acoustic impedance in cylindrical tube

[Fei]

I have 2 questions about a straight cylindrical tube placed along x axis, terminated (not rigidly) at one end x=L. (So the acoustic impedance ZL at x=L should normally be a complex number.)

Question 1: I wonder if ZL should be different under 2 cases, (both assuming wavelength is much larger than the tube diameter so plane wave travels in the tube):
first, tube is open at the other end x=0 with incident plane wave travelling into the opening of the tube along the tube's axis, assuming there is no interaction between the sound source and the tube (i.e. free field condition);
second, tube is blocked by a vibrating piston (having an acoustic impedance Z0) at the x=0 end and transmit plane wave into the tube.

Question 2: Under the free field condition as in Question 1, i.e. plane wave incident into the tube from opening at x=0. What would the acoustic impedance Z0 be at the opening x=0? Does Z0 depend on the tube length and ZL, or can it be calculated as the impedance of a piston vibrating in air? (I'm saying this because if tube is open at x=0 with a vibrating piston at x=L, then Z0 can be calculated as a piston vrbrating in air)

I've been puzzled for quite a long time. I'll be very happy to get any of your replies! Thank you!

 

[pheon]

1. The impedance is different in your two cases. Assume a left and right going plane wave in the tube and solve for the boundary conditions at the two ends.

2. I am a little confused about the way you phrased your question. The impedance (which impedance by the way, mechanical, acoustic or specific?) is the ratio of force to velocity, for a fixed velocity at x=0 there will be two waves, one traveling up the tube the other radiating into the air. The force will be the sum of the forces due to those two waves, so the impedance will be the sum of the two impedances. Calculating the effect of a plane wave arriving from the air may be a little more complicated as it is a coupled scattering problem and depends on the geometry.



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