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I have been reading a physicsbook, and I've come across harmonic waves, where Hookes law and Newtons 2. law are mentioned. They describe, mathematecly, how a harmonic wave is moving, and they come across differential equation that says (Wich are made by Hookes law and Newtons 2. law):

a(t) = -(k/m)*x(t) , where k is thespring constant.

Wich becomes:

x''(t) = -(k/m)*x(t) <---- differential equation

Then they say that mathematicly the differential equation will look like this when solved:

x(t) = A*sin(w*t) + B*cos(w*t) , where A, B and w are constants.

Then the rest of the proof is about how you end up with:

x(t) = A*sin(w*t) , that describes a harmonic wave.

My problem is that I have to, mathematecly, show how a muffled/deaden harmonic wave (I'm not sure what it is called in english. I'm danish, so the physics/mathematic words aren't my strong side. But I hope you know what I mean) is moving.

I know the differential equation should look like this:

x''(t) = -(k/m)*x(t)-a*v => x''(t) = -(k/m)*x(t)-a*x'(t) , where a is a constant.

The term (-a*x'(t)) should, according to my notes, be the frictional force. So the only thing different from the first differential equation is the term (-a*x'(t)).

And since the book didn't show how to solve the first differential equation, I'm actually kinda lost about how the differential equation x''(t) = -(k/m)*x(t)-a*x'(t) could come to an equation that describes a muffled/deaden harmonic wave.

I hope you can see what my problem is, and understand what I'm saying, and of course maybe help me.

Ylle

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# Harmonic waves and differential equations

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