So this is a pretty simple question that I just need a bit of clarification on where I am messing up:
$$\frac{∂w}{∂t} + 4 \frac{∂w}{∂x} = 0,~~~ w(0,t) = \sin(3t)$$
My method so far:
$$\frac{dx}{dt} = 4$$
$$x = 4t + x_0$$
$$x_0 = x-4t$$
This is where my problem comes up. Whats the next step? Usually with an initial condition I would simply substitute it into the condition, but how do I proceed from here?
We need a variable change to see the form $f$ has for the solution with the given initial conditions.
We know that $w(x,t)=f(x-4t)$, with $f$ some single argument, differentiable function, and $w(0,t)=\sin(3t)$, so,
$f(-4t)=\sin(3t)$, as you correctly saw.
Defining $u=-4t$, we have $t=-\dfrac{u}{4}$ and $3t=-\dfrac{3u}{4}$. From here we can know the form the formula for $f$ has to have: $f(u)=\sin\left(-\dfrac{3u}{4}\right)$. Finally,
$w(x,t)=\sin\left(-\dfrac{3}{4}(x-4t)\right)$