Harmonic waves and differential equations

[Ylle]

Hi !

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 the spring 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

 

[vincentchan]

try x(t)=e^{\lambda t}
any linear combination of the solution(s) is/are your solution(s)

Edit: no one say the solution is real...
a complex solution is acceptable in mathematics, but not in Physics, if you need a real solution, try apply an reasonable (real) initial condition..

 

[Crosson]

It sounds like you have taken a class or two of calculus, but not a class in Differential Equations. Hopefully this will make sense:

We have shown that a mathematical function which represents the displacement of a spring as a function of time X(t), must meet the following condition (at all times t) :

X''(t) + X(t) = 0

Only solutions that satisfy the equation are valid, for examply X(t) = 2t is not a solution because the second derivative X''(t) = (0) added to the function X(t) = 2t, gives us

2t = 0 which cannot be true for all t.

So, there are only a few types of functions which can satisfy the differential equation. Because of the type of equation (Second order derivative, linear, 0 on the right side) there are exactly two functions which work.

Here is the simplest way to solve the differential equation: Oh, look! X(t) = Sin(t) or X(t) = cos(t) would work! That is the General Method for solving a differential equation, use your knowledge of functions to spot one which is a solution.

Of course, there exist special techniques such as the one Vincent Chan noted, but in order to use that method you need to know something about complex variables.

To solve your dampened oscillator Diff EQ, verify that the following is a solution:

X(t) = (e^-t)( sin(t) + cos(t) )

You have to figure out where the physical constants a, k, and m go.