Diagnosing an Error in my String Wave Solution

In summary, the conversation discusses a problem with a clamped, uniform string under tension \(T\) and its initial transverse velocity distribution. The solution obtained is a sine function and the question arises whether it is correct and if the corresponding plot is also correct. The conversation also mentions the energy of each mode and a plot showing the amplitude growing over time, which prompts the question of whether it should actually spike and decay due to the nature of the initial velocity distribution.
  • #1
Dustinsfl
2,281
5
My plot seems wrong so I am not sure what the problem is: (a) mistake in sol (b) mistake in coding.

A clamped, uniform string under tension \(T\) has length \(\ell\). The string is struck in the middle, giving an initial tranverse velocity distribution
\[
\dot{u}(x, 0) = \delta(x - 1/2).
\]
So the solution I obtained:
\[
u(x, t) = \frac{2}{\pi}\sum_{n=1}^{\infty}\frac{1}{n} \sin\left(\frac{n \pi}{2}\right) \sin(n\pi x)\sin(n\pi t)
\]
Is the solution correct? If so, is the plot correct?

Also, how do I find the energy of each mode?

View attachment 1508
 

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  • #2
Here is the plot t = 0:0.01:0.11.

We can see that the amplitude grows as time grows. Shouldn't the amplitude spike since it is a delt function then decay?

View attachment 1510
 

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  • #3
Does the wording under tension \(T\) something to the equation I am missing?
 

1. What is a string wave solution?

A string wave solution is a mathematical model that describes the behavior of waves on a string, such as a guitar string or a jump rope. It takes into account factors such as the tension, length, and density of the string to predict the shape and movement of the wave.

2. How do I diagnose an error in my string wave solution?

To diagnose an error in your string wave solution, you should first check your calculations and equations for any mistakes. You should also compare your solution to known examples or solutions to see if they match. Additionally, you can run simulations or experiments to test the accuracy of your solution.

3. What are some common errors in string wave solutions?

Some common errors in string wave solutions include incorrect equations, incorrect assumptions about the properties of the string, and rounding errors in calculations. It is also possible to have errors in initial conditions or boundary conditions that can affect the accuracy of the solution.

4. How can I improve the accuracy of my string wave solution?

To improve the accuracy of your string wave solution, you can use more precise measurements for the properties of the string, such as tension and density. You can also use more advanced mathematical techniques, such as numerical methods, to solve the equations. Additionally, running multiple simulations or experiments and averaging the results can help reduce errors.

5. Can I use string wave solutions in real-world applications?

Yes, string wave solutions have many real-world applications in areas such as acoustics, music, and physics. They can be used to model the vibration of guitar strings, the sound waves produced by musical instruments, and the behavior of seismic waves in the earth's crust. They are also used in engineering and design to understand the behavior of structures under stress and to design efficient bridges and buildings.

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