find this 439.8. If you don't convert you're going to get 4.398 which is the wrong number. So just make sure that you convert notice right off the bat that we can find the amplitude, we can find the wave number and we can find the angular frequency just by looking at the equation so the amplitude done the way nothing else. We need to use these to find the wave length, the period and the speed so the wave number is related to the wave length all I have to do is multiply Pi up and divide the wave number over. So this is the wave length is 2 Pi divided by the wave number which is 2 Pi divided by 0.628 which is going to be 10 centimetres, so that's another one that we're done. The period is related to the angular frequency we can say that the angular frequency is 2 Pi over the period so if we multiply the period up and the angular frequency over then the period is 2 Pi over the angular frequency which is 2 Pi over 439.8 which is going to be 700 sorry reading the wrong part of my notes here 0.0143 seconds, that is the third thing that we need to find.
Finally we need to find the speed but we know the wavelength and we know the period so the speed is easy to find the speed is simply the wavelength divided by the period which is going to be 10 centimetres divided by 0.0143 seconds which is going to be 700 centimetres per second or 7 meters per second and that is all for things that we were asked to find really quickly guys notice this number 7 right here 7 meters per second and this number sorry 10 centimetres over there to the left. Those numbers appear here and here why do they appear there and there that's because the equation of the form that it's written. I'm going to minimise myself and put a little note right here the equation of the form that it's written is Y equals A cosine of 2 Pi over lambda X minus V T this is another very common way of writing the mathematical equation for a wave and if you notice you already have a lambda and the speed written right lambdas 10 the speed is 7 but since we didn't cover this explicitly we didn't cover this explicitly I didn't want to start from that point. Alright guys that wraps up our discussion on the mathematical description of a wave. Thanks for watching.
Example #1: Graphs Of Mathematical Representation of Wave
Hey guys let's do a quick example draw this placement versus position and the displacement versus time graphs of the transverse wave given by the following representation and they give us the mathematical representation of that wave. I'm going to draw the wave over a single period because there is no way to draw all of the wave these waves go on from negative infinity to positive infinity they go on forever so I'm just going to draw a single period you might be asked to draw two periods of the wave or three periods or four periods or whatever but if you're told just to draw the wave draw a full cycle doesn't really matter how much you draw so here are my graphs here's displacement verses time here is displacement verses position in order to graph them we need to know three things we're going to need to know the amplitude we're going to need to know the period and we're going to need to know the wavelength. Amplitude is easy right one and a half centimetres amplitude done, so on both of these graphs I'm going to write 1.5 centimetres and -1.5 centimetres and these are going to mark the boundaries that the waves are going to oscillate in between the waves are going to stay between the amplitude. Now remember this number right here represents the wave number and this number right here represents that angular frequency don't forget that so our wave number is 2.09 inverse centimetres and that's 2 pi over lambda so lambda is 2 pi over 2.09 which is about 3.0 centimetres
Now the angular frequency as we can see is 2 pi over 0.01 second and the angular frequency related to the period is 2 pi sorry little technical difficulty 2 pi over the period so relating these two equations together we can see that the period is simply 0.01 second. So now we have enough information to draw a full period or a full cycle of each of these waves we know that during the cycle the wave is going to take 0.01 seconds so if I draw 0.01 seconds right here on my displacement verses time graph I can just draw my sine wave right this is a sine wave if this was cosine we would start at a amplitude drop down to the negative amplitude and go back up to the amplitude for displacement versus position we know that a cycle takes 3 centimetres so I'm going to mark 3 centimetre and I'm going to draw the same sine wave and this is exactly the positions sorry the displacement verses time and the displacement verses position graphs for this function. Alright guys thanks for watching.
Practice: Write the mathematical representation of the wave graphed in the following two figures.
Concept #2: Phase Angle
Hey guys in this video we're going to delve a little deeper into the mathematical representation of waves and cover a related concept called phase angle, let's get to it.Now we said a wave that begins with no displacement is a sine wave right and a wave that begins with a maximum displacement a displacement at the amplitude is a cosine wave but what if we have a wave that begins with a displacement in between the two. Then what happens, well in this case the most complete description of a wave the best mathematical description of a wave is to combine all of those possibilities into one and we typically write that as A sine K X minus omega T plus phi where phi is what we call the phase angle. The phase angle tells us what that initial displacement is, the phase angle was determined by the initial displacement of the wave a wave that begins with no displacement has a phase angle of 0 degrees which is a pure sine wave. A wave that begins with a maximum displacement has a phase angle of pi over 2, which is a pure cosine wave an arbitrary wave one has displacement between 0 and its maximum is going to have a phase angle between 0 and pi over 2 ok its going to be a mixture of sine and cosine waves.
Let's do an example, a wave with a period of 0.5 seconds and a velocity of 25 meters per second has an amplitude of 12 centimeters at T equals 0 and X equals 0 the wave has a displacement of 8 centimeters. What is the mathematical representation of this wave? So we're saying Y is A sine K X minus omega T plus phi since X and T are both 0. We can ignore them and all that phi depends upon is the initial displacement so this is going to be 12 centimeters sine of phi and that equals 8 centimeters that's the initial displacement so sine of phi equals 8 centimeters over 12 centimeters and phi. Which is the inverse sine of 8 over 12 is simply 0.62. Very very quick very easy, alright guys this wraps up our discussion on the phase angle and the proper representation of a wave as a mathematical function. Thanks for watching.
Practice: A transverse wave is represented by the following function: y = (18 cm) sin [2π (x/2cm − t/5s + 1/4)]. What is the phase angle of this wave?
Concept #3: Velocity Of A Wave
Hey guys in this video we want to talk specifically about the velocity of a wave, let's get to it. Remember guys that there are two types of waves, right we have transverse and we longitudinal waves both of them have a propagation velocity both waves move right both of them propagate but transverse waves have a second type of velocity which we call a transverse velocity which is how quickly the medium is moving upwards. For instance a wave on a string, a wave on a string is not only moving horizontally at some propagation speed but the individual parts of the string are also moving up and down with some velocity, that's the transverse velocity. The propagation velocity of a wave depends upon two things, depends upon the type of wave and it depends upon the medium that the wave is in. This is an absolutely fundamental property of all waves the type of wave will tell you the equation to find the speed, the medium will tell you some number that goes into the equation alright and we're going to get to a type of wave on a sorry type of transverse wave called waves on a string where we will see that the type of wave ie wave on a string will determine the equation in the medium ie the tension on the string and the mass per unit length on the string will determine how fast it goes. So it's both it's the type of wave and it's the characteristics of the medium that its in the only wave that does not propagate in a medium is light all other waves propagate in a medium light can propagate in a vacuum. We can rewrite as I mentioned in a problem before we can rewrite the equation for a wave like so this is using the fact that K. is 2 pi over lambda and omega let me write that in different colour and omega is 2 pi over T. Where T I can relate to the speed using a regular old speed equation and this becomes 2 pi over the lambda times V. So I can pull this 2 pi over lambda out of the equation and I'm left with a V here. This form tells us the speed of the wave instantly and the direction that it's going in. When we have the X minus V.T. as our input. The wave is propagating in the positive direction when we have X plus V T as our input where v is a positive number in both of these cases, the wave is propagating in the negative direction. A wave is represented by the equation given here, what is the propagation velocity is it positive or is it negative. A really quick way to find given the mathematical representation of a wave the speed of the wave is that the provocation speed is always going to be omega over K this is a really quick equation that you can show is true very easily using these substitutions that I said omega is 2 pi V over lambda over 2 pi over lambda. Those 2 pi's over lambda are left sorry they cancel and all that is left is V you guys don't have to write that down by the way I'm just showing you that it's true. All right now the equation is written where we have omega right here and we have K right here so omega is 1.7 inverse seconds K is 0.2 inverse centimetres which is 8.5 centimetres per second and now the question is is it positive or is it negative well this is a negative sign so it is positive.
Now on a transverse wave we have a transverse component in the velocity and because the wave is going up and going down going up and going down going up and going down obviously the velocity is changing that transverse velocity right initially it's going up so its a positive velocity then its coming down so it is a negative velocity since that's changing there must be a transverse acceleration. So that's another thing to worry about in the transverse direction. Given our general equation for a wave. We have general equations for the transverse velocity and for the transverse acceleration in general the transverse velocity can be written as the amplitude times omega cosine of K X minus omega T plus pi whatever the phase angle happens to be in general the acceleration can be written as negative A omega squared sine K X minus omega T plus phi. None of the numbers change A omega K phi they're all the same in these equations as they would be in the irregular equation for the wave right where we have Y is A sine K X minus omega T plus phi it's all the same variables alright now cosine can get as big as positive 1 and as small as negative 1 so obviously the maximum transverse speed is just A omega likewise sine can get as big as positive 1 and as small as negative 1 so the largest transverse acceleration is just A omega squared just those coefficients of the trig functions and lastly we want to do one more example. A longitudinal wave has a wavelength of 12 centimetres and a frequency of 100 hertz what is the propagation speed of this wave and what is the maximum transverse velocity of this wave. Well we'll use our regular old wave speed equation this is the propagation speed right it goes forward some distance lambda in a period times the frequency of the wave. So as a wave length of 12 centimetres and a frequency of a 100 hertz, which is 1200 centuries per second or 12 meters per second and what about the maximum transverse velocity well there is no transverse velocity this is a longitudinal wave, longitudinal waves have no transverse velocity the transverse velocity is always 0 for longitudinal waves right there is no transverse oscillation all the oscillation occurs down the length sorry down the propagation distance of the wave there's no transverse component so there is no transverse velocity for longitudinal waves but there absolutely is still a propagation velocity for the wave and it follows the exact same equation that you would use for a transverse wave. Alright guys that wraps up our discussion on the velocity of waves. Thanks for watching.
Practice: The function for some transverse wave is 𝑦 = (0.5 m) sin [(0.8 m−1)x − 2𝜋(50 Hz)t + π/3 ]. What is the transverse velocity at t=2 s, x=7 cm? What is the maximum transverse speed? The maximum transverse acceleration?
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