Application of Fourier series to pressure waves

[Sam Harrison]

Homework Statement

Assume that a pressure wave produces a change in pressure at a point in space $$\Delta P(t)$$ which is proportional to a sawtooth function of frequency f = 1/2 Hz.

(i) If the amplitude of the pressure wave is $$\Delta P_{0}$$, write down an expression for $$\Delta P(t)$$.

(ii) Two oscillators, designed to respond to changes in pressure, resonate at a frequency f = 3/2 Hz and f = 5/4 Hz respectively. When the pressure wave encounters the oscillators, which of these will resonate and why?

Homework Equations

The Fourier expansion of a sawtooth function $$f(x) = x, -1 < x < 1$$ is given by

$$f(x) = \sum_{r=1}^{\infty} \frac{-2(-1)^{r}}{\pi r} \sin{\pi r x}$$

The Attempt at a Solution

Is the previous equation simply the answer to part (i) with the x's replaced with t's? Or is there a bit more to it? I think I'm missing something pretty obvious.

For part (ii), we know that $$r \pi t = 2 \pi ft$$ and hence $$r = 2f$$. For f = 3/2 Hz, r = 3 and for f = 5/4 Hz, r = 5/2. Would only the first resonate as it is the only one that produces r that is an integer and hence satisfies the expansion above?

 

[vela]

Homework Statement

Assume that a pressure wave produces a change in pressure at a point in space $$\Delta P(t)$$ which is proportional to a sawtooth function of frequency f = 1/2 Hz.

(i) If the amplitude of the pressure wave is $$\Delta P_{0}$$, write down an expression for $$\Delta P(t)$$.

(ii) Two oscillators, designed to respond to changes in pressure, resonate at a frequency f = 3/2 Hz and f = 5/4 Hz respectively. When the pressure wave encounters the oscillators, which of these will resonate and why?

Homework Equations

The Fourier expansion of a sawtooth function $$f(x) = x, -1 < x < 1$$ is given by

$$f(x) = \sum_{r=1}^{\infty} \frac{-2(-1)^{r}}{\pi r} \sin{\pi r x}$$

The Attempt at a Solution

Is the previous equation simply the answer to part (i) with the x's replaced with t's? Or is there a bit more to it? I think I'm missing something pretty obvious.

You have to get the amplitude right as well.

For part (ii), we know that $$r \pi t = 2 \pi ft$$ and hence $$r = 2f$$. For f = 3/2 Hz, r = 3 and for f = 5/4 Hz, r = 5/2. Would only the first resonate as it is the only one that produces r that is an integer and hence satisfies the expansion above?

Yes. The sawtooth pressure wave contains harmonics with frequencies equal to integral multiples of 1/2 Hz, so it has a 3/2-Hz component but lacks a 5/4-Hz component and will excite only the one resonator.