- #1

- 91

- 23

- Homework Statement
- See below.

- Relevant Equations
- ## \frac {\partial^2 \psi} {\partial t^2} = v^2 \frac {\partial^2 \psi} {\partial x^2} ##

## \frac {\partial^2 \psi} {\partial t^2} = v^2 \frac {\partial^2 \psi} {\partial x^2} ##

has solution

## \psi (x, t) = \sum_{m=0}^\infty A_m \sin(k_mx + \alpha_m)sin(\omegat + \beta_m) ##

The boundary conditions I can discern

$$ \psi (0, t) = 0 $$

$$ \frac {\partial \psi} {\partial x} (L, t) = 0 $$

The first boundary condition gives ##\alpha_m = 0##, since it is for all t and forces the first sine term to be 0. The second boundary condition gives...

$$ 0 = \sum_{m=0}^\infty k_mA_m \cos(k_mL) $$

$$ k_mL = (m - \frac {1} {2})\frac {\pi} {2} $$

$$ k_m = (m - \frac {1} {2})\frac {\pi} {2L} $$

And then by the dispersion relation, I just did ##\omega_m = v_m*k ## and ##v_m = \sqrt {\frac {T} {\rho}} (m - \frac {1} {2})\frac {\pi} {2L}##.

Clearly, I'm missing a lot here. The correct answer is given at https://ocw.mit.edu/resources/res-8-005-vibrations-and-waves-problem-solving-fall-2012/problem-solving-videos/standing-waves-part-ii-1/problems/#problem1

I think I might be missing a few boundary conditions? Well I suppose I am since I should have 4. My biggest obstacle to finding more was that it doesn't seem the problem specifies what happens initially and how the string is released.

Also, my answer seems radically different than the correct one, so I'm wondering if I may be neglecting something totally and have taken a very-off approach. Looking forward to your feedback.