Show that the function f(x, t) = x 2 + 4axt−4a 2 t 2 satisfies the wave equation if one assumes a certain relationship between the constant a and the wave speed u. What is this relationship?
The wave equation is
Given
Differentiating it with respect to x, we get
Differentiating it again with respect to x, we get
Differentiating the function with respect to time, we get
Differentiating it again with respect to time, we get
Substituting equation (2) and (3) into equation (1) we get
The relationship between constant a and the wave speed u is
or
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