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In courses for students in electrical engineering \ at Chalmers University College we have used sound as an experimental tool to \ investigate aspects of mathematics that otherwise tend to elude students.\n\n\ We give examples from two areas: ordinary differential equations and Fourier \ analysis.\nIn the first example, we let ", StyleBox["Mathematica", FontSlant->"Italic"], " solve a differential equation with a parameter and then play the sound of \ the solution for different values of that parameter. The object is to find \ beat and resonance by listening. In the second example, we expand different \ functions in Fourier series and then listen to the functions and partial sums \ of these series." }], "Text"], Cell[CellGroupData[{ Cell["Introduction", "Subtitle"], Cell[TextData[{ "There are two major aspects of learning mathematics: the technical aspect \ (algebraic manipulation, calculus skills) and synthetic understanding \ (formulation, interpretation). For example, on the one hand we want a student \ to be able to compute the derivative of a fairly complex function, using \ formulas and knowledge of the derivative of elementary functions. On the \ other hand we want her to be able to interpret the answer, to understand the \ concept of differentiation and use that understanding to solve problems in \ wider areas. \n\nOften, we focus on the technical aspect. When time is short \ it is much easier to present rules and procedures for doing mathematics. By \ choosing very specific and well-formulated problems and systematically split \ them into smaller parts, we avoid the more demanding task of showing what is \ behind the procedure. Instead of explaining the nature of the problem, we \ show the students a series of short steps: how to take them and the correct \ order in which to take them. Solving a mathematical problem is reduced to \ following a recipe. Understanding and reflection is neither needed nor \ encouraged.\n\nThis is unsatisfying in any circumstance but even more so now, \ when powerful systems like ", StyleBox["Mathematica ", FontSlant->"Italic"], "are available. If our teaching results in the students being able to do \ only a small part of what ", StyleBox["Mathematica", FontSlant->"Italic"], " can do, and with much less reliability, we should question our goals \ altogether.\n\nObviously, we need to shift the balance. And we believe that \ software like ", StyleBox["Mathematica", FontSlant->"Italic"], " can help us do that, not so much by replacing all the technicalities, but \ by providing us with an environment where we can conduct mathematical \ experiments. These experiments should give us a new and if possible radically \ different perspective on different mathematical areas. Hopefully this will \ inspire students to reflect and investigate.\n\nOne such different \ perspective comes from using the notion of sound.\nIn the math courses for \ students in Electrical Engineering at Chalmers University College we have \ concieved a few such experiments. \n\n\nWe like to give credit to the \ excellent book by Glynn and Gray [1], whose chapter on sound was a source of \ inspiration. \nThat sound is a good way to illustrate the notion of beat and \ resonance is pointed out in [2].\n\n\n\n" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "First example: Ordinary differential equations \n\n \"", StyleBox["the quest for the beat\"", FontSlant->"Italic"] }], "Subtitle"], Cell[TextData[{ "The first example is taken from the field of ordinary differential \ equations. This is a sort of field where students fairly quickly absorb the \ technique to solve a given equation but pay little attention to the behaviour \ of the solution and the significance to an underlying problem. The following \ hypothetical dialog illustrates our point:\n\nLecturer: Solve the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(y(t)\), "+", RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "(", "t", ")"}]}], "=", "0"}], TraditionalForm]]], "\nStudent: y=A cos(t)+B sin(t)\nLecturer: \ Solve the equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "(", "t", ")"}], "-", \(y(t)\)}], "=", "0"}], TraditionalForm]]], "\nStudent. ", Cell[BoxData[ \(TraditionalForm\`\(\ y = B\ \[ExponentialE]\^\(-t\) + A\ \[ExponentialE]\^t\)\)]], "\nLecturer: Characterize the solutions of these two equations.\nStudent: \ ?\n\nThe responses given are roughly what we would expect if we posed the \ questions to ", StyleBox["Mathematica", FontSlant->"Italic"], ".\nThe problem is of course that this student does not reflect. \n\nNow \ let us show an experiment that may inspire her to do just that. We call it \ \"the quest for the beat\"\n\nSolve the differential equation ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(a\ \(y(t)\)\), "+", RowBox[{ SuperscriptBox["y", "\[DoublePrime]", MultilineFunction->None], "(", "t", ")"}]}], "=", \(cos(2\ t) + cos(3\ t)\)}], TraditionalForm]]], " if ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["y", "\[Prime]", MultilineFunction->None], "(", "0", ")"}], "=", \(y(0) = 0\)}], TraditionalForm]]], ". \n(Here ", StyleBox["a", FontSlant->"Italic"], " is of course a parameter.) \nThis is the quest: find two values of ", StyleBox["a", FontSlant->"Italic"], ", in the neighbourhood of which a certain phenomenon will appear. Find \ those two values by listening to the solutions for different values of ", StyleBox["a", FontSlant->"Italic"], ".\n\nNote that we do not mention words like beat or resonance. Actually, \ the less the student know of what to expect, the better. On the other hand, \ in our experience, it is a rare student that figures out the correct values \ of the parameter by just looking at the equation. \n\nWe begin by bringing ", StyleBox["Mathematica", FontSlant->"Italic"], " to work:" }], "Text"], Cell[BoxData[ \(\(DSolve[{\(\(y'\)'\)[t] + a\ y[t] == Cos[2 t] + Cos[3 t], \n\t\t \(y'\)[0] == y[0] == 0}, \ y[t], t]; \)\)], "Input"], Cell[TextData[{ "We want the solutions expressed as a function which is dependent of both ", StyleBox["a", FontSlant->"Italic"], " and t." }], "Text"], Cell[BoxData[ \(\(solutionstoplay[a_, t_] = y[t]\ /. \ %[\([1]\)]\ \ // \ FullSimplify; \)\)], "Input"], Cell["\<\ We use FullSimplify since later on we want to be able to look at this \ expression. Although the solutions are periodical, the frequencies are much too low to be \ audiable. We therefore raise the pitch by an appropiate factor f. \ \>", "Text"], Cell[BoxData[ \(\(f = 220\ 2 Pi; \)\)], "Input"], Cell["\<\ Before we begin, we play the two tones on the right hand of our equation:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Play[Evaluate[Cos[2\ f\ t]], \ {t, 0, 3}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Sound \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Play[Evaluate[Cos[3\ f\ t]], \ {t, 0, 3}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Sound \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "These sounds should be kept at hand since they bear an important role in \ the experiment.\n\nNow the quest begins. As little guidance as possible \ should be given, although it is advisable to tell the students to avoid \ integral values of ", StyleBox["a.\n\n", FontSlant->"Italic"], "Below, ", StyleBox["a", FontSlant->"Italic"], "=2.2 is chosen. For every try, the students should listen for some sort of \ deviation, and also compare the rendered sound with the two \"reference tones\ \" cos(2 f t) and cos(3 f t)" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Play[Evaluate[solutionstoplay[2.2, \ f\ t]], \ {t, 0, 3}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Sound \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ Sooner or later, the student will stumble on a value close to 4:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Play[Evaluate[solutionstoplay[3.8, \ f\ t]], \ {t, 0, 3}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Sound \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell["\<\ This is a distinctly different sound. Moving closer to a=4, the effect is \ palpable.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Play[Evaluate[solutionstoplay[3.95, \ f\ t]], \ {t, 0, 3}]\)], "Input"], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Sound \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "Comparing this sound to the sound from cos(2 f t) and cos(3 f t), we find \ that although the sounds are different, the pitch of the sound we got with ", StyleBox["a", FontSlant->"Italic"], "=3.95 is clearly that of cos(2 f t). \n\nThis is also a good time to plot \ a solution and reflect on the graph. 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