1) Phase refers to the position of a wave within its oscillation cycle and is represented by the term inside the brackets of a sinusoidal wave equation. 2) The phase difference between two waves is the difference between their phases and can be calculated by subtracting the phases. 3) Waves are in phase or undergo constructive interference when their phase difference is an even multiple of pi, and they are out of phase or undergo destructive interference when their phase difference is an odd multiple of pi.

1. Jenny Lee, PHYS 101 LJ2
PHILIP (YOUR ULTIMATE
GUIDE ON WAVES)
…
BROOOO

2. • phase is NOT A FIXED QUANTITY!
• TEXTBOOK SAYS: “Phase is a variable that
depends on position and time.”
𝜑 𝑥, 𝑡 = 𝑘𝑥 ± 𝜔𝑡 + 𝜑
𝜑 𝑥, 𝑡 =
2𝜋
λ
𝑥 ± 2𝜋𝑓𝑡 + 𝜑
• PHILLIP SAYS: We know that the equation for a
sinusoidal wave is 𝐷 𝑥, 𝑡 = 𝐴𝑠𝑖𝑛(
2𝜋
λ
𝑥 − 2𝜋𝑓𝑡 + 𝜑).
The
2𝜋
λ
𝑥 component refers to position, and the
2𝜋𝑓𝑡 component refers to time. In the D(x,t)
equation, 𝜑 refers to the phase constant, which
you can think of as how much the sin/cos
equation has shifted horizontally from a graph of
sin(x) or cos(x), or the head start that a wave has.
Don’t you hate
complicated textbook
explanations? Well, with my
expertise as a surfer and
physicist I can help you.
Let’s talk about what phase
and phase difference is.

3. • In simple terms: phase is whatever is found inside the brackets of the sin
or cos equation.
• 𝑫 𝒙, 𝒕 = 𝑨𝒔𝒊𝒏 𝒑𝒉𝒂𝒔𝒆 𝒐𝒓 𝑨𝒄𝒐𝒔(𝒑𝒉𝒂𝒔𝒆)
• It is important to remember that D(x,t) forms a 3D plot, and so it is
dependent on two variables.
• If we observe 2 waves at the exact same position over time, we can say
that their phases are 𝜑1 = 𝑘𝑥0 + 𝜔𝑡1 + 𝜑 and 𝜑2 = 𝑘𝑥0 + 𝜔𝑡2 + 𝜑
respectively. The phases indicate at what point of the sinusoidal cycle
the function is currently at.
Pretty
sick,
huh?

4. • PHILIP SAYS: The phase difference, ∆𝜑, can be thought of as the value
between two different phases.
∆𝜑 = 𝜑2 − 𝜑1
• If we plug in the two phases we identified earlier, we can determine the phase
constant.
∆𝜑 = ( 𝑘𝑥0 + 𝜔𝑡2 + 𝜑) − ( 𝑘𝑥0 + 𝜔𝑡1 + 𝜑)
= 𝜔𝑡2 + 𝜑 − 𝜔𝑡1 +𝜑
This represents an instance where two sinusoidal waves with different
initial displacements are observed at one position over time, and we
would like to know what the difference between their phases are.

5. • After every race, Phillip graphs out the wave that he has
ridden on. Sometimes, these waves are in-phase:
(This is known as constructive interference)
Other instances, the waves are out of phase.
(This is known as destructive interference)
As a professional
surfer, I like to
know the
characteristics of
a wave I’ve
been racing on.
𝜑2 − 𝜑1 = 𝑛 λ
The phase difference of
such waves are an even
multiple of pi.
𝜑2 − 𝜑1 = (𝑛 + 0.5) λ
The phase difference of
such waves are an odd
multiple of pi.

6. (Phillip hopes that you will never confuse phase,
phase difference, and phase constant again.)