- #1
LinearWave
- 2
- 0
Homework Statement
Assume that the wavelength of acoustic waves in an organ pipe is long relative to the width of the pipe so that the acoustic waves are one-dimensional (they travel only lengthwise in the pipe). Therefore, the equation governing the pressure in the wave is:
∂2p/∂t2-c2*∂2p/∂x2 = 0
p is the acoustic pressure. At a closed end of the pipe the air velocity (u) must be zero. Note that:
-ρ*∂u/∂t = ∂p/∂x
So that if u = 0, ∂p/∂x = 0. At an open end, p is approximately 0
Find the acoustic modes in a pipe of length L with:
(a) 2 closed ends
(b) 2 open ends
(c) one closed end and one open end
The Attempt at a Solution
From lecture we started by using separation of variables on the wave function
∂2p/∂t2-c2*∂2p/∂x2 = 0
let y(x,t) = c2*∂2p/∂x2
ytt = f*g''
yxx = f''*g
f*g'' - c2*f''*g = 0
g''/g = c2*f''/f = constant = -ω2
ω = cn*∏/L k = n*∏/L
g'' + ω2*g = 0 --> g = A1*cos(ωt) + A2*sin(ωt)
f'' + ω2/c2*g = 0 --> f = B1*cos(kx + B2*sin(kx)
f(0) = B1 f(L) = B2*sin(n*∏*x/L)
y(0,t) = y(L,t) = 0
y(x,t) = f(x)*g(x)
= ∑ cn*cos(ωn*t+øn) * B2*sin(n*∏*x/L)
The biggest problem I'm having is we aren't given an initial value for x. I'm sure you can find one with the second partial differential equation given but I don't know how to go about that.
We have been covering Fourier series lately and this looks a lot like the form of Fourier series but I also don't know how to convert it. Finally, I don't know what a mode is. Does it have to do with the nodes and anti-nodes? Someone told me it's the amount of sets of different nodes/anti-nodes(if that makes sense) but wouldn't that be infinite?
thanks