So, let's say we have String from 0 to L.

If we define the displacement as $\xi (x,t)$, the equation that governs the system is

$$\big(\dfrac{\partial ^2}{\partial t^2}-c^2\dfrac{\partial ^2}{\partial x^2}\big) \xi(x,t)$$

Our professor gave us initial condition as following: $$\xi (x,t_o) = \sin\big(\dfrac{2 \pi x}{L}\big)$$ and $$\dot{\xi} (x,t_o) = \sin^2\big(\dfrac{2 \pi x}{L}\big)$$

But from what I have studied online: http://farside.ph.utexas.edu/teaching/315/Waves/node24.html

the most general solution for a stationary wave is $$\xi_ν = \xi_o \sin(ν\pi \dfrac{x}{L})\cos(\omega_ν t +\phi_ν)$$

from which I can't relate to my Initial Condition.

Since this is a Homework-Related question and I know it's not right and polite to ask for exact solutions, ANY advice to guide me to the solution is MUCH appreciated

So, let's say we have String from $0$ to $L$.

If we define the displacement as $\xi (x,t)$, the equation that governs the system is

$$\big(\dfrac{\partial ^2}{\partial t^2}-c^2\dfrac{\partial ^2}{\partial x^2}\big) \xi(x,t)=0$$

Our professor gave us initial conditions as following: $$\xi (x,t_o) = \sin\big(\dfrac{2 \pi x}{L}\big)$$ and $$\dot{\xi} (x,t_o) = \sin^2\big(\dfrac{2 \pi x}{L}\big)$$

But from what I have studied online: http://farside.ph.utexas.edu/teaching/315/Waves/node24.html

the most general solution for a stationary wave is $$\xi_ν = \xi_o \sin(ν\pi \dfrac{x}{L})\cos(\omega_ν t +\phi_ν)$$

from which I can't relate to my Initial Condition.

Since this is a Homework-Related question and I know it's not right and polite to ask for exact solutions, ANY advice to guide me to the solution is MUCH appreciated

So, let's say we have String from 0 to L.

If we define the displacement as $\xi (x,t)$, the equation that governs the system is

$$\big(\dfrac{\partial ^2}{\partial t^2}-c^2\dfrac{\partial ^2}{\partial x^2}\big) \xi(x,t)$$

Our professor gave us initial condition as following: $$\xi (x,t_o) = \sin\big(\dfrac{2 \pi x}{L}\big)$$ and $$\dot{\xi} (x,t_o) = \sin^2\big(\dfrac{2 \pi x}{L}\big)$$

But from what I have studied online: http://farside.ph.utexas.edu/teaching/315/Waves/node24.html

the most general solution for a stationary wave is $$\xi_ν = \xi_o \sin(ν\pi \dfrac{x}{L})\cos(\omega_ν t +\phi_ν)$$

from which I can't relate to my Initial Condition.

Since this is a Homework-Related question and I know it's not right and polite to ask for exact solutions, ANY advice to guide my to the solution is MUCH appreciated

So, let's say we have String from 0 to L.

If we define the displacement as $\xi (x,t)$, the equation that governs the system is

$$\big(\dfrac{\partial ^2}{\partial t^2}-c^2\dfrac{\partial ^2}{\partial x^2}\big) \xi(x,t)$$

Our professor gave us initial condition as following: $$\xi (x,t_o) = \sin\big(\dfrac{2 \pi x}{L}\big)$$ and $$\dot{\xi} (x,t_o) = \sin^2\big(\dfrac{2 \pi x}{L}\big)$$

But from what I have studied online: http://farside.ph.utexas.edu/teaching/315/Waves/node24.html

the most general solution for a stationary wave is $$\xi_ν = \xi_o \sin(ν\pi \dfrac{x}{L})\cos(\omega_ν t +\phi_ν)$$

from which I can't relate to my Initial Condition.

Since this is a Homework-Related question and I know it's not right and polite to ask for exact solutions, ANY advice to guide me to the solution is MUCH appreciated