WEBVTT
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The following two functions can be used to model two light waves.
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(i) π¦ equals two sin three π₯ plus π over two.
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(ii) π¦ equals five sin ππ₯ plus π over two.
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What value of π would make these two waves coherent?
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This question is essentially asking us to look at the different parts of each function and determine what the value of π should be in the second function.
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But there are a lot of different parts besides the one that just includes π.
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To help us understand what they are, we can compare them to the wave equation for a generic sine wave.
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π¦ equals π΄ sin ππ₯ plus π, where π΄ is the amplitude of the wave.
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π is related to the frequency, with higher values of π indicating a higher frequency.
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And the Greek letter π indicates the phase of a wave, which is typically expressed in degrees, like 90 degrees or 270 degrees, or expressed in radians, like π over two or π.
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Using this generic sine wave equation can help us determine the properties of the waves produced by these two functions so that we can determine the value of π that would make these two waves coherent.
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Waves are coherent when they share the same frequency and a constant phase difference.
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Any other properties of the waves do not matter when determining if they are coherent.
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This means then, looking back at the wave equation over here, weβll only care about the variable π, which is related to the frequency, and π, which is the phase.
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The variable π΄, amplitude, will not matter at all for determining whether these waves are coherent or not.
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So when we look back at the two functions, we can ignore the difference between two in function (i) and five in function (ii).
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They wonβt matter for whether or not they are coherent.
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Instead, letβs look at the values of π representing frequency in these functions.
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This is three for function (i) and π for function (ii).
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In order for these two functions to be coherent, they must have the same frequency, which means their values of π must be the same, which in this case means that π must be equal to three.
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This satisfies the condition that the waves produced by these functions have the same frequency.
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But even now that they do, we still have to make sure that they have a constant phase difference in order for them to be coherent.
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This is not a problem though, since the values of π, the phase, for both functions (i) and (ii) are the same, π over two.
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This means they have a constant phase difference because they have the same phase.
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And actually, even if the values of π were different for both of these functions, they would still be coherent because what matters is if itβs a constant phase difference.
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If there are only two waves and they are regular, meaning that they are given by functions like this, and generally follow a stable inconsistent wave pattern unlike the wave shown here, then the phase difference between them is constant and they are coherent with each other, provided of course they also have the same frequency, which means the value of π which would make the two waves given by these two functions coherent with each other is three.