What is the solution for a fixed string oscillating at both ends?

In summary, the conversation discusses a string of length L clamped at both ends and the equation that describes its oscillating displacement. The goal is to show that the displacement can be represented by a summation of sinusoidal functions. The method of separation of variables is mentioned as a way to achieve this goal.
  • #1
saturnsalien
2
0
This isn't actually a homework or coursework problem, but the style of the question is similar so I'm posting it here. Anyways, here goes. Consider a string of length L clamped at both ends, with one end at x=0 and the other at x=L. The displacement of the oscillating string can be described by the following equation:
[itex]\frac{\partial^2 \Psi}{\partial t^2}=\frac{\partial^2 \psi}{\partial x^2}[/itex]

[itex]
\textrm{Given that at t=0:}\\
\>\>\>\> \psi(x,0)=\frac{2xh}{L},0\leq x\leq \frac{L}{2}\\
\>\>\>\> \psi(x,0)=\frac{2xh}{L},(L-x),\frac{L}{2}\leq x\leq L\\
\textrm{Show:}\\
\>\>\>\> \psi(x,t)=\sum_{m=1}^\infty\sin\left(\frac{m\pi x}{L}\right)\cdot\cos\omega_mt\cdot\left(\frac{8h}{\pi^2m^2}\right)\cdot\sin\left(\frac{\pi m}{2}\right)
[/itex]

So, how do we go about doing that?
 
Last edited:
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  • #2
Do you know the method of separation of variables?
 
  • #3
haruspex said:
Do you know the method of separation of variables?

Yeah, that's just rearranging the equation so that different variables occur on opposite sides of the equation. You can also do this by defining a variable as some expression, substitute, and the separate them.
 
  • #4
So what does that give you for generic solutions of the PDE?
 
  • #5


The solution to a fixed string oscillating at both ends can be obtained by using the method of separation of variables. This involves separating the variables in the given equation, namely time and position, and then solving each part separately.

First, we can assume that the solution has the form of a product of two functions, one depending only on time and the other only on position. This can be written as \psi(x,t)=X(x)T(t).

Substituting this into the given equation, we get:

\frac{\partial^2}{\partial t^2}\left(X(x)T(t)\right)=\frac{\partial^2}{\partial x^2}\left(X(x)T(t)\right)

Dividing both sides by X(x)T(t), we get:

\frac{1}{T(t)}\frac{\partial^2 T(t)}{\partial t^2}=\frac{1}{X(x)}\frac{\partial^2 X(x)}{\partial x^2}

Since the left side depends only on time and the right side depends only on position, both sides must be equal to a constant, which we can denote as -\omega^2. This gives us two separate equations:

\frac{\partial^2 T(t)}{\partial t^2}+\omega^2T(t)=0

and

\frac{\partial^2 X(x)}{\partial x^2}+\omega^2X(x)=0

Solving the first equation, we get T(t)=A\cos\omega t+B\sin\omega t, where A and B are constants. Similarly, solving the second equation, we get X(x)=C\sin\left(\frac{m\pi x}{L}\right)+D\cos\left(\frac{m\pi x}{L}\right), where m is a constant.

Now, using the given initial conditions, we can determine the values of A, B, C and D. Replacing these values in the above equations, we get:

T(t)=\cos\omega t

and

X(x)=\frac{8h}{\pi^2m^2}\sin\left(\frac{m\pi x}{L}\right)

Therefore, the solution to the given equation is:

\psi(x,t)=\sum_{m=1}^\infty C_m\sin\left(\frac{m\pi x
 

What is Fixed String Oscillation?

Fixed string oscillation is a type of vibration that occurs when a string is fixed at both ends and is plucked or struck. It is a common phenomenon in musical instruments such as guitars and violins, and is also studied in physics and engineering.

What causes Fixed String Oscillation?

Fixed string oscillation is caused by the string's tension and its elasticity. When the string is plucked or struck, it is displaced from its equilibrium position and then pulled back by the tension, causing it to vibrate back and forth.

What factors affect the frequency of Fixed String Oscillation?

The frequency of fixed string oscillation is affected by the string's tension, length, and mass. Higher tension and shorter length result in higher frequencies, while a heavier string will vibrate at a lower frequency.

What is the relationship between frequency and pitch in Fixed String Oscillation?

In fixed string oscillation, the frequency of vibration is directly related to the pitch of the sound produced. Higher frequencies result in higher pitched sounds, while lower frequencies produce lower pitched sounds.

How is Fixed String Oscillation used in practical applications?

Fixed string oscillation has various practical applications, such as in musical instruments, stringed sports equipment like tennis rackets, and in the design of bridges and buildings to prevent structural damage from vibrations. It is also used in scientific research and experiments to study the properties of waves and oscillations.

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