Question

Standing waves on a string are generated by oscillations having amplitude 0.005 m, angular frequency 942 rad/s, and wave number 0.750π rad/m.
a.) What is the equation of the standing wave?
b.) At what distances from x=0 are the nodes and antinodes?
c.) What is the frequency of a point on the string at an antinode?
d.) If the string is 4m long, how many nodes are there?

Answer 1

the generall equation for a standing wave is
\[F(x,t)=\sin (kx)*\cos (wt)\]
if u look at this equation u see that for sin(0) F=0 for any time value
and also for sin(pi). these are points on the wave (a string lets say) that dont move these points r called nods! They r standing in one place (thats the reason for the name).
the distance between two nods is easy to spot and it is half the wave length!

now put in the numbers
\[f=w/2\pi\] (in one spin u have 2pi radians)
\[k*\lambda=2\pi\]

Enjoy!
good luck :)

Answer 2

Im not sure if his frequency equation is correct

Answer 3

Cause that for just frequency in general

Answer 4

But I need of a point on the string at an antinode

Answer 5

I've resolved that the equation for the standing wave is: y=.01sin(.750pix)cos942t

Answer 6

And think the location for nodes and antinodes can be expressed as n(lamba)/2 or n(lamda)/4

Answer 7

Where lambda is found using k=2pi/lamba, we know k is .750pi so lamba comes out to 3/8 after calculation

Answer 8

Everything that dobkowski said is correct. The locations of the nodes are
\[x = \frac{n\lambda}{2} \space \text{n = 1,2,3...}\]
and the locations of the antinodes are
\[x = \frac{(2n-1)\lambda}{4} \space \text{n = 1,2,3...}\]

Answer 9

What about c and d?

Answer 10

The tools to find the answers you are looking for are all here.\[F_n=\frac{v}{\lambda_n}\] where F is the frequency, v is the velocity, n is an integer (1,2,3...) and lamda is the wavelength. \[F_n=nf_1\] whis means that the frequency harmonic you are looking for is a number times the first frequency of that wave.
For your length you would just use\[L=\frac{n \lambda_n}{2}\] All you need is to apply what they have given you already in this thread and the answers are there.

Answer 11

Conceptually it might help to note that the frequency of all the points are the same. If they weren't, the string would not vibrate coherently and it wouldn't be much of a standing wave, would it?