PHY254 Lecture 6
Kinematics of Oscillations-continued
▯ Readings: Supplementary notes handout.
▯ Kinematics of oscillatory motion: phasor representation, constructive/destructive
interference, commensurate frequencies, beats.
z = re i▯ form for describing oscillations
▯ Last time we worked on a new formalism for describing oscillations.
The 1D oscillatory motion would be represented by circular motion
in 2D where the new axis was the imaginary axis. We therefore use
complex numbers to describe the circular motion and remember that
the REAL part of the complex number actually describes the physical
▯ For a general sinusoidal oscillation described by x = Acos(!t+▯), the
complex form of the motion is:
z = x + iy = A[cos(!t + ▯) + isin(!t + ▯)] = Aei(!t+▯:
We recover x by taking the \real" part of this:
x = Re(z) = Re[Ae i(!t+] = Acos(!t + ▯):
▯ We can use the polar form to easily calculate derivatives. E.g. x’s
! ▯ ▯
v = Re dz = Re i!Ae i(!t+▯) = ▯!Asin(!t + ▯)
dv! ▯ ▯
a = Re = Re i ! Ae i(!t+▯)= ▯! Acos(!t + ▯)
For complicated di▯erential equations, the exponential notation can
make solving the equation MUCH simpler. We will see these when we
get to the dynamics of oscillations.
1 Phasor Representation
▯ Notice that the exponential form could also be written: z = (Ae )(e i▯ i!).
In this representation, the time dependency is isolated in the exponen-
tial and the amplitude of the motion at t = 0 is given by Re(Ae ) = i▯
Thus we can consider z to be a vector with magnitude A and angle ▯
at t = 0, rotating at a rate !.
This is known as the phasor representation of sinusoidal motion.
Superposition of Signals
▯ We often deal with superposed (added up) sinusoidal signals. For ex-
ample think of music. Playing a chord on a piano means releasing 3
sound waves (oscillations) of di▯erent frequencies into the air at the
same time. If the resultant oscillation is a linear superposition of the
3 waves what properties does it have? This process of adding up sinu-
soidal signals is also going to be crucial when we use Fourier’s theorem
to investigate general oscillations.
▯ Suppose we have two distinct signals on the x axis:
x 1 A co1(! t + 1 ) an1 x = A c2s(! t 2 ▯ ) 2 2
The corresponding complex representation signals are
z = A e i(!1t+▯1)and z = A e i(2 t+▯2)
1 1 2 2
Consider the following cases of the superposition of these signals.
2 1. General case: If ! 61 ! an2 A 6= A1then 2here is not much algebra
to do and the signal might be complicated and not periodic.
Constructive and destructive interference: If A a1d A are 2eal and
positive, then the magnitude of the resultant vector z 1z i2 bounded
on both sides by:
jA1▯ A j2▯ jz +1z j 2 jA + 1 j 2 (1)
▯ Lower limit: occurs when the vectors have a phase di▯erence
▯1▯ ▯ =2▯. In this case they are antiparallel and ‘maximum
destructive interference’ occurs
▯ Upper limit: occurs when the vectors have a phase di▯erence of
0. In this case they are parallel and ‘maximum constructive
▯ In between: phase di▯erence somewhere between 0 and ▯ (i.e. not
parallel or antiparallel).
▯ Note that jx 1 x j 2 jz + 1 j be2ause we have to project the
resultant onto the real axis.
2| + 2
Real |x +x |