5 ms·

No? Hooks law is an approximation to a material (and spring) property that sets the PDE up. Sin(x) is the solution. But I could be wrong :S

capn_duck 5y ago
ASin(wx) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.
- tagoregrtst 5y ago
  I meant its a solution to a specific problem set up using Hooke’s law
kergonath 5y ago
Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator. It’s not surprising if you see a solid material that way: - it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum - therefore either extending it or compressing it increases its energy - therefore the second derivative of its energy is positive. On a high level, this sounds very much like a simple parabola, because it is one. For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator. Alternatively, to a physicist, almost everything looks like a parabola. If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law). The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well. I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).

capn_duck 5y ago

ASin(wx) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.

tagoregrtst 5y ago
I meant its a solution to a specific problem set up using Hooke’s law

kergonath 5y ago

Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator. It’s not surprising if you see a solid material that way: - it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum - therefore either extending it or compressing it increases its energy - therefore the second derivative of its energy is positive. On a high level, this sounds very much like a simple parabola, because it is one. For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator. Alternatively, to a physicist, almost everything looks like a parabola. If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law). The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well. I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).