Wednesday:
We examined superposition of waves. Using the trigonometric identity for added two sines,
sin ( α ) + sin (β ) = 2 sin [ ( α + β )/2 ] cos [ ( α - β )/2 ]
we added two traveling waves with the same amplitude, finding
y ( x, t) = 2 A sin [ (k1 + k2) x/2 ±
(ω1 ± ω2) t/2 + (φ1 + φ2) /2]
cos [ (k1 - k2) x/2 ±
(ω1 m ω2) t/2 + (φ1 - φ2) /2]
Note the first ± in the sine term indicates if the first wave (k1, ω1)
is moving in the negative (the +) or positive (the -) direction and the second ± in the sine term indicates
if the two waves move in the same direction (the +) or opposite directions (the -). Similarly, the ± in the cosine term indicates if the
first wave (k1, ω1) is moving in the negative (the +) or positive
(the -) direction, while the m in the
cosine term indicates if the two waves move in the same direction (the -) or opposite
directions (the +).
We considered a special case of two waves with the same amplitude, same wavelength, same frequency, same direction
and different phase.
y ( x, t) = 2 A cos [ (φ1 - φ2) /2]
sin [ k x ± ω t + (φ1 + φ2) /2]
The term 2Acos[(φ1-φ2)/2] serves as a "new amplitude."
Particular cases include
- Constructive interference: when φ1-φ2 = 2 n π with n = 0, ±1, ±2, ... (even multiples
of pi)
- Destructive interference: when φ1-φ2 = (2 n +
1) π with n = 0, ±1, ±2, ... (odd
multiples of pi)
The first condition for destructive interference (φ1-φ2 =
π (radians) = 180 °) coincides with our notion
of being 180° "out of phase" or "out of sync."
We considered a second special case in which the waves had the same amplitude, wavelength, frequency but
were in opposite directions.
y ( x, t) = 2 A sin [ k x + (φ1 + φ2) /2]
cos [ ω
t + (φ1 - φ2) /2]
The result is a standing wave in which the spatial and temporal behaviors "decouple."
The different colors above represent different times. The wave does not travel
but oscillates "in place." There are special points called nodes that do not move
at all. As the picture above shows the distance between nodes is
half of a wavelength.
Next, we discussed the behavior of a wave pulse at a fixedboundary – that at a general
boundary part of the wave is reflected and part of the wave transmitted (and in turn part
of the energy is reflected and part of the energy transmitted), but at a fixed boundary the wave is
entirely reflected. Conservation of energy implies that the amplitude of the reflected wave is equal
to the original pulse.
