Physics 102 HW page

The figure represents the situation in the air column at time t=0. The solid curve is a plot of the displacement of the air at the time depicted. Where the graph is above 0 the air is displaced to the right over the whole crosssection of the column, and where it is below 0 the air is displaced to the left. The little arrows drawn inside the column also represent the displacement of the air. The long-dashed curve shows the displacement of the air at the later time t=T/2, i.e. one half period later.

From the displacement patterns one can deduce where the pressure is high and where it is low. Places away from which air has displaced are low in pressure, while areas where air is converging from both sides have higher than background pressure. Places where the air displacement has reached a maximum and are beginning to decrease have no net gain or loss of air and so are at the background pressure. This information concerning the pressure can all be put together in a single plot, the dotted one in the top figure. Note that this curve is also a sinusoidal wave, but it is 90¡ out of phase with curve that describes the air displacement. Having drawn the pressure curve, one can identify the maxima, minima and nodes at time t = 0. Note that there are pressure nodes at the ends of the tube. Since air flows freely in and out at the ends, the pressure at open ends stays at atmospheric pressure. At a closed end, which this column does not have, there would be no allowed displacement and this would correspond to a pressure maximum or minimum depending on where in the cycle one is.

Note that in this case the wavelength of the oscillation is exactly the same as the tube length. Since c = lf, so f = c/L, where L is the column length. For a column 0.33 m in length, with c = 330 m/s, we get that f=330m/s/0.33m=1000Hz, an audible frequency.

Your assignment is to draw the situation at a later time, at t=T/2, i.e. one half period later in time. You should include this sketch: (1) a curve showing the displacement variation along the tube (essentially copying the long-dashed curve from the above figure); (2) the pressure variation as a dotted curve; (3) little arrows representing the displacement vs. position along the column; and (4) the location of the pressure maxima, minima and nodes.

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Physics 102 HW page
Solution

Physics 102 HW page