A Differential Equation Problem

[SLD]

Two people are holding a string 10 units long along the x axis. The person at point 0 moves the string up and down between 1 and -1 at a fixed rate. The person at the other end moves it left and right between 1 and -1 at the same rate. Describe the motion of the string in terms of x, y, z, and t.

If just one person was moving the string then there would be a sine wave traveling down the string and the solution is basically a partial differential equation that I learned in undergraduate days. I looked it up. Not too difficult. I figured if the string were suspended in a liquid and hit by transverse waves at the same frequency and along the axes described, it's a bit more difficult but doable.

But it’s the boundary problems here that are exceptionally difficult. The wave is pinched off at each end. At the origin the string only moves up and down and at the other end it only moves right and left. That constraint will affect the movement all along the axes.

Thoughts?

SLD

 

[steve_bank]

xx

 

[steve_bank]

In general linear superposition applies. The defection at any point is the linear sum of effects of the two sources taken independently.

Imagine a mechanical strong machine driving the string which is attached to a wall. The wave propagates forward and is reelected back to the source. At any point the amplitude is the sum of the forward and reverse energy. If the machine puts out a burst on a long string and stops the burst bounces back and forth until friction consumes the energy. Like an electrical transmission line.

For the question you would end up with a partial differential equation that takes into account frequency and amplitude of the sources, length and the mechanical properties of the string. Search on vibrating string equation.

I imagine it is a practical problem on things like long suspended power transmission lines.

 

[steve_bank]

In general linear superposition applies. The defection at any point is the linear sum of effects of the two sources taken independently.

Imagine a mechanical strong machine driving the string which is attached to a wall. The wave propagates forward and is reelected back to the source. At any point the amplitude is the sum of the forward and reverse energy. If the machine puts out a burst on a long string and stops the burst bounces back and forth until friction consumes the energy. Like an electrical transmission line.

For the question you would end up with a partial differential equation that takes into account frequency and amplitude of the sources, length and the mechanical properties of the string. Search on vibrating string equation.

I imagine it is a practical problem on things like long suspended power transmission lines.

The boundary conditions would be the attachment points to a machine driving the string. You could assume 100% reflection of energy at the attachment points. Analogous to an infinite potential well in QM.

 

[SLD]

In general linear superposition applies. The defection at any point is the linear sum of effects of the two sources taken independently.

Imagine a mechanical strong machine driving the string which is attached to a wall. The wave propagates forward and is reelected back to the source. At any point the amplitude is the sum of the forward and reverse energy. If the machine puts out a burst on a long string and stops the burst bounces back and forth until friction consumes the energy. Like an electrical transmission line.

For the question you would end up with a partial differential equation that takes into account frequency and amplitude of the sources, length and the mechanical properties of the string. Search on vibrating string equation.

I imagine it is a practical problem on things like long suspended power transmission lines.

I am familiar with the equation for a vibrating string from my under grad days. When I first thought of this problem several years ago I figured out the solution to the problem of a vibrating string suspended ina fluid being hit by two orthogonal waves. That’s a similar problem, without the boundary issues. It’s the addition of the boundaries that stopped me from going further.

 

[steve_bank]

In general linear superposition applies. The defection at any point is the linear sum of effects of the two sources taken independently.

Imagine a mechanical strong machine driving the string which is attached to a wall. The wave propagates forward and is reelected back to the source. At any point the amplitude is the sum of the forward and reverse energy. If the machine puts out a burst on a long string and stops the burst bounces back and forth until friction consumes the energy. Like an electrical transmission line.

For the question you would end up with a partial differential equation that takes into account frequency and amplitude of the sources, length and the mechanical properties of the string. Search on vibrating string equation.

I imagine it is a practical problem on things like long suspended power transmission lines.

I am familiar with the equation for a vibrating string from my under grad days. When I first thought of this problem several years ago I figured out the solution to the problem of a vibrating string suspended ina fluid being hit by two orthogonal waves. That’s a similar problem, without the boundary issues. It’s the addition of the boundaries that stopped me from going further.

Yea. It has been a while since I thought about this stuff. Way back I read a good little book called All About Waves.

The other boundary condition is the fact that at the connection points of the string the velocity of the string relative to the string's rest position is zero. A stringer fixed to a point on a wall.

Suspension in a fluid adds to damping. Air and water are both fluids with different viscosity. or friction losses. A parameter in the PDE would change value.