dot product of standing wave
L to 0 ∫sin (nπx/L) * sin (mπx/L)dx = 0 if n≠m
= L/2 if n=m
superposition of standing waves
y(x,0) =∑ An sin(2πx/L)
what does Am = ?
when y(x,0) =∑ An sin(2πx/L)
Am=2/L (L to 0) ∫ y(x,0) sin(mπx/L)dx
what is a periodic function
repeates regularly over a given interval
what can you tell about the Fourier series if f(x) is even
contains only cos terms
what can you tell about the Fourier if f(x) is odd
contains only sin terms
what happens when a periodic function is symmetric on the x axis
1/2 a_0 is even
what is a_n in a Fourier series
a_n= 2/2π (2π to 0) ∫ f(x) cos (nx) dx
what is b_n in a Fourier series
a_n= 2/2π (2π to 0) ∫ f(x) sin (nx) dx
what are the integrals and results to find an and bn
(2π to 0) ∫ cos(mx) cos(nx) dx
= 0 if m≠n = π if m=n
(2π to 0) ∫ sin(mx) sin(nx) dx
= 0 if m≠n = π if m=n
(2π to 0) ∫ sin(mx) cos(nx) dx= 0
