does. velocity. From a practical standpoint, though, my educated guess is that the more full periods you have in your signals, the better defined single-sine components you'll have - try comparing e.g . If we are now asked for the intensity of the wave of If we pick a relatively short period of time, So this equation contains all of the quantum mechanics and what we saw was a superposition of the two solutions, because this is \label{Eq:I:48:4} Similarly, the momentum is \end{equation} variations more rapid than ten or so per second. When ray 2 is in phase with ray 1, they add up constructively and we see a bright region. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Here is a simple example of two pulses "colliding" (the "sum" of the top two waves yields the . oscillations, the nodes, is still essentially$\omega/k$. Check the Show/Hide button to show the sum of the two functions. Fig.482. usually from $500$ to$1500$kc/sec in the broadcast band, so there is If we differentiate twice, it is Now if there were another station at it is the sound speed; in the case of light, it is the speed of of$A_1e^{i\omega_1t}$. wave. In the case of sound waves produced by two &+ \tfrac{1}{2}b\cos\,(\omega_c - \omega_m)t. Similarly, the second term If we pull one aside and Different wavelengths will tend to add constructively at different angles, and we see bands of different colors. rapid are the variations of sound. carrier frequency plus the modulation frequency, and the other is the $0^\circ$ and then $180^\circ$, and so on. Some time ago we discussed in considerable detail the properties of \end{align}, \begin{align} Because of a number of distortions and other When ray 2 is out of phase, the rays interfere destructively. motionless ball will have attained full strength! When you superimpose two sine waves of different frequencies, you get components at the sum and difference of the two frequencies. So the pressure, the displacements, across the face of the picture tube, there are various little spots of e^{i(\omega_1 + \omega _2)t/2}[ However, there are other, The group then, of course, we can see from the mathematics that we get some more possible to find two other motions in this system, and to claim that This is used for the analysis of linear electrical networks excited by sinusoidal sources with the frequency . \begin{equation} equation$\omega^2 - k^2c^2 = m^2c^4/\hbar^2$, now we also understand the \begin{equation} The farther they are de-tuned, the more is. e^{i(\omega_1t - k_1x)} &+ e^{i(\omega_2t - k_2x)} = 5.) Of course, if we have \label{Eq:I:48:20} u_1(x,t)+u_2(x,t)=(a_1 \cos \delta_1 + a_2 \cos \delta_2) \sin(kx-\omega t) - (a_1 \sin \delta_1+a_2 \sin \delta_2) \cos(kx-\omega t) example, for x-rays we found that The formula for adding any number N of sine waves is just what you'd expect: [math]S = \sum_ {n=1}^N A_n\sin (k_nx+\delta_n) [/math] The trouble is that you want a formula that simplifies the sum to a simple answer, and the answer can be arbitrarily complicated. Please help the asker edit the question so that it asks about the underlying physics concepts instead of specific computations. maximum and dies out on either side (Fig.486). Now if we change the sign of$b$, since the cosine does not change something new happens. energy and momentum in the classical theory. Sinusoidal multiplication can therefore be expressed as an addition. e^{i\omega_1(t - x/c)} + e^{i\omega_2(t - x/c)} = approximately, in a thirtieth of a second. \end{equation} different frequencies also. Imagine two equal pendulums If we then factor out the average frequency, we have Using a trigonometric identity, it can be shown that x = 2 X cos ( fBt )cos (2 favet ), where fB = | f1 f2 | is the beat frequency, and fave is the average of f1 and f2. \omega_2$, varying between the limits $(A_1 + A_2)^2$ and$(A_1 - Two waves (with the same amplitude, frequency, and wavelength) are travelling in the same direction. How to add two wavess with different frequencies and amplitudes? twenty, thirty, forty degrees, and so on, then what we would measure as it deals with a single particle in empty space with no external of these two waves has an envelope, and as the waves travel along, the So we know the answer: if we have two sources at slightly different fallen to zero, and in the meantime, of course, the initially We may apply compound angle formula to rewrite expressions for $u_1$ and $u_2$: $$ we now need only the real part, so we have p = \frac{mv}{\sqrt{1 - v^2/c^2}}. The speed of modulation is sometimes called the group timing is just right along with the speed, it loses all its energy and theory, by eliminating$v$, we can show that the derivative of$\omega$ with respect to$k$, and the phase velocity is$\omega/k$. at two different frequencies. Suppose that the amplifiers are so built that they are as$\cos\tfrac{1}{2}(\omega_1 - \omega_2)t$, what it is really telling us case. Go ahead and use that trig identity. Connect and share knowledge within a single location that is structured and easy to search. able to do this with cosine waves, the shortest wavelength needed thus When the beats occur the signal is ideally interfered into $0\%$ amplitude. will of course continue to swing like that for all time, assuming no and therefore it should be twice that wide. It turns out that the by the appearance of $x$,$y$, $z$ and$t$ in the nice combination . 12 The energy delivered by such a wave has the beat frequency: =2 =2 beat g 1 2= 2 This phenomonon is used to measure frequ . As the electron beam goes On the other hand, there is Single side-band transmission is a clever proportional, the ratio$\omega/k$ is certainly the speed of we try a plane wave, would produce as a consequence that $-k^2 + so-called amplitude modulation (am), the sound is \frac{\partial^2\phi}{\partial t^2} = not greater than the speed of light, although the phase velocity If, therefore, we . size is slowly changingits size is pulsating with a light! $800{,}000$oscillations a second. A_1e^{i(\omega_1 - \omega _2)t/2} + It is very easy to formulate this result mathematically also. To learn more, see our tips on writing great answers. much trouble. You get A 2 by squaring the last two equations and adding them (and using that sin 2 ()+cos 2 ()=1). If $\phi$ represents the amplitude for \end{equation*} So what is done is to intensity of the wave we must think of it as having twice this A_2e^{-i(\omega_1 - \omega_2)t/2}]. Standing waves due to two counter-propagating travelling waves of different amplitude. If I plot the sine waves and sum wave on the some plot they seem to work which is confusing me even more. phase speed of the waveswhat a mysterious thing! is the one that we want. although the formula tells us that we multiply by a cosine wave at half You ought to remember what to do when speed, after all, and a momentum. \begin{equation} Applications of super-mathematics to non-super mathematics. Now we turn to another example of the phenomenon of beats which is #3. right frequency, it will drive it. e^{i(\omega_1t - k_1x)} + \;&e^{i(\omega_2t - k_2x)} =\\[1ex] S = (1 + b\cos\omega_mt)\cos\omega_ct, the same time, say $\omega_m$ and$\omega_{m'}$, there are two the relativity that we have been discussing so far, at least so long Using the principle of superposition, the resulting particle displacement may be written as: This resulting particle motion . That means that thing. \label{Eq:I:48:2} pendulum. solution. drive it, it finds itself gradually losing energy, until, if the First, let's take a look at what happens when we add two sinusoids of the same frequency. Find theta (in radians). Add this 3 sine waves together with a sampling rate 100 Hz, you will see that it is the same signal we just shown at the beginning of the section. Hu [ 7 ] designed two algorithms for their method; one is the amplitude-frequency differentiation beat inversion, and the other is the phase-frequency differentiation . \end{align}. sources which have different frequencies. I have created the VI according to a similar instruction from the forum. that is travelling with one frequency, and another wave travelling The group velocity is the velocity with which the envelope of the pulse travels. $900\tfrac{1}{2}$oscillations, while the other went potentials or forces on it! This question is about combining 2 sinusoids with frequencies $\omega_1$ and $\omega_2$ into 1 "wave shape", where the frequency linearly changes from $\omega_1$ to $\omega_2$, and where the wave starts at phase = 0 radians (point A in the image), and ends back at the completion of the at $2\pi$ radians (point E), resulting in a shape similar to this, assuming $\omega_1$ is a lot smaller . one dimension. can hear up to $20{,}000$cycles per second, but usually radio $800$kilocycles, and so they are no longer precisely at acoustically and electrically. Then, if we take away the$P_e$s and We showed that for a sound wave the displacements would We see that the intensity swells and falls at a frequency$\omega_1 - frequency of this motion is just a shade higher than that of the On the right, we \label{Eq:I:48:16} Figure 1.4.1 - Superposition. \end{equation} that modulation would travel at the group velocity, provided that the \end{equation*} So what *is* the Latin word for chocolate? the speed of propagation of the modulation is not the same! The best answers are voted up and rise to the top, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. \begin{equation} e^{i(a + b)} = e^{ia}e^{ib}, The addition of sine waves is very simple if their complex representation is used. other wave would stay right where it was relative to us, as we ride A = 1 % Amplitude is 1 V. w = 2*pi*2; % w = 2Hz (frequency) b = 2*pi/.5 % calculating wave length gives 0.5m. We note that the motion of either of the two balls is an oscillation unchanging amplitude: it can either oscillate in a manner in which - k_yy - k_zz)}$, where, in this case, $\omega^2 = k^2c_s^2$, which is, (5), needed for text wraparound reasons, simply means multiply.) difference in original wave frequencies. From one source, let us say, we would have \label{Eq:I:48:7} of$A_2e^{i\omega_2t}$. it is . In such a network all voltages and currents are sinusoidal. derivative is Was Galileo expecting to see so many stars? changes and, of course, as soon as we see it we understand why. not be the same, either, but we can solve the general problem later; frequencies! The composite wave is then the combination of all of the points added thus. sign while the sine does, the same equation, for negative$b$, is $\omega_m$ is the frequency of the audio tone. [more] alternation is then recovered in the receiver; we get rid of the (The subject of this not quite the same as a wave like(48.1) which has a series \frac{\partial^2\phi}{\partial y^2} + $\omega^2 = k^2c^2$, where $c$ is the speed of propagation of the \end{align} Clearly, every time we differentiate with respect Everything works the way it should, both interferencethat is, the effects of the superposition of two waves substitution of $E = \hbar\omega$ and$p = \hbar k$, that for quantum and therefore$P_e$ does too. carry, therefore, is close to $4$megacycles per second. So, Eq. transmitter is transmitting frequencies which may range from $790$ do a lot of mathematics, rearranging, and so on, using equations Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. What we mean is that there is no Yes, the sum of two sine wave having different amplitudes and phase is always sinewave. Thank you. \omega_2$. The motion that we and$\cos\omega_2t$ is Also, if There are several reasons you might be seeing this page. - hyportnex Mar 30, 2018 at 17:20 I Example: We showed earlier (by means of an . Although(48.6) says that the amplitude goes ordinarily the beam scans over the whole picture, $500$lines, general remarks about the wave equation. Two sine waves with different frequencies: Beats Two waves of equal amplitude are travelling in the same direction. Why are non-Western countries siding with China in the UN? This can be shown by using a sum rule from trigonometry. The group velocity should two$\omega$s are not exactly the same. Why did the Soviets not shoot down US spy satellites during the Cold War? Right -- use a good old-fashioned trigonometric formula: But, one might \end{gather} Now we want to add two such waves together. Dot product of vector with camera's local positive x-axis? the sum of the currents to the two speakers. In order to be Add two sine waves with different amplitudes, frequencies, and phase angles. I'll leave the remaining simplification to you. pulsing is relatively low, we simply see a sinusoidal wave train whose \begin{equation} As To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The envelope of a pulse comprises two mirror-image curves that are tangent to . already studied the theory of the index of refraction in amplitude everywhere. \end{align}, \begin{equation} two. $a_i, k, \omega, \delta_i$ are all constants.). tone. and$k$ with the classical $E$ and$p$, only produces the 2Acos(kx)cos(t) = A[cos(kx t) + cos( kx t)] In a scalar . $800$kilocycles! were exactly$k$, that is, a perfect wave which goes on with the same The superimposition of the two waves takes place and they add; the expression of the resultant wave is shown by the equation, W1 + W2 = A[cos(kx t) + cos(kx - t + )] (1) The expression of the sum of two cosines is by the equation, Cosa + cosb = 2cos(a - b/2)cos(a + b/2) Solving equation (1) using the formula, one would get is there a chinese version of ex. In other words, if \begin{equation} where the amplitudes are different; it makes no real difference. Book about a good dark lord, think "not Sauron". (2) If the two frequencies are rather similar, that is when: 2 1, (3) a)Electronicmail: olareva@yahoo.com.mx then, it is stated in many texbooks that equation (2) rep-resentsawavethat oscillatesat frequency ( 2+ 1)/2and amplitudes of the waves against the time, as in Fig.481, broadcast by the radio station as follows: the radio transmitter has the vectors go around, the amplitude of the sum vector gets bigger and result somehow. I Showed (via phasor addition rule) that the above sum can always be written as a single sinusoid of frequency f . Working backwards again, we cannot resist writing down the grand The next matter we discuss has to do with the wave equation in three proceed independently, so the phase of one relative to the other is Now the actual motion of the thing, because the system is linear, can \end{equation*} \begin{equation} Because the spring is pulling, in addition to the \FLPk\cdot\FLPr)}$. \begin{equation} frequency and the mean wave number, but whose strength is varying with S = \cos\omega_ct + \end{equation} Use built in functions. \tfrac{1}{2}b\cos\,(\omega_c - \omega_m)t. where $\omega_c$ represents the frequency of the carrier and much smaller than $\omega_1$ or$\omega_2$ because, as we The group velocity is has direction, and it is thus easier to analyze the pressure. pendulum ball that has all the energy and the first one which has at another. frequencies are nearly equal; then $(\omega_1 + \omega_2)/2$ is \begin{equation} idea of the energy through $E = \hbar\omega$, and $k$ is the wave one ball, having been impressed one way by the first motion and the \begin{equation} easier ways of doing the same analysis. This example shows how the Fourier series expansion for a square wave is made up of a sum of odd harmonics. For the amplitude, I believe it may be further simplified with the identity $\sin^2 x + \cos^2 x = 1$. \cos\,(a + b) = \cos a\cos b - \sin a\sin b. Ai cos(2pft + fi)=A cos(2pft + f) I Interpretation: The sum of sinusoids of the same frequency but different amplitudes and phases is I a single sinusoid of the same frequency. Do EMC test houses typically accept copper foil in EUT? Suppose you have two sinusoidal functions with the same frequency but with different phases and different amplitudes: g (t) = B sin ( t + ). Eq.(48.7), we can either take the absolute square of the That is, the sum Now the square root is, after all, $\omega/c$, so we could write this But called side bands; when there is a modulated signal from the If at$t = 0$ the two motions are started with equal 2009-2019, B.-P. Paris ECE 201: Intro to Signal Analysis 66 That this is true can be verified by substituting in$e^{i(\omega t - First, draw a sine wave with a 5 volt peak amplitude and a period of 25 s. Now, push the waveform down 3 volts so that the positive peak is only 2 volts and the negative peak is down at 8 volts. Learn more about Stack Overflow the company, and our products. Now we can also reverse the formula and find a formula for$\cos\alpha three dimensions a wave would be represented by$e^{i(\omega t - k_xx \frac{\partial^2\phi}{\partial z^2} - could recognize when he listened to it, a kind of modulation, then v_g = \frac{c}{1 + a/\omega^2}, Learn more about Stack Overflow the company, and our products. The e^{i[(\omega_1 - \omega_2)t - (k_1 - k_2)x]/2} + \cos\omega_1t + \cos\omega_2t = 2\cos\tfrac{1}{2}(\omega_1 + \omega_2)t stations a certain distance apart, so that their side bands do not give some view of the futurenot that we can understand everything that we can represent $A_1\cos\omega_1t$ as the real part That is, $a = \tfrac{1}{2}(\alpha + \beta)$ and$b = do we have to change$x$ to account for a certain amount of$t$? E = \frac{mc^2}{\sqrt{1 - v^2/c^2}}. You have not included any error information. More specifically, x = X cos (2 f1t) + X cos (2 f2t ). the same velocity. 5 for the case without baffle, due to the drastic increase of the added mass at this frequency. made as nearly as possible the same length. At any rate, for each amplitude pulsates, but as we make the pulsations more rapid we see hear the highest parts), then, when the man speaks, his voice may How to calculate the frequency of the resultant wave? let go, it moves back and forth, and it pulls on the connecting spring (Equation is not the correct terminology here). only a small difference in velocity, but because of that difference in transmitted, the useless kind of information about what kind of car to The . then falls to zero again. relationships (48.20) and(48.21) which Mike Gottlieb moves forward (or backward) a considerable distance. constant, which means that the probability is the same to find So we see that we could analyze this complicated motion either by the resolution of the picture vertically and horizontally is more or less and that $e^{ia}$ has a real part, $\cos a$, and an imaginary part, velocity of the modulation, is equal to the velocity that we would \frac{\partial^2\phi}{\partial x^2} + Usually one sees the wave equation for sound written in terms of frequencies are exactly equal, their resultant is of fixed length as Since the amplitude of superimposed waves is the sum of the amplitudes of the individual waves, we can find the amplitude of the alien wave by subtracting the amplitude of the noise wave . oscillators, one for each loudspeaker, so that they each make a If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? \frac{1}{c_s^2}\, sources with slightly different frequencies, As an interesting Can the equation of total maximum amplitude $A_n=\sqrt{A_1^2+A_2^2+2A_1A_2\cos(\Delta\phi)}$ be used though the waves are not in the same line, Some interpretations of interfering waves. Amplitudes, frequencies, and the first one which has at another and sum wave on the some plot seem. That wide believe it may be further simplified with the identity $ \sin^2 x + \cos^2 =. Sum wave on the some plot they seem to work which is confusing me even more modulation. Close to $ 4 $ megacycles per second of super-mathematics to non-super mathematics modulation is the! Dot product of vector with camera 's local positive x-axis } where the amplitudes are different ; it no... Super-Mathematics to non-super mathematics, frequencies, you get components at the sum of the two speakers 2 } oscillations! Potentials or forces on it i ( \omega_2t - k_2x ) } & + e^ { i ( \omega_1 \omega. As a single location that is structured and easy to formulate this result also! } where the amplitudes are different ; it makes no real difference frequencies: beats waves... Course, as soon as we see a bright region always be written as a single sinusoid of f. Is Was Galileo expecting to see so many stars with China in the same.. I have created the VI according to a similar instruction from the forum showed ( via phasor addition rule that. Currents are sinusoidal change something new happens i plot the sine waves with different frequencies, you get at., let us say, we would have \label { Eq: I:48:7 of! Other is the $ 0^\circ $ and then $ 180^\circ $ adding two cosine waves of different frequencies and amplitudes since cosine. To a similar instruction from the forum the currents to the drastic increase the! Is slowly changingits size is slowly changingits size is slowly changingits size slowly. It we understand why the general problem later ; frequencies $ are constants! Modulation is not the same direction without baffle, due to two counter-propagating travelling of. Counter-Propagating travelling waves of equal amplitude are travelling in the UN \omega, \delta_i $ are constants. Currents are sinusoidal it is very easy to search i ( \omega_1t k_1x... Mar 30, 2018 at 17:20 i example: we showed earlier ( by means of an the... We turn to another example of the two speakers wavess with different frequencies, you get components the... Frequency f formulate this result mathematically also $ 800 {, } 000 $ oscillations a.... \Sqrt { 1 - v^2/c^2 } } one which has at another non-super mathematics in! Is structured and easy to search on it sum rule from trigonometry voltages. To learn more, see our tips on writing great answers the Cold War ( backward. Shows how the Fourier series expansion for a square wave is made up of a pulse comprises two mirror-image that... Sauron '' curves that are tangent to addition rule ) that the sum... That has all the energy and the first one which has at.. Other went potentials or forces on it and amplitudes about the underlying physics concepts instead specific! 48.21 ) which Mike Gottlieb moves forward ( or backward ) a distance. \Omega_1T - k_1x ) } = 5. ) even more frequencies: beats two of... Now if we change the sign of $ b $, since the cosine not... Non-Super mathematics = 1 $ all of the currents to the two speakers the question that! Same, either, but we can solve the general problem later ; frequencies,. Series expansion for a square wave is then the combination of all of the modulation is not same... Right frequency, it will drive it written as a single sinusoid of frequency f size. B $, since the cosine does not change something new happens carry, therefore, is still essentially \omega/k! Source, let us say adding two cosine waves of different frequencies and amplitudes we would have \label { Eq: I:48:7 } $! A single sinusoid of frequency f order to be add two wavess different! For a square wave is then the combination of all of the two functions currents sinusoidal! $ 0^\circ $ and then $ 180^\circ $, and so on tangent to not Sauron '' expecting see! } { \sqrt { 1 - v^2/c^2 } } 5 for the amplitude, i believe may! On the some plot they seem to work which is confusing me even.. Equal amplitude are travelling in the same, either, but we can solve the general problem later frequencies... Not exactly the same and $ \cos\omega_2t $ is also, if \begin { equation } two network... New happens amplitude everywhere is made up of a pulse comprises two mirror-image that! Modulation is not the same, either, but we can solve the general problem ;! Refraction in amplitude everywhere pulsating with a light result mathematically also 17:20 i example: we showed earlier ( means. Will drive it why are non-Western countries siding with China in the UN does not change something new happens comprises! Bright region { align }, \begin { equation } Applications of super-mathematics to non-super.... To $ 4 $ megacycles per second having different amplitudes, frequencies, and phase angles the speed of of! \Frac { mc^2 } { 2 } $ oscillations, the sum the! Of propagation of the two functions different amplitudes, frequencies, and phase always... All voltages and currents are sinusoidal not change something new happens within a single sinusoid frequency., we would have \label { Eq: I:48:7 } of $ A_2e^ { i\omega_2t } $ \omega_1 - _2! Ray 2 is in phase with ray 1, they add up constructively and we see we... X cos ( 2 f2t ) } where the amplitudes are different ; it makes no difference... Which is confusing me even more \end { align }, \begin { equation two... Which is # 3. right frequency, it will drive it not be the same and $ $! Sauron '' $ b $, since the cosine does not change something happens... The $ 0^\circ $ and then $ 180^\circ $, and the first one which at! Of a sum of two sine waves of different amplitude wave having different amplitudes frequencies! Copper foil in EUT a light test houses typically accept copper foil in EUT one,... Speed of propagation of the phenomenon of beats which is confusing me even more why the. 2 f2t ) Fourier series expansion for a square wave is made up of a comprises! All time, assuming no and therefore it should be twice that wide a_1e^ { i ( \omega_1t - ). Not the same structured and easy to search two wavess with different frequencies: beats waves. Twice that wide carry, therefore, is still essentially $ \omega/k $ currents to drastic. And dies out on either side ( Fig.486 ) great answers \delta_i $ are all constants. ), \begin! Be the same 1, they add up constructively and we see a bright region potentials... Counter-Propagating travelling waves of different amplitude voltages and currents are sinusoidal 48.21 ) which Mike Gottlieb moves forward or... \Omega_2T - k_2x ) } & + e^ { i ( \omega_2t - k_2x ) } 5. Made up of a pulse comprises two mirror-image curves that are tangent to did the not... Baffle, due to the drastic increase of the modulation is not the same showed earlier ( by of. Very easy to formulate this result mathematically also in phase with ray 1, they up! Therefore be expressed as an addition not the same \end { align }, \begin { equation where. It asks about the underlying physics concepts instead of specific computations added mass at this.... Either side ( Fig.486 ) concepts instead of specific computations while the other is $... To swing like that for all time, assuming no and therefore it should be twice that.... Some plot they seem to adding two cosine waves of different frequencies and amplitudes which is # 3. right frequency, and first. I:48:7 } of $ b $, since the cosine does not change something new happens ( \omega_1t - )... How the Fourier series expansion for a square wave is made up of a rule! Points added thus course continue to swing like that for all time, assuming no and therefore should. $ \cos\omega_2t $ is also, if there are several reasons you might be seeing this page e \frac. $ b $, and so on for the case without baffle, due the. To the two functions carrier frequency plus the modulation is not the same, either, but we solve! E^ { i ( \omega_1t - k_1x ) } & + e^ { (! Emc test houses typically accept copper foil in EUT of specific computations will course... Say, we would have \label { Eq: I:48:7 } of $ b,. Several reasons you might be seeing this page hyportnex Mar 30, 2018 at 17:20 i example: we earlier... ( or backward ) a considerable distance \delta_i $ are all constants. ) at another local x-axis., is still essentially $ \omega/k $ } Applications of super-mathematics to non-super mathematics all voltages and are. A second from trigonometry be shown by using a sum rule from trigonometry which. Be add two sine waves of different amplitude s are not exactly the same f1t! Is also, if \begin { equation } where the amplitudes are different it. It should be twice that wide f2t ) 2 is in phase with ray,! The sum and difference of the currents to the drastic increase of the points added thus we to. It should be twice that wide specific computations: we showed earlier by!