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## Homework Statement

Hi!

I'm suppose to find the points x on the "string" 1-D wave which are not moving during the vibrations, i.e., 0<x<1 such that u(x,t) = 0 for all times t >0

## Homework Equations

[tex]\left.u(x,t) = sin( \pi x)cos(\pi t) + \frac{1}{2}sin(3\pi x)cos(3\pi t) + 3sin(7\pi x) cos(7\pi t)[/tex]

[tex]\left.u(x,0) = sin( \pi x) + \frac{1}{2}sin(3\pi x) + 3sin(7\pi x) [/tex]

[tex]\left.\frac{du}{dx}(x,0) = \pi cos( \pi x) + \frac{3}{2}\pi sin(3\pi x) + 21\pisin(7\pi x) [/tex]

## The Attempt at a Solution

Here's what I have so far,

I reasoned that if there are stationary points, it doesn't matter at what t I'm at, I'll still achieve the same x values and therefore, I simplified the problem by using t = 0.

Now the next thing I said was for a x value to be non moving throughout the vibrations, this means that the velocity would be zero therefore, I took the derivative of the function.

so, to find the x value I simply set the derivative to zero so:

[tex]\left.\pi cos( \pi x) + \frac{3}{2}\pi sin(3\pi x) + 21\pi sin(7\pi x) = 0[/tex]

I divided out the [tex]\pi[/tex] to simplify things:

[tex]\left. cos( \pi x) + \frac{3}{2} sin(3\pi x) + 21sin(7\pi x) = 0[/tex]

At this point, I'm lost on how to approach this problem in order to find all the non moving points.

I asked my professor whether I can approximate the points by looking at the graph but he wants to points to be solved in a systematic fashion so...any type of suggestion would be greatly appreciated! Thank you!!!!!