Introductory Physics Volume Two

7.3 Describing Waves 141
For a harmonic oscillator the phase increases at a constant rate:
dφ
dt = ω −→ φ = ωt + φ 0
and so the displacement is
x = A cos(ωt + φ 0 )
⊲ Problem 7.2
An oscillator has an amplitude of 3.2. At this instant the displacement
of the oscillator is 1.4. What are the two possible phases of the oscillator
at this instant?
⊲ Problem 7.3
You have a mass connected to a spring.
0 x
You start the mass oscillating in the following ways. What is the initial
phase φ 0 in each case.
(a) Stretch the spring to the right, at t = 0 release the mass.
(b) Compress the spring to the left, at t = 0 release the mass.
(c) At t = 0 strike the mass so that it begins moving to the left.
(d) At t = 0 strike the mass so that it begins moving to the right.
(e) At t = 0 the mass is at the position x 0 = 2.0m and has a velocity
of v 0 = 3.0 m rad
s
(assume that ω = 5.0
s ).
⊲ Problem 7.4
It takes a time of T = 0.025s in order for an oscillator to complete one
cycle. What is the angular frequency (ω) of the oscillator?
§ 7.3 Describing Waves
Consider dropping a rock into a pool of still water. Ripples spread
out from the point at which the rock enters the water. If you examine
the motion of the water at one fixed point, the surface of the water
moves up and down as successive waves move past. The height of
the water at our fixed point is an oscillation. This is true at other
points a well, at each location the height of the water oscillates. Since
an oscillation is so simple, the only thing that can differ between one
point and another is the amplitude of the oscillation and the initial
phase φ 0 . Let us write the height of the water, at position ⃗r and time
t, as y(⃗r, t). Since the oscillation at each point is a harmonic oscillation
we can write:
y(r, t) = A(⃗r) cos [ωt + φ 0 (⃗r)]