Understanding Standing Wave Sign Conventions

AI Thread Summary
The discussion centers on the confusion surrounding the sign conventions for amplitudes when combining incoming and reflected waves to form standing waves. It highlights that the reflected wave's amplitude must always have the opposite sign to that of the incoming wave, which is a consistent rule across examples. The standing wave is derived through algebraic manipulation, utilizing trigonometric identities to combine the two waves effectively. The examples provided illustrate how to apply these principles, showing that the resulting amplitude can be calculated using specific sine and cosine identities. Understanding these conventions is crucial for accurately forming standing wave equations.
AmandaWoohoo
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Hi.
Okay, this has been driving me crazy. When combining two given waves into a standing wave equation, how do you know which sign to put in front of the amplitude? All the examples I've been finding seem to contradict each other. Here are three examples from my textbook:

1. Incoming Wave:
y=4sin[2pi(t)-6pi(x)]

Reflected Wave:
y=-4sin[2pi(t)+6pi(x)]

Standing Wave:
y=-8cos3(pi)t*sin6(pi)x

---------------------------

2. Incoming Wave
y=-4sin[2pi(t)+6pi(x)]

Reflected Wave:
y=4sin[2pi(t)-6pi(x)]

Standing Wave:
y=8cos3(pi)t*sin6(pi)x

---------------------------

3. Incoming:
y=-8sin[2(pi)t-7(pi)x]

Reflected Wave:
y=8sin[2(pi)t+7(pi)x]

Standing:
y=16cos2(pi)t*sin7(pi)x
---------------------------

Help?!?
How do I know when it's positive or when it's negative?
 
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Well there is at least one common denominator to all there cases. In each of them, the reflected wave's amplitude is of opposite sign to the incoming one. The resulting amplitude in the standing wave is a result of algebra. For exemple, for the first:

1. Incoming Wave:
y=4sin[2pi(t)-6pi(x)]

Reflected Wave:
y=-4sin[2pi(t)+6pi(x)]

Standing = Incoming + Reflected = 4sin[2pi(t)-6pi(x)] - 4sin[2pi(t)+6pi(x)] = 4{sin[2pi(t)-6pi(x)]+sin[-2pi(t)-6pi(x)]}

I used the fact that -sin(x) = sin(-x). Now I'll use the identity sin(A-B)+sin(A+B)=2sinAcosB with A= -6pi(x) and B=-2pi(t):

Standing = 8sin[-6pi(x)]cos[-2pi(t)] = -8sin[6pi(x)]cos[2pi(t)]

I used again the identity -sin(x) = sin(-x) as well as cos(x) = cos(-x).
 
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