Many students are apparently confused about how to
deal with interference phenomena.
Before discussing waves and interference, there are a few important
definitions, facts and relationships that must become totally familiar to
you. For this purpose, we use sound
waves as the example. All waves you
need to know about behave the same way, so that what is learned about
interference in sound waves can be carried over to apply to interference in
light waves.
Assume that we have an electronic oscillator that
generates a specific sinusoidal frequency f in cycles per second
(Hertz). When a speaker is connected to
this oscillator, the speaker cone is driven back and forth. As a result, a sound wave is emitted as a
pure tone at the given frequency f.
The sound wave propagates through air as a sinusoidally varying pressure
wave. The wave travels with velocity v
and has a wavelength λ that satisfies the relationship λ
= v/f. Since the frequency f
has the units of cycles/second, the inverse of the frequency must have the
units seconds/cycle. The quantity 1/f is called the period T,
i.e., the time interval for one cycle.
As the wave falls incident on a listening device, such as your eardrum,
the oscillating pressure is processed by your inner ear and interpreted by your
brain as sound at the frequency f.
Now
assume that two speakers are connected to the same oscillator. What this means is that when one speaker
cone is moving outward during its vibrational motion, the other one is
performing the exact same motion. The two speakers are said to be in
phase with each other. This implies
that the two emitted waves at the respective positions of the speakers are
exactly in phase with each other. The
intensity of the two sounds can also be adjusted so that they are the same or
nearly so, although is not a requirement.
But now let us look at the consequences of placing a
receiver that “hears” the sound from each speaker simultaneously at a specific
point in space. To understand the
effect, we need only know what happens when two waves are superimposed on each
other. Experiment shows that when two
waves meet at the same point in space, the two waves simply add algebraically
point-by-point. Sound waves are said to
obey the superposition principle. But
when waves travel different distances to merge at a given point in space, any
phase relationship they had at the speakers is modified. The new phase relationship can be determined
easily if we take into account the fact that the sound travels at a velocity v. If a receiver is located equidistant from
each speaker, then the sound from each speaker takes exactly the same time to
reach the receiver. Since each wave
oscillates at the same frequency and travels at the same speed, the phase at
the position of the receiver is maintained exactly the same as at the speakers.
Next, suppose our two identical speakers are
positioned such that speaker A is farther from the receiver than speaker B by
an amount Δs. This means
that a given pulse from A takes longer to reach the receiver than the same
pulse from B, and the delay time is Δt = Δs/v. In general this delay will cause the two
waves to be out of phase with respect to each other when they reach the
receiver. But if Δt is
exactly one period, then by the very definition of a period T, the two
waves will be back in step. This is
equivalent to saying that sin(z) = sin(z + 360o). A
shift by one period or by any whole number of periods leaves the phase
unchanged. Times can be restated in
terms of distances. Thus, a shift of
one period in times is equivalent to a difference in distance Δs = vT =
v/f = λ. Of course, a shift in
two periods is equivalent to a difference of Δs = 2λ,
etc.
In summary, if two sources are in phase with each
other, and one source is an integral number of wavelengths farther from the
listener than the other source, the waves will be in phase with each other at
the position of the listener, and the sound will be relatively loud
(constructive interference). Similarly,
if two sources are in phase with each other, and one source is half of a
wavelength farther from the listener than the other source (or 3/2 wavelength,
5/2 wavelength, etc.), the waves will be out of phase with each other at the
position of the listener, and the sound will be relatively soft (destructive
interference).
This summary is the basis of essentially all the
problems in the chapter 28!