# Exponential Solution to Wave Equation

#### vancouver_water

I just learned how to derive the wave equation and now I have some questions about it. In my physics text (first year) it simply says (without reason) that the solution to the wave equation is y(x,t) = Acos(kx-wt), where A is the amplitude of displacement, k is the wave number and w is the angular frequency. This makes sense and fits the equation, but why isn't the solution y(x,t) = exp(x+ct), where c is the speed of the wave, valid? It doesn't make sense physically because there is no exponential growth in the amplitude of the wave, but does this come in anywhere? Is there any physical significance to this solution?

PS. I have not taken classes in differential equations yet so it might get explained there, but as of right now I don't know the answer.

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#### Orodruin

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why isn't the solution y(x,t) = exp(x+ct), where c is the speed of the wave, valid?
Any function f(x-ct)+g(x+ct) is a solution to the wave equation. Exponentials in general are not looked at much - they usually do not fit boundary or initial conditions. Plane waves (sines and cosines) are commonly studied as you can build up a large subgroup of the solutions (typically the physically interesting one) using Fourier transforms.

#### hunt_mat

Homework Helper
Simply plug the solution they give you into the wave equation.

#### vancouver_water

Any function f(x-ct)+g(x+ct) is a solution to the wave equation. Exponentials in general are not looked at much - they usually do not fit boundary or initial conditions. Plane waves (sines and cosines) are commonly studied as you can build up a large subgroup of the solutions (typically the physically interesting one) using Fourier transforms.
I think this makes sense to me. So if the wave were along a rope, the boundary conditions would be the values of y(x,t) at certain times and x-positions? and the exponential solution would not fit these boundary values?