progressive wave equation is represented by = a sin( \( 150 \pi t-\pi x / 4) \). if the phase difference a progressive wave is \( 45^{\circ} \), the value of \( x \) in e equation is \( \begin{array}{ll}\text { A) } 3 \mathrm{~cm} \\ \text { D) } 4 \mathrm{~cm} \\ \text { D) } 2 \mathrm{~cm}\end{array} \)

To find the value of \( x \) in the given progressive wave equation \[ y = a \sin\left(150 \pi t - \frac{\pi x}{4}\right), \] we need to analyze the phase difference. The phase of the wave is given by \[ \phi = 150 \pi t - \frac{\pi x}{4}. \] The phase difference of \( 45^\circ \) can be converted to radians: \[ 45^\circ = \frac{\pi}{4} \text{ radians}. \] For two points on the wave to have a phase difference of \( \frac{\pi}{4} \), we can set up the equation: \[ \left(150 \pi t - \frac{\pi x_1}{4}\right) - \left(150 \pi t - \frac{\pi x_2}{4}\right) = \frac{\pi}{4}. \] This simplifies to: \[ -\frac{\pi x_1}{4} + \frac{\pi x_2}{4} = \frac{\pi}{4}. \] Dividing through by \( \pi \): \[ -\frac{x_1}{4} + \frac{x_2}{4} = \frac{1}{4}. \] Multiplying through by 4 gives: \[ -x_1 + x_2 = 1. \] This means: \[ x_2 = x_1 + 1. \] Now, we can assume \( x_1 = x \) and \( x_2 = x + 1 \). The distance between the two points is 1 unit (in the same units as \( x \)). To find the value of \( x \) that corresponds to the given options, we can substitute \( x \) with the options provided (in cm): 1. If \( x = 2 \) cm, then \( x_2 = 3 \) cm. 2. If \( x = 3 \) cm, then \( x_2 = 4 \) cm. 3. If \( x = 4 \) cm, then \( x_2 = 5 \) cm. Since we are looking for a phase difference of \( 45^\circ \) (or \( \frac{\pi}{4} \)), we can check the values: - For \( x = 2 \) cm, the phase difference is \( \frac{\pi}{4} \). - For \( x = 3 \) cm, the phase difference is \( \frac{\pi}{4} \). - For \( x = 4 \) cm, the phase difference is \( \frac{\pi}{4} \). Thus, the possible values of \( x \) that yield a phase difference of \( 45^\circ \) are \( 2 \) cm and \( 3 \) cm. However, since the question asks for a specific value, we can conclude that the answer is: **C) 2 cm**.