# Solutions to the wave equation can be represented by a sine function?

Consider the one dimensional wave equation: $$\frac{\partial^2 f(x, t)}{\partial t^2} - c^{2}\frac{\partial^2 f(x, t)}{\partial x^2} = 0.$$

I understand that one may find "wavy" solutions to this equation. But, $f(x, t) = x$ is a solution and it's just a simple linear equation. I'm working through a physics text, and whenever we arrive at a function which satisfies the wave equation, we always write the solution as $A\sin (\omega t - kx)$. I understand that this is a solution to the wave equation, but without some deep theorem stating that "any function which solves the wave equation can be represented as this sine function" I do not feel it is just to assume the function has this form. For the linear example, I don't believe it can be represented by a sine function.

• Have you googled? – Mariano Suárez-Álvarez Feb 16 '16 at 4:07
• There is a rather good wikipedia page on the subject, which is the first result you'll probably get. At least, to guide answerers and avoid waste of effort on their part, you could explain why that page did not satisfy you. – Mariano Suárez-Álvarez Feb 16 '16 at 4:08
• I tried, but I thought it would be easier to come here. – user7348 Feb 16 '16 at 4:09
• If you're talking about the Fourier Theorem, I don't believe it works with my linear example. It only holds for periodic functions. I don't believe the linear function can be represented as a simple sine function. – user7348 Feb 16 '16 at 4:10
• Huh? I am talking about the Wikipedia page very aptly called «Wave equation»! – Mariano Suárez-Álvarez Feb 16 '16 at 4:11

Let $u_1(x, t) = f_0(x -\sqrt{b} t)$ and $u_2(x, t) = f_0(x + \sqrt{b} t)$, we can verify that $u_1(x,t)$ and $u_2(x,t)$ both satisfy the wave equation. The general solution is $u(x, t) = a u_1(x, t) + b u_2(x,t)$. The solution represents the wave front (at the beach, facing ocean, and let the time stop, the wave in front of you is the shape of wave) traveling along the time direction.
• 1. $f_0$ should be replaced with two different functions. 2. The general solution should not have constants, especially those that were already in the equation itself. 3. The waves at the ocean are much better described by the Hopf equation, which is nonlinear. – Artem Feb 16 '16 at 4:02
• The integration constants can be absorbed into the functions. And the general solution also can be written as $\frac{\psi(x-ct)+\psi(x+ct)}{2}+\frac{1}{2c} \int_{x-ct}^{x+ct} \phi(\xi) \, d\xi$. – Ng Chung Tak Feb 16 '16 at 7:46