Final answer:
The linear combination of two wave functions that are solutions to the wave equation is also a solution, due to the principle of superposition. This can be demonstrated by applying trigonometric identities to the sum of two functions, confirming they satisfy the wave equation as a result of its linear characteristic.
Step-by-step explanation:
Proving that a linear combination of two different wave functions is a solution to the wave equation involves showing how these functions behave according to the principle of superposition. For two wave functions y1 (x, t) and y2 (x, t) that are solutions to the linear wave equation, the superposition of these waves is given by Ay1 (x, t) + By2 (x, t), where A and B are constants. This sum must satisfy the same wave equation to be considered a solution.
Let's consider the example of two sine waves y1 (x, t) = A sin (kx - wt) and y2 (x, t) = A sin (kx - wt + φ) with a phase difference. Using trigonometric identities, we can show that the sum, y1 (x, t) + y2 (x, t), results in a new wave function that is also a solution to the wave equation. This is due to the linear nature of the wave equation, which allows for linear combinations of solutions to be solutions as well.
The interference of waves such as this results in complex waveforms that can be analyzed by breaking them down into their component sine waves, each of which is a solution to the wave equation. Thus, the principle of superposition ensures that their sum also solves the wave equation.