Step 1: Identify the general form of a progressive wave equation and compare it with the given equation. The general equation for a progressive wave moving in the positive x-direction is: $$ y = A \sin(\omega t - kx) $$ The given wave equation is: $$ y = a \sin\left(150 \pi t - \frac{\pi x}{4}\right) $$ By comparing these two equations, we can see that the term representing the phase constant due to position $x$ is $\frac{\pi x}{4}$. This term represents the phase difference from the origin at a given position $x$. Step 2: Convert the given phase difference from degrees to radians. The problem states that the phase difference is $45^\circ$. To use this in the equation, we must convert it to radians: $$ \text{Phase difference} = 45^\circ \times \frac{\pi \text{ radians}}{180^\circ} $$ $$ \text{Phase difference} = \frac{\pi}{4} \text{ radians} $$ Step 3: Equate the phase constant from the equation to the given phase difference. The phase constant in the wave equation is $\frac{\pi x}{4}$. We are given that this phase difference is $\frac{\pi}{4}$ radians. $$ \frac{\pi x}{4} = \frac{\pi}{4} $$ Step 4: Solve for $x$. To solve for $x$, we can multiply both sides of the equation by 4: $$ \pi x = \pi $$ Then, divide both sides by $\pi$: $$ x = \frac{\pi}{\pi} $$ $$ x = 1 $$ Step 5: State the units. The options are given in cm, so the value of $x$ is $1 \text{ cm}$. The final answer is $\boxed{\text{D}}$. Last free one today — make it count tomorrow, or type /upgrade for unlimited.