Derivation of d’Alembert’s Formula (Wave Function)

The one-dimensional wave equation is

In such the characteristic equation



By solving, 


By partial differential,



Similarly,



Substitute back to the PDE, it results in:


After arriving the general solution, consider the initial values

By integrating the second equation,



Therefore


By substituting back to the general solution, 






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Evaporation (in Conical flask)

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 By definition of the evaporation, the rate of volume change shall be directly proportional to that of the surface area exposed: In this case assume that the water is evaporating in a conical flask. The evaporation is done by the upper surface exposed to the air (the surface will always be a circle).  The volume of water at any time t would be enclosed in the shape of a frustum as shown. The volume is hence: Subject to the needs of the differential equation, we rearrange it in the form of Area as a function of Volume, By substitution, In which 
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Canonical Forms and General Solutions of 2nd PDE

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Consider the second-order equation in which the derivatives of second-order all occur linearly, with coefficients only depending on the independent variables: Step 1: Solving the characteristic equation is: Step 2: Solve the 1 st order differential equation. Step 3: Define μ(x, y) and η(x, y) as the two “constant” function as the result of the Step 2 Step 4: u = u(μ(x, y), η(x, y)) Step 5: Conduct partial differentiation and substitute back the result into the original PDE, resulting in a Canonical Form. Step 6: Integrate (partial) the simple transformed Canonical Form to arrive General Solution (Note that if all coefficients are Constants, final answer: u =  μ(x, y) +  η(x, y), as the general solution)
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