Video Transcript
The following five functions can be used to model five light waves. (i) 𝑦 equals three sin two 𝑥 plus two 𝜋. (ii) 𝑦 equals two sin two 𝑥 minus two 𝜋. (iii) 𝑦 equals sin two 𝑥 plus four 𝜋. (iv) 𝑦 equals 0.4 sin 𝑥 over two plus 𝜋 over two. And (v) 𝑦 equals 1.8 sin two 𝑥 minus eight 𝜋 over two. Which of the five waves is not coherent with the other four?
To determine this, let’s recall what it takes for two or more waves to be coherent
with each other. They must have both the same frequency and a constant phase difference. We may be able to determine these by looking at the light waves modeled by these five
functions on a graphing program. But an easier way would be to look at these five functions and relate them to the
generic wave equation for a sine wave. 𝑦 is equal to 𝐴 sin 𝑘𝑥 plus 𝜙, where 𝐴 is the amplitude of the wave or how high
it goes. 𝑘 is related to frequency, with a higher 𝑘 meaning a wave with higher
frequency. And the Greek letter 𝜙 is the phase difference of the wave. We can use this generic form of the sine wave equation to determine which of these
five functions would model waves that have the same frequency and a constant phase
difference.
Any wave that does not have the same frequency or a constant phase difference will
not be coherent with the other four waves. And to determine the properties of these waves, we can look at the related variables
and the wave equation for each function: 𝐴, 𝑘, and 𝜙 for amplitude, frequency,
and phase difference. Notably, 𝐴, representing amplitude, does not affect whether waves are coherent at
all. So the values of 𝐴, which are different for each of the five functions, will not
play a role for this question, only 𝑘 related to the frequency and 𝜙 the phase
difference.
Now then, looking first at 𝑘, the values related to frequency in these functions, we
see that functions (i), (ii), (iii), and (v) all have the same value of 𝑘, which is
two. The only one that is different is in function (iv), which has 𝑥 over two instead of
two 𝑥, meaning its 𝑘-value is one-half. So it seems like the light wave coming from function (iv) would not have the same
frequency as the light waves coming from the other four functions, which means it
would not be coherent with the other four waves.
But we’re not done yet. We still have to look at the phase difference. The phase difference 𝜙 is the number that is added or subtracted inside of the
brackets for the sine function. We see that the phase difference for each of these five functions is completely
different, which would imply that there is a nonconstant phase difference between
each of these five functions, which means that none of the waves from these
functions would be coherent with each other, let alone the other four.
The wording of this question makes it seem like we’re still just looking for a single
wave. We know that the wave given by function (iv) not only has a nonconstant phase
difference, but also a different frequency. So maybe it is the most not coherent out of the five. Well, it doesn’t work this way. Waves are either coherent or not; it is not a scale. So it seems that we are stuck until we remember what phase actually is.
When we are looking at a wave, its phase can be expressed in different ways, usually
with degrees, radians, or portions of total wavelength. It looks like the phase differences being expressed in these functions are using
radians. So let’s stick with those. A typical sine wave starts at the midpoint of the wave going up, with a value of zero
radians. The peak of a wave is 𝜋 over two. Midpoint of a wave going down is just 𝜋 by itself. The lowest point or value of a wave is three 𝜋 over two. And the end of a wave is two 𝜋. Or since the end of a wave is the start of a new wave, we can also express it as just
zero. And if we continue the wave, it would be relative to the two 𝜋 or the zero. Both are correct. What really matters for the value of phase is whether we’re looking at the same part
of the wave. It doesn’t matter whether we say 𝜋 over two or five 𝜋 over two so long as we know
that we’re referring to the peak of a wave.
In the case of functions (i), (ii), (iii), and (v), we see that all of the phase
differences are divisible by two 𝜋, which means they are all occurring at the start
of a wave cycle, which means that they can all be expressed as a zero, with function
(v) in particular, negative eight 𝜋 over two, being equal to negative four 𝜋,
which since it’s divisible by two 𝜋 is also equal to zero. The phases of functions (i), (ii), (iii), and (v) all refer to the same part of a
light wave, the start or end of one, which means they all actually have the same
phase, which means they have a constant phase difference, which is unlike function
(iv), which has a phase difference of 𝜋 over two, indicating a peak of a wave, not
the start or end of one.
So we see that the waves given by functions (i), (ii), (iii), and (v) are all
coherent with each other since they all have the same value of 𝑘, two, and the same
phase difference, zero, very much unlike function (iv), which has a different
frequency and a nonconstant phase difference. So the function that gives a wave that is not coherent with the other four waves is
function (iv) 𝑦 equals 0.4 sin 𝑥 over two plus 𝜋 over two.
