Video Transcript
The following two functions can be
used to model two light waves. (i) 𝑦 equals two sin three 𝑥 plus
𝜋 over two. (ii) 𝑦 equals five sin 𝑘𝑥 plus
𝜋 over two. What value of 𝑘 would make these
two waves coherent?
This question is essentially asking
us to look at the different parts of each function and determine what the value of
𝑘 should be in the second function. But there are a lot of different
parts besides the one that just includes 𝑘. To help us understand what they
are, we can compare them to the wave equation for a generic sine wave. 𝑦 equals 𝐴 sin 𝑘𝑥 plus 𝜙,
where 𝐴 is the amplitude of the wave. 𝑘 is related to the frequency,
with higher values of 𝑘 indicating a higher frequency. And the Greek letter 𝜙 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.
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 𝑘 that would make these two waves
coherent. Waves are coherent when they share
the same frequency and a constant phase difference. Any other properties of the waves
do not matter when determining if they are coherent. This means then, looking back at
the wave equation over here, we’ll only care about the variable 𝑘, which is related
to the frequency, and 𝜙, which is the phase. The variable 𝐴, amplitude, will
not matter at all for determining whether these waves are coherent or not.
So when we look back at the two
functions, we can ignore the difference between two in function (i) and five in
function (ii). They won’t matter for whether or
not they are coherent. Instead, let’s look at the values
of 𝑘 representing frequency in these functions. This is three for function (i) and
𝑘 for function (ii). In order for these two functions to
be coherent, they must have the same frequency, which means their values of 𝑘 must
be the same, which in this case means that 𝑘 must be equal to three. This satisfies the condition that
the waves produced by these functions have the same frequency.
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. This is not a problem though, since
the values of 𝜙, the phase, for both functions (i) and (ii) are the same, 𝜋 over
two. This means they have a constant
phase difference because they have the same phase. And actually, even if the values of
𝜙 were different for both of these functions, they would still be coherent because
what matters is if it’s a constant phase difference.
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 𝑘 which would
make the two waves given by these two functions coherent with each other is
three.
