The graph above was obtained from the National Climatic Data Center (NCDC) Web site -- a good source for data and information about climate and weather. Notice that in Washington, D. C. the hottest time of the year is in July and the coldest time of the year is in January. If you think about your own experience this is probably not surprizing but if you think about what causes the seasonal variation in temperature it may be somewhat unexpected.
Over the course of the year the daily number of hours of sunshine varies. The longest days are around the summer solstice, usually June 22 in the northern hemisphere. The shortest days are around the winter solstice, usually December 22 in the northern hemisphere. Thus, a city like Washington, D.C. in the northern hemisphere receives the most energy from the sun around June 22 and the least energy from the sun around December 22. This might lead you to expect the hottest time of the year to be around June 22 and the coldest time around December 22. But the hottest month of the year is July and it is hotter in August than in June. Similarly the coldest month of the year is January and it is colder in February than in December.
(Aside) The remarks above can lead to an interesting discussion about models and testing models against real-world data. Many people are confused about the cause of the difference between summer and winter temperatures. They believe that the difference is caused by the earth's elliptical orbit, which is closer to the sun at some points than at other points. Their mental model explains the difference between summer and winter temperature by saying that summer is hotter than winter because the earth is closer to the sun during the summer. This model fails its reality check because while the northern hemisphere is enjoying summer the southern hemisphere is enjoying winter. Since the entire earth is at the same point on its orbit, this mistaken mental model would predict that summer occurs at the same time in both hemispheres.
You can observe a similar phenomenon in the way that temperature varies during the day. This variation is caused by the sun's position in the sky. Around noon the sun is almost directly overhead but earlier and later in the day the sun's light arrives at an angle and the sun "feels weaker." During the night we receive no energy from the sun. Thus, you might expect the hottest daily temperatures to be around noon. The buttons below lead to instructions and programs for the TI graphing calculators that will enable you to set up a TI-CBL and temperature probe to record the temperature. With the TI-83, TI-85, TI-86, or TI-92 you can collect up to 512 temperature readings easily. We suggest that you record the temperature over the course of one week, making a total of 504 temperature readings at 20 minute intervals. With the TI-82 we suggest that you record the temperature over the course of two days, making a total of 96 temperature readings at 30 minute intervals. The temperature can change quite dramatically for a variety of reasons -- for example, if a cold front or a warm front passes through. So you may need to do several experiments to begin to see the more general temperature variation over the course of a day.
Click here to open a new window with the March 1997 Tide Table from the Somethin' Fishy home page. Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active. The map below was obtained from the Tiger Map Service maintained by the United States Census Bureau. It will help you locate the area in New Jersey covered by the tide table.
Tides are caused primarily by the gravitational effects of the moon. The gravitational effects of the sun are also a factor. The most extreme tides occur when the moon and the sun are lined up so that their two effects reinforce each other.
The tide table here gives the times of high and low tides in Barnegat Bay or Little Egg Harbor. This table was compiled for fishermen who might be fishing at other sites -- in particular, in nearby inlets or creeks. The tides out in the open water cause tides elsewhere but there is a delay similar to the delay we saw above in the effects of the daily or yearly variation of the sun's effects on temperature. The tide table gives the differences between the high and low tides in Barnegat Bay and in several nearby locations.
After you've examined these differences you should close the tide table window.
So far we've looked at three examples where the variation of one quantity -- air temperature, or the water level in a creek, is driven by the variation of another quantity -- the energy arriving from the sun or the tides in the ocean. In all three examples, the variation of the driven quantity lags behind the variation of the driving quantity. This is an extremely common phenomenon. We can gain some insight into this phenomenon by taking another look at Newton's model of cooling. In our first look at Newton's model of cooling we looked at the discrete dynamical system
where A is the constant ambient temperature and k is a positive constant that is less than one and whose value depends on the object that is cooling (or warming).
Since the temperature is changing continuously, it is most naturally described by a continuous dynamical system or differential equation like the equation below.
In this equation A is still the constant ambient temperature and k is a positive constant whose value depends on the object that is cooling (or warming). If the object is small and poorly insulated then k will be quite large and the temperature of the object will change rapidly. If the object is large and well insulated then k will be small and the temperature of the object will change more slowly.
This differential equation has a linear function on the right hand side and we understand it very well from our earlier work. The constant A is an attracting equilibrium point. This is not surprizing. We know from personal experience that cold objects warm up to room temperature or ambient temperature and that warm objects cool down to room or ambient temperature
Now we are interested in a new twist -- the ambient temperature is not constant. This leads to the differential equation.
in which the constant A has been replaced by a function that depends on time. As an example we look at the differential equation
in which the ambient temperature is oscillating with frequency w.
This differential equation is particularly suitable for the classroom -- not only does it describe important and interesting phenomena but it also illustrates the power of three different ways of working with differential equations -- qualitative (or graphical) methods, numerical methods, and symbolic methods. We begin by looking at this differential equation qualitatively.
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The graph at the right shows the function sin t.
We want to investigate the differential equation
The x-axis in this graph is used for the variable t and the y-axis is used for the variable T. The x-axis runs from 0 to 2 pi and the y-axis runs from -2 to +2. For each value of t and T the right hand side of this differential equation indicates the slope of the solution that passes through this point. Click anyplace inside the graph. A dot will appear at the point and a line will be drawn indicating the slope determined by the right hand side of the differential equation. Notice if you click on the curve then the slope is zero; if you click below the curve the slope is positive; and if you click above the curve the slope is negative. You can click on as many different points as you want. You can clean the graph by clicking on the grey border. |
Based on your work above, draw sketches showing how you think this system will behave over the long term. Draw several different sketches showing how you think the value of the constant k affects the long term behavior and how you think the initial temperature, T(0), affects the long term behavior. Write a few paragraphs about the same questions.
After you have completed the work above open your computer algebra system window and use the numerical methods described in that window to investigate the same questions.
If necessary revise your earlier comments above in view of your numerical experimentation.
So far we have used graphical methods and numerical methods to investigate this situation. Now we apply a little calculus to look at the problem symbolically. For simplicity we look at the equation
but it makes a good exercise for you and your students to use the same methods for the original equation
Based on your numerical experimentation above, it is reasonable to guess that the long term solution of our differential equation
is a function of the form
This function looks like the ambient temperature except that its amplitude is smaller and it is delayed by delta as shown in the graph below.
We want to do two things.
Substituting into the original differential equation we see that
If we substitute t = delta in this equation we see that
and if we substitute t = delta + pi/2 we see that
Hence,
and, looking at the triangle below
we see that
Notice that when k is very large, delta is close to zero and there is very little delay. Also notice that B is close to 1 and the amplitude of the oscillations of the temperature T is close to the amplitude of the oscillations of the ambient temperature.
On the other hand when k is close to zero B is very small and delta is close to pi/2 so the T(t) has very small oscillations and they lag behind the ambient temperature by close to 1/4 of the cycle.
We still need to check that the function
that we found on the basis of our numerical experimentation really does satisfy the original differential equation. This is a matter of straightforward calculation.
The last line above is the left hand side of our differential equation. Now we compute the right hand side.
Since the left hand side and the right hand side are equal, the function T(t) that we found does satisfy our differential equation. This is still not quite a proof that this is the long term behavior of this system. To complete the proof we would have to show that with any given initial conditions the system always approaches this function. We do not have the tools we need to complete the proof but it is a straightforward application of the tools developed in a differential equations course.
One of our most important goals is to recreate in our classrooms the same excitement about mathematics that we live in our careers. One of the most exciting and contentious days in my life as a mathematician occurred at a meeting of mathematicians, including many authors of calculus books, held at Carroll College in 1996. The vigorous part of the discussion began when Ross Finney (Thomas and Finney) presented a slide showing what happened in the walls of a building as the exterior temperature varied. He said that adobe houses were commonly built using bricks whose thickness resulted in a twelve hour lag between the oscillations of the exterior temperature and the oscillations of the interior temperature. A well-known mathematician in the audience leapt to his feet to object that this was impossible. Drawing on his knowledge of Newton's model of cooling with an oscillating ambient temperature --
-- he remarked that the oscillations of the interior temperature could not lag more than six hours behind the oscillations of the exterior temperature. A full and frank discussion ensued, lasted well into the night, and was followed by a flurry of email over the next few days. Think about this question -- Do you think it is possible to build an adobe house whose walls produce a lag of 12 hours between the daily variation of the exterior temperature and the interior temperature?
