HARMONIC MOTION AND WAVE MOTION
Objectives: The student will
1. show the relationships between vibratory motion and circular motion
2. show addition of waves for interference, standing waves, resonance, and beats.
References: Halliday and Resnick Ch 13,16, and 17. Serway Ch 13.5-13.8, LGB
I. Relationship between harmonic motion and uniform circular motion
A. Describing harmonic motion
1. We saw in Hooke’s Law and Harmonic Oscillators that for harmonic motion, we can use x = A cos(t)
a. A is the amplitude,  is the angular frequency in radians/sec
b. x is the distance from equilibrium, and t is the time, in seconds
c. Note that the units for t is radians, where 2 radians = 1 cycle
d. Then T is (rad/sec)sec = 2 radians = 1 cycle
e. Just for the record, 360o = 2 radians = 1 cycle
2. Introduce a phase factor: suppose x is not A at t = 0
Q
Q
a. Then, we can define , where xo = A cos
b. This ‘‘ is called the phase factor
c. Therefore, in general, x = A cos(t +)
3. A couple of useful trigonometric identities
a. cos(a  b) = cos a cos b -/+ sin a sin b
x
P
P
b. sin(a  b) = sin a cos b  cos a sin b

o
4. Suppose the phase is 90 = /2
a. Then x = A cos(t + /2 )
b. Or x = A[cost cos/2 - sint sin/2) because of A.3.a. above
c. Therefore x = -A sint
y
d. And if the phase is -/2, x = A sint
B. Circular motion: adding motion at right angles
1. Suppose a particle is moving on a circle of radius A
2. Suppose  is the constant angular speed
3. Let P be the perpendicular projection of Q on the horizontal diameter, the x axis, and P is the x value of the point Q
4. Call Q the reference point and the circle the reference circle
5. Then as Q traverses the circle, P moves back and forth across the horizontal diameter
6. Let the angle of the radius at t = 0 be 
7. Then at another time, the angle is (t +)
8. Therefore x = A cos(t + )
9. And therefore vx = -A sin(t +)
10. And ax = -A2 cos(t +)
C. Now for two dimensions
1. We have x = A cos(t +)
2. But, because Q is on a circle, y = A sin(t +)
3. So that vy = A cos(t + )
4. And ay = -A2sin(t +)
5. But, notice that x2 + y2 = A2
6. And vx2 + vy2 = v2 = 2A, because cos2x + sin2x = 1
7. And likewise, the magnitude of a = 2A
D. Adding the displacements vectorially
1. Suppose we have two harmonic oscillators perpendicular to each other
a. x = Axcos(t + x)
b. y = Aycos(t + y)
c. That is, the motions may have different amplitudes and different phases, but they have the same frequency
2. Suppose x = y, that is, the same phase
a. To simplify things, let this phase be zero
b. Then x = Axcost and y = Aycost
c. So that y = (Ay/Ax)x, a straight line through the origin of slope Ay/Ax
3. Now, another case, suppose y = x - /2 = - /2
a. Then y = Aycos (t +  - /2)
b. Or y = Aysin(t + )
c. While x = Axcos(t +)
d. But it is just circular motion if Ax = Ay
e. And elliptical if they are not equal
II. Adding waves along the same direction - scalar addition
A. Basic equations for harmonic waves
1. For a wave traveling to the right, y = A sin(kx - t)
t + 
HARMONIC MOTION AND WAVE MOTION Page 2
a. We are located at position x, looking at the displacement y that the particles located at x are experiencing
b. 'k' is called the wavenumber, related to the wavelength of the wave, defined below
c. So if we are located at x, as time passes, the particle moved up and down with angular frequency 
d. Conversely, if we stop time (say by use of a camera), then we can move along the wave, along the 'x' direction, and see the positions of the
particles vary according to sin kx
e. After we have traveled a distance , such that k = 2, the particles are at the same displacement as when we started with x = 0
f. Consequently, k = 2/ , so that k is essentially a 'spatial' frequency,  is called the wavelength
g. Therefore, if (kx - t) is constant, the displacement y is constant
h. That is, as time goes on, we must travel along in space with x such that (kx - t) is constant, because y is to remain constant to stay at the
same spot 'on the wave'
i. Therefore, as t increases, x must increase for this argument to be constant, so the argument for a wave traveling to the right
j. (kx - t) is the argument for a wave traveling to the left
2. Or for a wave traveling to the left, y = A sin(kx + t)
3. Consider adding two waves of the same amplitudes and frequencies, but of different phases
a. Let y1 = A sin(kx - t), phase = 0
b. And let y2 = A sin(kx - t - ), where  is the phase shift of y2 relative to y1
c. But we can write y2 = A sin[k(x - /k) - t]
d. Or we can write y2 = A sin[kx - (t + /)]
e. So that at fixed time t, y2 is shifted spatially by (/k) relative to y1
f. And at fixed space, y2 is shifted in time by (/)
B. Now let's add two waves moving in the same direction, so to the right
1. Then y = y1 + y2 = A[sin(kx - t) + sin(kx - t - )]
2. But another identity: sin B + sin C = 2[sin(1/2)(B + C)][cos(1/2)(C - B)]
3. Therefore y = 2A[sin(1/2)(kx - t -  + kx - t)][cos(1/2)(kx - t -  - kx + t)]
4. So that y = [2A cos(/2)][sin(kx - t - /2)]
C. Interpretation
1. The amplitude of the resulting wave is 2A cos(/2)
a. If  = 0, then they are in phase, the amplitude doubles. This is called CONSTRUCTIVE INTERFERENCE.
b. If  = n(180o), they are out of phase, and the amplitude of the resulting wave is zero, and this is destructive interference
c. Notice the kx and t dependence. The space and time dependence of the resulting wave is the same as that of the component waves
d. /k = the path difference between the two waves
e. Constructive interference is when  = 0, 2, 4, ..., 2n
f. Destructive interference is when  = ", 3, 5, ..., (2n - 1)
D. Standing waves: Adding waves moving in opposite directions
1. Let y1 = A sin(kx - t), y2 = A sin(kx + t)
2. Then y = y1 + y2 = 2A sinkx cost
a. The maximum displacement is 2A sin kx
b. It is space dependent
c. It is zero when kx = 0, , 2, ... n. These are called NODES
d. Is maximum when kx = /2, 3/2, ... (2n -1)/2, the ANTINODES
E. Beats, adding two waves of slightly different frequencies
1. Consider one point in space, so drop the x dependence
2. y1 = A cos1t, y2 = A cos 2t
3. y = y1 + y2 = A[cos1t + cos2t]
a. But cos a + cos b = 2 cos(1/2)(a - b)cos(1/2)(a + b)
b. So y = 2A[cos(1/2)(1 - 2)t][cos (1/2)(1 + 2)t]
c. So y = 2Acos(ampt)cos(avgt)
d. The amplitude varies as amp = (1/2)(1 - 2)
e. This cosampt is zero twice per cycle, hence the beat frequency is 1 - 2
f. The resulting wave frequency is avg = (1/2)(1 + 2)
g. If the two frequencies are equal, a steady tone results at twice the amplitude of
either component
HARMONIC MOTION AND WAVE MOTION Page 3
III. Problems
1. How could you prove experimentally that energy is associated with a wave?
2. When two waves interfere, does one alter the progress of the other?
3. Suppose a particle is moving on a circle of radius 15 cm at a speed of 35 cm/sec. (a) What is the angular speed of this particle? (b) What are the
equations of motion for the x and y directions? (c) What is the speed of this particle along each axis? (d) What are the equations for the
accelerations experienced by the particle? (e) What are the maximum speeds in each direction? (f) What are the maximum accelerations
experienced by the particle? (g) How do these compare for the acceleration which we called 'centripetal' in mechanics? (h) What is the phase of
the y motion relative to the x motion?
4. Suppose the x-direction harmonic oscillation has an amplitude of 10 cm, and the y-direction harmonic oscillation has an amplitude of 15 cm.
Suppose they have the same frequency of 20 rad/sec. (a) Write the equations for these two oscillators, tht is, their displacement equations. (b)
Create a table and sketch a plot of the sum of these displacements. (This is what we do when we use an oscilloscope to investigate periodic
motion such as sound. The x and y directions are the 'horizontal' and 'vertical' signals into the scope.
5. Suppose the x-direction motion had double the frequency of the y-direction motion. Use the numbers from problem #4 and plot this motion.
6. (HR16-3) The equation of a transverse wave traveling in a rope is given by
y = 10 cm sin(0.0314 x - 6.28 t),
where x and y are expressed in centimeters, and t in seconds. (a) Find the amplitude, frequency, velocity, and wavelength of the wave. (b) Find
the maximum transverse speed of a particle in the rope.
7. Write the equation for a wave traveling in the negative direction along the x axis and having an amplitude of 0.01 m, a frequency of 550 vib/sec,
and a speed of 330 m/sec.
8. (HR16-6) A wave of frequency 500 cy/sec has a velocity of 350 m/sec. (a) How far apart are two points 60o out of phase? (b) What is the phase
difference between two displacements at a certain point at times 10 -3 sec apart?
9. Determine the amplitude of the resultant motion when two sinusoidal motions having the same frequency and traveling in the same direction are
combined, if their amplitudes are 3.0 and 4.0 cm, and they differ in phase by 90 o.
10. Two sound waves with the same frequency, 540 cy/sec, travel at a speed of 330 m/sec. What is the phase difference of the waves at a point that is
4.40 m from one source, and 4.00 m from the other source if the sources are in phase?
11. (HR17-39) A tuning fork of unknown frequency makes three beats/sec with a standard fork of frequency 384 vib/sec. The beat frequency
decreases when a small piece of wax is put on a prong of the first fork. What is the frequency of this fork?