What type of wave is described by y'' = -k*y^2

[jaydnul]

A sinusoid can be described by the differential equation y'' = -k*y, where the force y'' is proportional to how far away from the center it is.

However in many physical systems the force between two bodies decreases with distance squared. So would we still classify the differential equation y'' = -k*y^2 as sinusoid as well?

 

[kuruman]

If you substitute ##y=\sin(\sqrt{k}~t)## in the first equation, you can verify that you have a solution. Is this also the case for the second equation?

 

[jaydnul]

Hmm, no you end up with $$-k\sin(\sqrt{k}~t) =-k\sin^2(\sqrt{k}~t)$$

So the second equation doesn't have sinusoidal solution? If not then what do we call that type of wave?

 

[Baluncore]

If not then what do we call that type of wave?

A fundamental sinusoid with odd harmonics ?

 

[jaydnul]

Sorry could you explain a littler further?

 

[Baluncore]

Thinking more about it y'' would be asymmetric, so even harmonics.
Since y^2 is always positive the force will always act in the same direction of -k .
Could it actually oscillate with that condition ?

 

[Fred Wright]

$$
y^{"}=-ky^2
$$
is the Emden Fowler differential equation whose solution in this case is,
$$
y=-\frac{6}{kx^2}
$$
which diverges at ##x=0## and is obviously not periodic.

 

[jaydnul]

Oh ok thanks! I realized i wrote the wrong equation anyway. What about:

$$y’’=\frac{-k}{y^2}$$

Whats the solution for this one, and would it oscillate?

 

[kuruman]

Oh ok thanks! I realized i wrote the wrong equation anyway. What about:

$$y’’=\frac{-k}{y^2}$$

Whats the solution for this one, and would it oscillate?

What do you think? If you rewrite constant ##k## as the product ##k=GM## where ##G## is the gravitational constant and ##M## is whatever is needed given the value of ##k##, you get ##a=-\frac{GM}{y^2}##. Does this equation look familiar? Does it have oscillatory solutions?

 

[jaydnul]

Right, and that's kind of the origins of my question. If you consider a proton and an electron as classical particles they will behave like this. And i assume the electron will oscillate back and forth as it passes through the proton. But this oscillation would technically not be sinusoidal correct? Since the force isn't proportional to distance, its proportional to the inverse square of the distance.

 

[kuruman]

If I were to consider a proton and an electron as classical particles, say two oppositely charged spheres, why would the electron pass through the proton and not just stick to it? The devil is in the details of this "pass through".

 

[jaydnul]

Ok then just consider an extremely dense ring of mass in space that pulls a small mass through it, so the force would be approximated as 1/y^2, would this oscillate?

 

[PeroK]

Ok then just consider an extremely dense ring of mass in space that pulls a small mass through it, so the force would be approximated as 1/y^2, would this oscillate?

You could consider a WIMP (weakly interacting massive particle) that is gravitationally attracted to a large spherical mass but passes straight through, owing to the lack of any electromagnetic interaction. The solution for the distace ##r## as a function of time, given the particle is initially at rest at a distance ##r_0## from the centre of the mass ##M## is:
$$t = \sqrt{\frac{r_0^3}{2GM}}[cos^{-1} \sqrt{\frac{r}{r_0}} + \sqrt{\frac{r}{r_0}(1- \frac{r}{r_0})}]$$The full derivation is here, for example:

https://www.physicsforums.com/threads/falling-to-a-star-with-varying-acceleration.866218/

 

[vela]

Ok then just consider an extremely dense ring of mass in space that pulls a small mass through it, so the force would be approximated as 1/y^2, would this oscillate?

You didn't change the math at all, so you still have the problem of what happens when ##y=0##.

In any case, you can solve ##y''=-k/y^2## by multiplying both side by ##y'## and then integrating. Then you can integrate once more using separation of variables.

 

[vanhees71]

Right, and that's kind of the origins of my question. If you consider a proton and an electron as classical particles they will behave like this. And i assume the electron will oscillate back and forth as it passes through the proton. But this oscillation would technically not be sinusoidal correct? Since the force isn't proportional to distance, its proportional to the inverse square of the distance.

Well, the equation of motion
$$\ddot{x}=-G M/x^2$$
describes a planet moving radially in the gravitational field of the Sun. For this there are no periodic motions. Depending on the total energy it either falls finally into the Sun (##E<0##) or it moves to infinity (##E \geq 0##).

For the periodic planetary motion you need solutions with non-vanishing angular momentum. Then you have periodic motion on an Kepler ellipse for ##E<0## and non-periodic motions to infinity along parabolae (##E=0##) or hyperbolae (##E>0##).

 

[Fred Wright]

A particular solution of ##y^{"}=-\frac{-k}{y^2}## is,
$$
y=\left ( \frac{9k}{2} \right )^{\frac{1}{3}}x^{\frac{2}{3}}
$$
No matter how hard you contrive to bend physical reality to your desire, you won't succeed. At the risk of violating a PF rule, I render an English meaning of qur'an 23:71
"And should the Truth have followed their desires, surely the heavens and the Earth and all those therein would have perished..."