Sinusoidal wave (not sure what to do for the last part)and have test tommoro

[~christina~]

[SOLVED] sinusoidal wave (not sure what to do for the last part)and have test tommoro

Homework Statement

A sinusoidal wave representing transverse oscillations on a string has the wave function

$$y(x,t)= (0.40m)cos[(10.0\frac{rad} {s})t- 5.00\frac{rad} {m}x]$$

The linear mass density of the string is 0.015kg/m
a) what is the phase velocity of the wave, the tension in the string and the period of oscillation?
b) Find the transverse velocity and acceleration of the wave when x= 3.60m and t= 2.00s
c) If the ends of the string are fixed, a second sinusoidal wave function is given by

$$y(x,t)= (0.40m)cos[(10.0\frac{rad} {s})t- 5.00\frac{rad} {m}x]$$

travels on the string. Give a description of the resulting interference of these waves.
d) sketch the resulting wave pattern

Homework Equations

The Attempt at a Solution

for
a) what is the phase velocity of the wave, the tension in the string and the period of oscillation?
I think I'd use these equation:
phase velocity: $v=\omega /k$
tension: [itex]v= \sqrt{T/ \mu}[/tex]<br /> <b><i>period of oscillation: (not sure what the period of oscillation is though) </i></b>$\omega= 2\pi / T$<br /> <br /> <b>b) Find the transverse velocity and acceleration of the wave when x= 3.60m and t= 2.00s</b><br /> I think I'd find the derivative of the wave equation thus having in general<br /> $$v_y= \frac{d_y} {d_t} |_{x= constant} = \frac{\delta y} {\delta t}= -\omega A cos(kx-\omega t)$$<br /> <br /> <b>d)</b><b>For the last part I'm not sure how to determine what type of interference it is. I think that since one wave is + and one is - one is going to the right(-) and one to the left(+) and both have the same amplitude and such but how would I describe it? <br /> The funny part is that what is "different direction" as it pertains to the signs exactly?<br /> In my book it shows destructive interference:(one wave up and one wave down and both coliding) HOWEVER both are going in different directions (towards each other and that's what I think of when they say "different directions but is this wrong?). <br /> Constructive interference: (based on book) both waves are up and both collide. What's confusing is since they collide, are they from "different directions"? <br /> </b><br /> basically need the most help with the last part<br /> <br /> Thank you<img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" /><br /> I'd REALLY REALLY appreciate if if someone could help me out with the last part at least.[/itex]

 

[Mindscrape]

Yes, I agree with what you have done so far. Partial differentials are \partial in TeX code, FYI.

Wait, are they really the same wave or did you just type up the wrong thing? I mean, if they are the same wave then the interference is easy, right? Of course you will need some kind of initial conditions, which I don't see, to be precise, but you can tell generally what will happen.

This is kind of a cool simulation for 1D waves (take damping to zero and play with pulse or oscillation):
http://phet.colorado.edu/new/simulations/sims.php?sim=Wave_on_a_String

Were you taught euler's formula and complex notation? If not, you should teach yourself at some point because it is cool and really useful for playing with waves.

 

[~christina~]

Yes, I agree with what you have done so far. Partial differentials are \partial in TeX code, FYI.

Oh..I'll remember that next time.

Wait, are they really the same wave or did you just type up the wrong thing? I mean, if they are the same wave then the interference is easy, right? Of course you will need some kind of initial conditions, which I don't see, to be precise, but you can tell generally what will happen.

typed incorrect. one has a sign difference (that's what I was confused about)
$$y(x,t)= (0.40m)cos[(10.0\frac{rad} {s})t- 5.00\frac{rad} {m}x]$$

$$y(x,t)= (0.40m)cos[(10.0\frac{rad} {s})t+ 5.00\frac{rad} {m}x]$$
(second one has + kx)
not sure if the wave is going in the different direction and how a wave would look if it was upside down on a string (formula wise) [book doesn't explain this and just mentions "different direction" leads to destructive interference.

This is kind of a cool simulation for 1D waves (take damping to zero and play with pulse or oscillation):
http://phet.colorado.edu/new/simulations/sims.php?sim=Wave_on_a_String

Were you taught euler's formula and complex notation? If not, you should teach yourself at some point because it is cool and really useful for playing with waves.

no I wasn't taught that formula. I may teach myself after the exam tomorow when I have time.
Thanks for the link but it got sort of confusing (simulation).
and about oscillation (if they ask for period of oscillation that would be the same as "period T" right?)

Thanks Mindscrape

 

[Mindscrape]

Maybe just think about if on a string you made a couple pulses of waves (the whole wave would be that pulse repeated over and over again) and so one of the pulses will go one way, and then the other will go the other way. Bouncing off walls makes the waves flip, and so if one immediately bounces off the left wall while the other does its thing, what is going to happen? Now what would happen if they were offset a little? What kinds of interference can you create? Once the positions, the offsets are set, will the interference change?

Period of oscillation is the same as T.

 

[~christina~]

Maybe just think about if on a string you made a couple pulses of waves (the whole wave would be that pulse repeated over and over again) and so one of the pulses will go one way, and then the other will go the other way. Bouncing off walls makes the waves flip, and so if one immediately bounces off the left wall while the other does its thing, what is going to happen?

so the sign just says where it's going, not if it's flipped or not right?(but my book says it does matter) like if there is another person at the other end of the string creating a pulse coming fom the other end...will the equation still be the same and just have a + instead of -?
well they will have destructive interference if it is reflected off the wall and hits an incoming pulse going in the direction it was before. The 2 pulses going in different directions one + and other - will cancel out

Now what would happen if they were offset a little? What kinds of interference can you create? Once the positions, the offsets are set, will the interference change?

what do you mean if they were offset a little? do you mean if one was not the same but smaller or larger? If this is what you're speaking of they will do the same as the above one except, they will collide and partially cancel out then they will go past each other and reform themselves as if they did not collide at all.

Period of oscillation is the same as T.

okay.

 

[Mindscrape]

Yes, the k in cos(kx-wt) is called the propagation vector. In particular if k is positive then the wave propagates to the right. Notice that since cos is even then it could be written equivalently as cos(wt-kx).

Right, so if the waves hit going different directions then they will cancel out. What I am getting at with the waves being offset a little is that in general waves always interfere, always destruct, or some mixture of the two. To actually figure it out you literally have to add the two waves together.

new wave = wave 1 + wave 2

Sometimes, and with practice, you can reason through how the interference will look, like this case, and other times you just have to go through the math.

You seem to stuck in the idea that waves traveling opposite will cause destructive interference. That will only happen in certain cases when the phases are exactly opposite. Waves traveling together don't always constructively interfere. This stuff is pretty confusing the first time you go through it all, especially suddenly seeing a multivariate function, so don't worry.

If you actually want to add the two together this identity will help you (along with cos being even and sin being odd)
cos(a+b) = cos(a)cos(b)-sin(a)sin(b)

 

[~christina~]

Yes, the k in cos(kx-wt) is called the propagation vector. In particular if k is positive then the wave propagates to the right. Notice that since cos is even then it could be written equivalently as cos(wt-kx).

Right, so if the waves hit going different directions then they will cancel out. What I am getting at with the waves being offset a little is that in general waves always interfere, always destruct, or some mixture of the two. To actually figure it out you literally have to add the two waves together.

new wave = wave 1 + wave 2

Sometimes, and with practice, you can reason through how the interference will look, like this case, and other times you just have to go through the math.

You seem to stuck in the idea that waves traveling opposite will cause destructive interference. That will only happen in certain cases when the phases are exactly opposite. Waves traveling together don't always constructively interfere. This stuff is pretty confusing the first time you go through it all, especially suddenly seeing a multivariate function, so don't worry.

no, actually that's what I was confused about in the first place.
situation 1: wave is moving right. and another wave moves left
problem I have?=> which way is the pulse facing? if both are up then they have constructive interference.
situation 2: same as situation one except one pulse is up and the other one going left is facing down.

Question is: Are you supposed to know whether the pulse is up or down by the equation given or is it from the situation such as in this question where the string ends are fixed?

If you actually want to add the two together this identity will help you (along with cos being even and sin being odd)
cos(a+b) = cos(a)cos(b)-sin(a)sin(b)

I saw something like this but it was with the phase angles and they based that on how you would know if the interference was destructive/constructive or etc.

would I have to have numbers for that equation you gave? I assume so since the equation for the wave has the constants..

 

[Mindscrape]

Well you can make anything a and b in that equation. For example, you could make a=10t and b=-5x.

The only time a wave is going to flip is if it hits a boundary, one of the fixed ends.

 

[~christina~]

Well you can make anything a and b in that equation. For example, you could make a=10t and b=-5x.

The only time a wave is going to flip is if it hits a boundary, one of the fixed ends.

Okay, that answers the questions I had.

Thanks a lot for your help Mindscrape