Boundary conditions of a forced oscillator (string)

In summary, the conversation discusses how to find the frequencies at which a string with fixed ends can oscillate by solving the wave equation with Dirichlet boundary conditions. The solution also takes into account the initial shape and speed of the string. However, the problem becomes more complex when a force is acting on the string, requiring further specification of the cause and potential changes in frequency. The conversation also touches on the enduring solution and free vibration of the string, as well as the effects of damping on the system. Overall, the same principles apply to both discrete and continuous systems.
  • #1
cromata
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-If we have string of length L that has fixed ends, then we can easily find frequencies with which this string can oscillate:
We just need to solve wave equation: ∂2y/∂x2=1/c2*∂2∂t2 (c is determined by strings properties (linear density and tension), with Dirichlet boundary conditions (y(0,t)=0, y(L,t)=0) Of course to determine how the string is oscillating we also need to know initial shape/speed of string (but that only tells us amplitude of each harmonic)

-But what happens when some force is acting on the string? Let's say that some force F(t) is acting at some distance xo from one end of the string? How do we find solution to this problem?
Can it be treated like some sort of boundary condition or should that force be added to wave equation or something else?
 
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  • #2
The enduring (lasting) solution is the particular solution; the free vibration will die away sooner or later due to unavoidable damping. The particular solution will oscillate at the frequency of the excitation.If this is a stick-slip situation, such as a violin string, then it is going to be a bit messy, depending on the frequency at which slipping is starting.
 
  • #3
cromata said:
-But what happens when some force is acting on the string?
The problem has instantly got a lot harder. You would need to specify what causes this force. If you are hanging a mass on the string then the natural frequency of oscillation would change. If you use a spring, the force will vary with displacement so the frequency would change. If you have a rocket engine, applying a constant force then I cannot see how the frequency would change.
 
  • #4
Dr.D said:
The enduring (lasting) solution is the particular solution; the free vibration will die away sooner or later due to unavoidable damping. The particular solution will oscillate at the frequency of the excitation
I know that this is the case when there is forced discrete oscillating system (like masses connected with springs), and it can easily be shown for discrete systems that enduring solution is particular solution. But I wasn't sure that same thing happens with continuous system.
 
  • #5
The same processes are at work in the continuuous system as were in the discrete system. Air drag and internal hysteresis still serve to induce damping, so the homogeneous solution will die away, leaving only the particular solution.
 
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1. What are boundary conditions for a forced oscillator on a string?

The boundary conditions for a forced oscillator on a string refer to the constraints that must be satisfied at the two ends of the string. These boundary conditions determine how the string will behave when subjected to external forces.

2. What is the significance of boundary conditions in a forced oscillator?

The boundary conditions in a forced oscillator are important because they dictate the behavior of the string at its fixed ends. They also determine the possible modes of vibration and the frequencies at which the string will resonate.

3. How do boundary conditions affect the motion of a forced oscillator on a string?

Boundary conditions have a direct impact on the motion of a forced oscillator on a string. They determine the amplitude and shape of the oscillations, as well as the energy transfer between the string and the external force.

4. What happens if the boundary conditions are not satisfied in a forced oscillator?

If the boundary conditions are not satisfied in a forced oscillator, the string may not vibrate at the desired frequency or amplitude. This can result in a distorted or irregular motion, and the string may not produce the expected sound or resonance.

5. Can boundary conditions be adjusted in a forced oscillator?

Yes, boundary conditions can be adjusted in a forced oscillator by changing the properties of the string or the external force applied. Altering the tension, length, or mass of the string, or adjusting the frequency or amplitude of the external force, can all impact the boundary conditions and the resulting motion of the oscillator.

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