Why use infinite series in Laplace equation with Fourier coefficients

[Mike Jonese]

Homework Statement

Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V₀(y) when x=0 iv) V=0 when x app infinity.
I understand and follow this problem (separating vars and eliminated constants) until the potential
is found to be V(x,y) = Ce^(-kx)*sin(ky)

Condition ii implies that V(0,a) = Csin(k*a) = 0 so sin(ka) must =0 and k=n*π/a
Question 1) Why can't we just set k=π/a so that V(0,y) = Csin(πy/a) where V(0,a) = 0 and V(0,0) =0? This would not screw up the boundary condition and you wouldn't have to deal with infinite solutions because of n?

Homework Equations

Moving on,
Book give a more general solution that includes all the different solutions due to n
V(x,y) = (n=1---->inf)∑C_n*e^(-nπx/a)*sin(nπy/a)
V(0,y) = (n=1---->inf)∑C_n*sin(nπy/a) = V₀(y)
and now we are trying to find the Coefficient C's that make this possible
I'm including the section here to save typing

I didn't realize the forum would kill the resolution

3.Attempt at solution
I tried, I don't understand this in general ( I haven't taken diff eqn yet)
Question 2. How can you solve the integral that leads to (3.32)
Question 3 How does V0(y) pop into the integrand in (3.32)
Question 4 how can the answer be 0 or a/2 based on n and n' (is this some type of delta fxn?)
Question 5 Can someone describe as you would to a 3rd grader why 3.36 is an answer and what it means. Is this just an approximation?
Question 6. What makes something orthogonal in the sense of 3.32
I extremely appreciate and help or insight (or direction to external reference material) anyone might have. Thank you very much! -Mike

 

[NFuller]

Question 1) Why can't we just set k=π/a so that V(0,y) = Csin(πy/a) where V(0,a) = 0 and V(0,0) =0? This would not screw up the boundary condition and you wouldn't have to deal with infinite solutions because of n?

Because you need the whole family of solutions in order to match the boundary condition $V(0,y)=V_{0}(y)$. There is no way to make $V(0,y)=C\text{sin}(\pi y/a)=V_{0}(y)$ unless $V_{0}$ happens to be that specific sine function.

Question 2. How can you solve the integral that leads to (3.32)

The point is that the sine functions are orthogonal so the integral is zero if the n's are different and $a/2$ if the n's are the same number.

Question 3 How does V0(y) pop into the integrand in (3.32)

We know that at $x=0$
$$V(0,y)=\sum_{n=1}^{\infty}C_{n}\text{sin}(n\pi y/a)=V_{0}(y)$$
The "trick" referenced is to first multiply both sides by $\text{sin}(n^{\prime}\pi y/a)$
$$\sum_{n=1}^{\infty}C_{n}\text{sin}(n\pi y/a)\text{sin}(n^{\prime}\pi y/a)=V_{0}(y)\text{sin}(n^{\prime}\pi y/a)$$
then integrate from 0 to $a$
$$\sum_{n=1}^{\infty}C_{n}\int_{0}^{a}\text{sin}(n\pi y/a)\text{sin}(n^{\prime}\pi y/a)dy=\int_{0}^{a}V_{0}(y)\text{sin}(n^{\prime}\pi y/a)dy$$

Question 4 how can the answer be 0 or a/2 based on n and n' (is this some type of delta fxn?)

Again, this is because sine functions are orthogonal. This means that when you integrate two of them over some number of half periods, then integral evaluates to zero if they have different frequencies or $a/2$ if they have the same frequency.

Question 5 Can someone describe as you would to a 3rd grader why 3.36 is an answer and what it means. Is this just an approximation?

I don't think I could explain Fourier analysis to a 3rd grader but Griffiths says just above this that he is now assuming that the potential is constant at $x=0$ so $V_{0}(y)=V_{0}$. This causes $V_{0}$ to just pop out of the integral allowing you to just integrate the sine function as usual.

Question 6. What makes something orthogonal in the sense of 3.32

See answer to question 4.

I tried, I don't understand this in general ( I haven't taken diff eqn yet)

I was actually in the same position when I was learning E&M from Griffiths. I had only taken up to Calc III and spent hours carefully rereading this chapter until it finally hit me as to what was going on.

 

[Mike Jonese]

Thank you for taking the time to respond to that, that definitely shed some light on this. Do you happen to remember any good lecture notes online or youtube channels that you came across when you were taking the class that were good references? I found a youtube guy that goes through the same textbook but the quality is hit or miss. Anyways thanks again!

 

[NFuller]

Thank you for taking the time to respond to that, that definitely shed some light on this. Do you happen to remember any good lecture notes online or youtube channels that you came across when you were taking the class that were good references? I found a youtube guy that goes through the same textbook but the quality is hit or miss. Anyways thanks again!

From what i remember I learned mostly from the book and my professor. It sounds like your main difficulty is the math, so I would suggest looking for books or websites on Fourier analysis.