JSFH
<Text-field style="Heading 1" layout="Heading 1">An unheated house and its temperature through the day</Text-field>Suppose you have a well-insulated but unheated house. How will indoor temperature vary over the day when outside temperature ranges in a sine wave between 35 and 75 degrees? 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Maple doesn't organize things to best advantage here. If you separate out the constant part, you see that T2 is actually 55-A1*cos(w*t)-A2*sin(w*t), where A1:=180/(9+Pi^2) and A2:=60/(9+Pi^2). The overall amplitude of the wave is sqrt(A1^2+A2^2). The wave is phase shifted some. The trigonometry goes like this: we have cos(w*t-phase)=cos(w*t)cos(phase)+sin(w*t)sin(phase). So, the phase shift has sin(phase)/cos(phase)=A2/A1, so phase=arctan(A2/A1). QyQ+SSZ0aGV0YUc2Ii1JJ2FyY3Rhbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUkiKkdGKTYkLCRJI1BpR0YpIiM3IyIiIiIjT0YzQyQ+SSd0aGV0YTJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2I0kmdGhldGFHRiUiIiI=That's a phase shift in terms of radians. We want it in hours. QyQtSSZldmFsZkclKnByb3RlY3RlZEc2IyomSSZ0aGV0YUc2IiIiIkkid0dGKSEiIkYqThe phase shift thus is a matter of a bit over 3 hours. The house warms up and cools down 3 hours behind the outside temperature, and its temperature varies by an amplitude which is sqrt(A1^2+A2^2). QyQ+SSNBMUc2IiwkKiQsJiIiKiIiIiokSSNQaUclKnByb3RlY3RlZEciIiNGKiEiIiIkIT1GKg==QyQ+SSNBMkc2IiwkKiZJI1BpRyUqcHJvdGVjdGVkRyIiIiwmIiIqRioqJEYoIiIjRiohIiIiI2dGKg==QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJXNxcnRHNiI2IywmKiRJI0ExR0YpIiIjIiIiKiRJI0EyR0YpRi5GL0YvThe house moderates the outside temperature swings by to a bit less than 70% of the swings that occur outside. 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Now, let's take up this question of "robust". What happens if the day is shorter? That is, how does the house moderate swings of temperature that go back and forth over a lesser span of time? We'll just reexecute all this, except with a larger value of w. Say we double w. Qyo+SSNtMEc2IiIjYiIiIj5JImtHRiUjRiciIiVGJz5JImJHRiUiIz9GJz5JIndHRiUsJEkjUGlHJSpwcm90ZWN0ZWRHI0YnIiInRic=QyQ+SSJURzYiLCZJI20wR0YlIiIiKipJImtHRiVGKEkiYkdGJUYoLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComRipGKEkidEdGJUYoISIiRigtSSRpbnRHRi42JComLUYtNiNGM0YoLUkkY29zR0YuNiMqJkkid0dGJUYoRjRGKEYoRjRGKEY1Rig=QyQ+SSNUMkc2Ii1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJIlRHRiUiIiI=Qyg+SSNBM0c2IiwkKiQsJiIiKiIiIiokSSNQaUclKnByb3RlY3RlZEciIiMiIiUhIiIiJCE9Rio+SSNBNEdGJSwkKiZGLEYqRihGMCIkPyJGKi1JJmV2YWxmR0YtNiMtSSVzcXJ0RzYkRi1JKF9zeXNsaWJHRiU2IywmKiRGJEYuRioqJEYzRi5GKkYqQyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJ2FyY3Rhbkc2JEYlSShfc3lzbGliRzYiNiMqJkkjQTRHRisiIiJJI0EzR0YrISIiRi8=QyY+SShvdXRzaWRlRzYiLCZJI20wR0YlIiIiKiZJImJHRiVGKC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjKiZJIndHRiVGKEkidEdGJUYoRighIiJGKC1JJXBsb3RHRi02JDckRiRJI1QyR0YlL0YzOyIiISIjW0YoThe variation in response to a quicker wave is a smaller fraction of the original amplitude.