Question

a) By direct substitution determine which of the following functions satisfy the wave equation.
1. g(x, t) = Acos(kx − $\omega$t) where A, k, $\omega$ are positive constants.
2. h(x, t) = Ae^{$-(kx-\omega t)^2$}
where A, k, $\omega$ are positive constants.
3. p(x, t) = Asinh(kx − $\omega$t) where A, k, $\omega$ are positive constants.
4. q(x, t) = Ae^{$i(-ax^2+\omega t)$} where A, a, $\omega$ are positive constants.
5. An arbitrary function: f(x, t) = f(kx−$\omega$t) where k and $\omega$ are positive constants. (Hint: Be careful
with your derivative rules.)
b) What similarities among functions that satisfy the wave equation do you observe? Do you notice any differences between functions that satisfy the wave equation and those that do not?
c) Verify by direct substitution that an arbitrary linear combination of two functions that satisfy the
wave equation will also satisfy the wave equation. That is, if f(x, t) and g(x, t) both satisfy the wave
equation you are to show that the function F(x, t) = $\alpha$f(x, t)+ $\beta g(x, t)$ also satisfies the wave equation
provided $\alpha$ and $\beta$ are constants.

Answer 1