How did they reach this conclusion regarding the coefficients?

[kirito_01]

TL;DR Summary

i am learning a course about waves we started by solving a problem of infinite num of masses on a spring then we moved to a finite number we so we got the same set solution space but with some conditions so we applied the conditions on the solution we found

the solution for the infinite num case

the problem is that i only could reach the condition that the coefficients are zero when i substituted n=0 , i am reaching two independent variables i am not sure what am i doing wrong that's preventing me from getting a similar result to them "that they are equal in value opposite in sign". , i am certain that they only used the condition n=0 for this , because i used the n=N condition elsewhere to write the solution in relation to modes

here is what they have reached

 

[kuruman]

Did the problem specify the initial condition? Specifically, note that $$\psi_n(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{i\delta_n}.$$If the system starts so that $\psi_n(0)=0$, then $\left[A_{\delta}^{+}+A_{\delta}^{-}\right]=0$ because the phase $e^{i\delta n}\neq 0$ for any $n.$

 

[kirito_01]

Did the problem specify the initial condition? Specifically, note that $$\psi_n(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{i\delta_n}.$$If the system starts so that $\psi_n(0)=0$, then $\left[A_{\delta}^{+}+A_{\delta}^{-}\right]=0$ because the phase $e^{i\delta n}\neq 0$ for any $n.$

oh do you mean since it is zero at all times it is also zero at t=0 so i can deal with a nicer version of the problem and reach the result ?

 

[kirito_01]

oh do you mean since it is zero at all times it is also zero at t=0 so i can deal with a nicer version of the problem and reach the result ?

since then i can understand but dealing with t as a variable i am not reaching anywhere

 

[kuruman]

oh do you mean since it is zero at all times it is also zero at t=0 so i can deal with a nicer version of the problem and reach the result ?

No. I mean that if $\psi_n(t)$ represents the displacement of the $n$th mass from equilibrium at any time $t$, at time $t=0$ all masses are instantaneously at the equilibrium position.

I am guessing here. It would help if you provided a full statement of the problem that you are solving instead of part of the solution.

 

[kirito_01]

No. I mean that if $\psi_n(t)$ represents the displacement of the $n$th mass from equilibrium at any time $t$, at time $t=0$ all masses are instantaneously at the equilibrium position.

I am guessing here. It would help if you provided a full statement of the problem that you are solving instead of part of the solution.

i will try to phrase the problem the $\psi_n(t)$ indeed represent the displacement of the $n$th mass from equilibrium at any time $t$, the full problem is as follows we had a set of masses n masses (an infinite number ) that are coupled on a spring with spring constant k for all and they are only allowed to move upwards and downwards where the displacement of the vertically is represented by $\psi$ after since we have infinite number of masses for each mass there is a mass before it and a mass after it we assumed a solution that looks like a function of time multipied by an exponential that has a frequency delta and n representing the place of the mass n after solving this i got the first and second equations up

 

[kirito_01]

then we where asked to move to finite number of masses we already the space of the solutions without conditions upon the system we added conditions on mass 0 and N not small n it was a problem on my part so the condition up are on 2 masses only the one at the start of the spring and at the end on all times

 

[kirito_01]

then we where asked to move to finite number of masses we already the space of the solutions without conditions upon the system we added conditions on mass 0 and N not small n it was a problem on my part so the condition up are on 2 masses only the one at the start of the spring and at the end on all times

then my book provided a result from the boundary conditions and this is where i reached a different result from them

 

[kuruman]

You have not told us what the boundary conditions on mass 0 and mass N are.

 

[kirito_01]

You have not told us what the boundary conditions on mass 0 and mass N are.

they are the ones written in the equation i provided 0 for mass zero and N for mass N thats all what was provided

 

[kuruman]

OK, so as we have seen in post #2 that at $t=0$ the general expression reduces to $$\psi_n(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{i\delta n}.$$ This is true for any value of $n$ at $t=0$. If you substitute $n=0$ in the above equation, you get $$\psi_0(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{0}=A_{\delta}^{+}+A_{\delta}^{-}.$$ But the boundary condition says that $\psi_0(t)=0$ for all times $t$ and that includes $t=0.$ Therefore, $$0=\psi_0(0)=A_{\delta}^{+}+A_{\delta}^{-}.$$It's that simple.

 

[kirito_01]

OK, so as we have seen in post #2 that at $t=0$ the general expression reduces to $$\psi_n(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{i\delta n}.$$ This is true for any value of $n$ at $t=0$. If you substitute $n=0$ in the above equation, you get $$\psi_0(0)=\left[A_{\delta}^{+}+A_{\delta}^{-}\right]e^{0}=A_{\delta}^{+}+A_{\delta}^{-}.$$ But the boundary condition says that $\psi_0(t)=0$ for all times $t$ and that includes $t=0.$ Therefore, $$0=\psi_0(0)=A_{\delta}^{+}+A_{\delta}^{-}.$$It's that simple.

thank you it is much simpler than i expected