Understanding Standing Wave Sign Conventions

[AmandaWoohoo]

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?

 

[quasar987]

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).

 

[kyle8921]

I understand that it can be frustrating when different sources seem to contradict each other. However, it is important to remember that different conventions and notations may be used by different authors or in different fields of study. In the case of standing wave equations, the sign convention for the amplitude is dependent on the specific context and setup of the problem.

In general, the sign in front of the amplitude represents the direction of the wave's displacement. For example, in the first example you provided, the incoming wave has a positive amplitude because it is moving in the positive direction (to the right), while the reflected wave has a negative amplitude because it is moving in the negative direction (to the left).

In the second example, the sign convention is reversed because the incoming wave is now moving in the negative direction (to the left) and the reflected wave is moving in the positive direction (to the right).

In the third example, the sign convention is different because the waves are now traveling in the opposite direction (from left to right instead of right to left).

In summary, the sign in front of the amplitude is dependent on the direction of the wave's displacement and can vary depending on the specific problem or setup. It is important to carefully consider the context and use consistent notation throughout your work. If you are unsure about a particular convention, it is always best to clarify with your instructor or consult multiple sources.