Complex-valued functions are commonly used to solve differential equations for one-dimensional motion of a harmonic oscillator with linear damping, a sinusoidal driving force, or both. However, the usual approach treats complex functions as an algebraic shortcut, neglecting geometrical representations of those functions and discarding imaginary parts. This article emphasizes the benefit of using diagrams in the complex plane for such systems, in order to build intuition about harmonic motion and promote spatial reasoning and the use of varied representations. Examples include the analysis of exact time sequences of various kinematic events in damped harmonic motion, sense-making about the phase difference between a driving force and the resulting motion, and understanding the discrepancy between the resonant frequency and the natural undamped frequency for forced, damped harmonic motion. The approach is suitable for supporting instruction in undergraduate upper-division classical mechanics.

1.
See, for example,
S.
Goldin-Meadow
and
S.
Beilock
, “
Action's influence on thought: The case of gesture
,”
Perspect. Psychol. Sci.
5
(
6
),
664
674
(
2010
);
[PubMed]
R. E.
Scherr
,
H. G.
Close
,
E. W.
Close
,
V. J.
Flood
,
S. B.
McKagan
,
A. D.
Robertson
,
L.
Seeley
,
M. C.
Wittmann
, and
S.
Vokos
, “
Negotiating energy dynamics through embodied action in a materially structured environment
,”
Phys. Rev. Spec. Top. Phys. Educ. Res.
9
,
020105-1
18
(
2013
);
R.
Lindgren
and
M.
Johnson-Glenberg
, “
Emboldened by embodiment: Six precepts for research on embodied learning and mixed reality
,”
Educ. Res.
42
(
8
),
445
452
(
2013
).
2.
P. B.
Kohl
,
D.
Rosengrant
, and
N. D.
Finkelstein
, “
Strongly and weakly directed approaches to teaching multiple representation use in physics
,”
Phys. Rev. Spec. Top. Phys. Educ. Res.
3
,
010108-1
10
(
2007
).
3.
A.
Van Heuvelen
and
X.
Zou
, “
Multiple representations of work-energy processes
,”
Am. J. Phys.
69
(
2
),
184
194
(
2001
).
4.
R. B.
Kozma
, “
The material features of multiple representations and their cognitive and social affordances for science understanding
,”
Learn. Instrum.
13
(
2
),
205
226
(
2003
).
5.
P. B.
Kohl
and
N. D.
Finkelstein
, “
Patterns of multiple representation use by experts and novices during physics problem solving
,”
Phys. Rev. Spec. Top. Phys. Educ. Res.
4
,
010111-1
13
(
2008
).
6.
B. S.
Ambrose
, “
Investigating student understanding in intermediate mechanics: Identifying the need for a tutorial approach to instruction
,”
Am. J. Phys.
72
(
4
),
453
459
(
2003
).
7.
R. E.
Pepper
,
S. V.
Chasteen
,
S. J.
Pollock
, and
K. K.
Perkins
, “
Observations on student difficulties with mathematics in upper-division electricity and magnetism
,”
Phys. Rev. Spec. Top. Phys. Educ. Res.
8
,
010111
(
2012
).
8.
C. A.
Manogue
and
E.
Gire
, “
Cognitive development at the middle-division level
,”
AIP Conf. Proc.
1179
,
19
(
2009
);
C. A.
Manogue
,
P. J.
Siemens
,
J.
Tate
,
K.
Browne
,
M. L.
Niess
, and
A. J.
Wolfer
, “
Paradigms in Physics: A new upper-division curriculum
,”
Am. J. Phys.
69
(
9
),
978
990
(
2001
).
9.
C.
Singh
, “
Interactive learning tutorials on quantum mechanics
,”
Am. J. Phys.
76
(
4
),
400
405
(
2008
).
10.
S. V.
Chasteen
,
S. J.
Pollock
,
R. E.
Pepper
, and
K. K.
Perkins
, “
Transforming the junior level: Outcomes from instruction and research in E&M
,”
Phys. Rev. Spec. Top. Phys. Educ. Res.
8
,
020107-1
18
(
2012
).
11.
J. B.
Marion
and
S. T.
Thornton
,
Classical Dynamics of Particles and Systems
, 3rd ed. (
Harcourt Brace Jovanovich
,
San Diego
,
1988
).
12.
J. R.
Taylor
,
Classical Mechanics
(
University Science Books
,
Herndon, VA
,
2005
).
13.
Taylor, Ref. 12, p. 184.
14.
D.
Halliday
,
R.
Resnick
, and
J.
Walker
,
Fundamentals of Physics
, 9th ed. (
Wiley
,
Hoboken, NJ
,
2011
);
R.
Knight
,
Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
, 3rd ed. (
Pearson
,
New York
,
2012
).
15.
A.
Sfard
, “
On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin
,”
Educ. Stud. Math.
22
(
1
),
1
36
(
1991
).
16.
D. L.
Schwartz
and
J. B.
Black
, “
Shuttling between depictive models and abstract rules: Induction and fallback
,”
Cogn. Sci.
20
,
457
497
(
1996
).
17.
E.
Tufte
,
Beautiful Evidence
(
Graphics Press
,
Cheshire, CT
,
2006
), p.
129
.
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.