Text books often give an expression like Asin(kx-ωt) as a solution of the wave equation, but they don’t show how to arrive at this solution. Other textbooks, which go through the complete solution process of the wave equation, determine the coefficients using Fourier series. My goal was to get directly from ∂2y/∂x2 = 1/v2 ∂2y/∂t2 to y(x,t) = Asin(kx-ωt) without determining the coefficient A, because this is the solution given in many texts and it is very useful to solve problems. So I went through the separation of variables process and used the boundary conditions and arrived at the solution y(x,t) = Csin(nkx)(Asin(nωt) + Bcos(nωt)), n = 1,2,3... If I am only interested in the fundamental mode I can drop the n-terms and get y(x,t) = Csin(kx)(Asin(ωt)+Bcos(ωt)) = Asin(kx)sin(ωt) + Bsin(kx)cos(ωt). Using the product formulas for sine and cosine I finally get y(x,t) = A(cos(kx-ωt) - cos(kx+ωt)) + B(sin(kx-ωt) + sin(kx+ωt)). (I absorbed the constant C into the constants A and B, and then I replaced A/2 and B/2 by A and B, because anyway they are just arbitrary constants, so it does not matter what I call them). Now I reason that this last solution is a linear combination of individual solutions. For instance, Asin(kx-ωt) is a wave moving to the right, Asin(kx+ωt) is a wave moving to the left, and the combination of the two is a standing wave. The cosine terms are just shifted versions of the sine terms, so in order to represent a wave moving in one direction the equation y(x,t) = Asin(kx-ωt) is all that is needed. Is this reasoning correct? I also wonder if the n-terms in the complete solution above should not always be equal to 1 in the case of light waves. Light waves don’t have modes, do they?