Video Transcript
The following five functions can be
used to model five light waves. (i) 𝑦 equals sin 𝑥, (ii) 𝑦
equals two sin 𝑥, (iii) 𝑦 equals sin two 𝑥, (iv) 𝑦 equals three sin 𝑥, and (v)
𝑦 equals 0.75 sin 𝑥. Which of the five waves is not
coherent with the other four?
We’ve drawn out all five of the
functions’ light waves in order to see what they look like. We can tell just by looking at them
that almost all of these waves have a different amplitude and that the wave given by
function (iii) has a different frequency. But how do we know how to draw out
these different light waves? Well, we can look at the equation
for a generic wave, 𝑦 equals 𝐴 sin 𝑘𝑥, where 𝐴 is the amplitude of the wave and
𝑘 represents the frequency. High values of 𝐴, like we see in
the wave given by function (iv), give high values of amplitude. And smaller values of 𝐴, like we
see in function (v), produce waves with smaller amplitudes. But we’re not looking for
differences in amplitude. We’re looking for differences in
whether these waves are coherent or not.
Recall that in order for waves to
be coherent, all they need is to have the same frequency and a constant phase
difference. Amplitude does not matter for
determining whether waves are coherent or not. So, when we look at these five
functions, the value of 𝐴, or the number in front of the sine function, does not
matter for determining whether these waves would be coherent or not. What does matter though is the
variable 𝑘 inside of the sine function. Since 𝑘 represents the frequency,
the functions which have the same value of 𝑘 must have the same frequency. And in the case where there is no
number in front of the 𝑥 inside of the sine function, then we know that 𝑘 must be
equal to one since one times 𝑥 is just 𝑥.
And we see that for most of these
functions, the value of 𝑘 is just one except for (iii) which has two 𝑥 inside of
the sine function. So, its value of 𝑘 is equal to
two, meaning that the wave given by this function will have a different frequency
than the others, which will ensure that it will not be coherent with the other
four. So, we note that the frequency of
the wave given by function (iii) is different. But what about the phase
difference? Well, if we want to represent a
phase difference using one of these functions, we would do it by adding some number
inside of the sine function after the 𝑘𝑥. But we see that none of these
functions have an addition inside of the sine function. So, there is no phase difference,
meaning it’s safe just to rely on the variable 𝑘 representing the frequency.
So since function (iii) has a
different value of 𝑘, meaning a different frequency from the other functions, it
cannot be coherent with the other four waves. The correct answer is function
(iii).
