Transverse waves on a string - People.fas.harvard.edu

ψ t
Figure 8
ψ i
10 CHAPTER 4. TRANSVERSE WAVES ON A STRING
the case of reflection above, although the ψi function isn’t applicable for positive x, we can
mathematically imagine ψi still proceeding to the right. The situation is shown in Fig. 7. As
above, let’s say that v2 = 3v1, which means that the 2v2/(v2 + v1) factor equals 3/2. Note
that in any case, this factor lies between 0 and 2. We’ll talk about the various possibilities
below.
Figure 7
ψ t
v 2
not really
there
(v2 = 3v1)
ψ i
v 1
x = 0
ψ i
actual
<strong>waves</strong>
(ψ r not shown)
In the first picture in Fig. 7, the incident wave is moving in from the left, and the
transmitted wave is also moving in from the left. The transmitted wave doesn’t actually
exist to the left of x = 0, of course, but it’s convenient to imagine it coming in as shown.
With v2 = 3v1, the transmitted wave is 3/2 as tall and 3 times as wide as the incident wave.
In the second picture in Fig. 7, the incident wave has passed the origin and continues
moving to the right, where it doesn’t actually exist. But the transmitted wave is now located
on the right side of the origin and moves to the right. This is the real piece of the wave.
For simplicity, we haven’t shown the reflected ψr wave in these pictures, but it’s technically
also there.
In between the two times shown in Fig. 7, things are easier to deal with than in the
reflected case, because we don’t need to worry about taking the sum of two <strong>waves</strong>. The
transmitted wave consists only of ψt. We don’t have to add on ψi as we did in the reflected
case. In short, ψL equals ψi + ψr, whereas ψR simply equals ψt. Equivalently, ψi and ψr
have physical meaning only to the left of x = 0, whereas ψt has physical meaning only to
the right of x = 0.
Fig. 8 shows some successive snapshots that result from the same square wave we con-
sidered in Fig. 6. The bold-line wave indicates the actual wave that exists to the right of
x = 0. We haven’t drawn the reflected wave to the left of x = 0. We’ve squashed the x
axis relative to Fig. 6, to make a larger region viewable. These snapshots are a bit boring
compared with those in Fig. 6, because there is no need to add any <strong>waves</strong>. As far as ψt
is concerned on the right side of x = 0, what you see is what you get. The entire wave
(on both sides of x = 0) is obtained by juxtaposing the bold <strong>waves</strong> in Figs. 6 and 8, after
expanding Fig. 8 in the horizontal direction to make the unit sizes the same (so that the ψi
<strong>waves</strong> have the same width).
ψ t