Normal modes of continuous systems

[hobmarcus]

Homework Statement

A room has two opposing walls which are tiled. The remaining walls, floors, and ceiling are lined with sound-absorbent material. The lowest frequency for which the room is acoustically resonant is 50Hz.

(a) Complex noise occurs in the room which excites only the lowest two modes, in such a way that each mode has its maximum amplitude at t=0. Sketch the appearance, fro each mode separately, of the displacement versus at t=0,t=1/200sec, and 1/100 sec.

(b) It is observed that the maximum displacement of dust particles in the air (which does not necessarily occur at the same time at each position.) at various points between walls is as follows:

what are the amplitude of each of the two separate modes?

Homework Equations

[/B]

The Attempt at a Solution

for part a, the lowest two modes are just simply 50hz and 100hz?
if yes, the equation of the complex noise is it just x=2Acos(50πt)cos(150πt)
Since ω=2πf and x=2Acos((ω₁+ω₂)/2)cos((ω₁-ω₂)/2)

for part b, I really have no clue how to approach.

Any tips or explanations, please. I appreciate any helps.

 

[BvU]

Hello Marcus, welcome to PF

Yes, the lowest mode is 50 Hz; the exercise text says so. So the next is 100 Hz.
I do wonder if you have a good idea of what happens.
I thought white noise has a flat frequency spectrum and complex noise has a Gaussian frequency spectrum -- anyway, you aren't after the equation of the noise but after the displacement as a function of (missing in problem statement. x ?) for three moments in time.

The tiles reflect the sound waves and for 50 and 100 Hz there is constructive interference of the waves going back and forth and a standing wave pattern emerges. You get displacements $\xi(x,t) = \xi_0 \; \cos(\omega t)\; sin(kx)$ for $\omega = 2\pi \;50$ rad/s and similarly for $\omega = 2\pi \;100$ rad/s (with its own $\xi_0$ !) and the exercise wants you to fill in the three times for each of the modes separately and sketch. Can you post your sketches ?

Once part a) is clear and understood, part b) will become easier to deal with.