What Can Be Done With the Expression for Total Energy in a Vibrating String?

[member 428835]

Hi PF!

SO we have defined energy per unit mass as $$E(t) = \int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx$$. We are given a vibrating string that exhibits ##u_x(0,t) = 0## and ##u(L,t)=0##. I am trying to figure out what is happening with total energy, ##E(t)##. My work is $$\int_0^L \frac{1}{2} u_t^2 + \frac{c^2}{2} u_x^2 dx = \int_0^L \frac{1}{2} u_t^2 dx + \int_0^L \frac{c^2}{2} u_x^2 dx=\int_0^L \frac{1}{2} u_t^2 dx + \frac{c^2}{2} u_x u \bigg|_0^L - \int_0^L u u_{xx} dx\\
=\int_0^L \frac{1}{2} u_t^2 dx - \frac{c^2}{2}\int_0^L u u_{xx} dx = \int_0^L \frac{1}{2} u_t^2 - \frac{1}{2} u u_{tt} dx$$ where the last quantity appears from ##u## satisfying the wave eq, ##u_{tt} = c^2 u_{xx}##. Is there anything more I can/should do?

Thanks!

 

[jvicens]

Yes, there is more that can be done with the expression for total energy, ##E(t)##. First, it is important to note that the expression you have derived is the total energy of the string at a specific time, ##t##. This means that the total energy will change as the string vibrates over time.

To further analyze the behavior of the total energy, it may be helpful to plot ##E(t)## as a function of time. This will give you a visual representation of how the energy changes over time. Additionally, you can use the wave equation, ##u_{tt} = c^2 u_{xx}##, to simplify the expression for total energy.

By substituting ##u_{tt}## into the expression, you can simplify it to ##E(t) = \frac{1}{2}\int_0^L u_t^2 + c^2 u_{xx}^2 dx##. This form of the expression may be more useful for further analysis.

Furthermore, you can also investigate the boundary conditions given for the string, ##u_x(0,t) = 0## and ##u(L,t)=0##. These conditions will impact the behavior of the string and its total energy. For example, if the string is fixed at both ends, the total energy will remain constant over time. However, if the string is free at both ends, the total energy will decrease over time due to energy being dissipated through the boundaries.

I hope this helps in your further analysis of the total energy of the vibrating string. Let me know if you have any other questions.