PDE (wave equation) used to find acoustic pressure in a a pipe

[LinearWave]

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

y_tt = f*g''

y_xx = f''*g

f*g'' - c²*f''*g = 0

g''/g = c²*f''/f = constant = -ω²

ω = c_n*∏/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)

= ∑ c_n*cos(ω_n*t+ø_n) * 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

 

[AlephZero]

What you did so far looks OK.

I don't know what a mode is.

Keeping it simple, an "acoustic mode" is just a non-zero solution of the PDE that satisfies the boundary conditions.

The biggest problem I'm having is we aren't given an initial value for x.

The question says the pipe is of length L, so you want to find solutions for 0 <= x <= L.

Or you could take -L/2 <= x <= L/2 if you prefer - it won't make the math any harder, or easier.

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?

You are told the boundary conditions at a closed and open end of the pipe. If you put those into your solution for f, you will only get non-zero values of B₁ and B₂ for certain values of ω, or k. There will be an infinite number of those values of ω = ω_i, i = 1, 2, 3, ..., so the general solution of the PDE will be an infinite series.

When you have done that, you will probably see the relation between all this and Fourier series, and how it relates to nodes and antinodes.

If you know about eigenvalues and eigenvectors, you will probably also see that the ω_i are rather like eigenvalues and the corresponding solutions for f are rather like eigenvectors.

 

[LinearWave]

Sorry for some of the confusion, by initial value for x i meant there isn't a y(x,0) function given like similar wave functions that use a string.

Also, how will having open versus closed ends affect the boundary conditions?

 

[Orodruin]

The initial conditions will not affect the possible modes. The general solution is a superposition of the modes and can be adjusted to the initial conditions.

Already in your first post you discussed the boundary conditions for a closed pipe, what changes if the pipe end is open?

Note that the modes must fulfill both the differential equation and the boundary conditions (both homogenous). Thus, if you change the boundary conditions, the modes will change.