Waves in opposite directions in a wire

[Karol]

Homework Statement

2 crosswise cosine waves of amplitude 2.5 cm and wavelength 5 cm are traveling in opposite directions in a wire with velocity 6 mm/s. draw graphs of the wire at times t=0, 2 and 4 sec.

Homework Equations

Waves propagating to the right: $y=A\cos 2\pi \left( \frac{x}{\lambda}-\frac{t}{T} \right)$
Waves propagating to the left: $y=A\cos 2\pi \left( \frac{x}{\lambda}+\frac{t}{T} \right)$

The Attempt at a Solution

Isn't it necessary to give the initial conditions at t=0? i mean what was the starting angle of each wave, or is it obvious that they both started with $\theta=0$ but in opposite directions, like in the drawing?

 

[Simon Bridge]

If you are given no other information than what you wrote above, then you can safely treat the wire as infinitely long ... so you are free to place the origin for time and length anywhere you like: it makes no difference to the physics anyway so you may as well make a choice that provides the least work ;)

OTOH: if you have other physical information - i.e. the length of the wire, the presence of a wall, etc - then you have to take that into account when you make your choices: don't just assume something - work it out. i.e. at the wall the amplitude is forced to a specific value.

Note: you have been asked for a graph of the wire, not the waves.

 

[Karol]

I was asked a graph of the shape of the wire, but isn't it the only graph possible? what is a graph of the wire and what is the graph of the waves? aren't they the same thing? i didn't learn anything else but the shape of the waves in a wire

 

[Simon Bridge]

You could do a graph of each wave separately or of the superposition of the two waves.
You could also graph properties other than the transverse displacement.
There's lots of graphs you could do - but the wave vs wire thing is a likely confusion.

A particular wave on a wire may be composed of a large number of components - in this case there are two.

 

[HalfLostAndSearching]

As a scientist, it is important to provide all necessary information and initial conditions in order to accurately depict and analyze a physical phenomenon. In this case, it would be helpful to know the initial phase, or starting angle, of each wave in order to accurately draw the graphs at different times. However, if it is assumed that both waves started with a phase of 0 and are traveling in opposite directions, then the graphs can be drawn as shown in the image provided. At t=0, both waves are at their maximum amplitude, with one wave traveling to the right and the other traveling to the left. At t=2 seconds, the waves have moved half a wavelength and have switched positions, with the wave that was initially traveling to the right now traveling to the left and vice versa. At t=4 seconds, the waves have moved one full wavelength and are back in their original positions. It is important to note that the amplitude and wavelength remain constant throughout the motion, and the only change is in the phase and direction of propagation.