A travelling wave travelled in string in `+x` direction with `2 cm//s`, particle at `x = 0` oscillates according to equation y (in mm) `= 2 sin (pi t+pi//3)`. What will be the slope of the wave at `x = 3cm` and `t = 1s` ?

Explanation

To find the slope of the wave at \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \), we will follow these steps:

### Step 1: Write down the wave equation
The given wave equation for the particle at \( x = 0 \) is:
\[ y(t) = 2 \sin\left(\pi t + \frac{\pi}{3}\right) \]
Since the wave travels in the \( +x \) direction with a speed of \( 2 \, \text{cm/s} \), we can express the wave equation as:
\[ y(x, t) = 2 \sin\left(\pi t + \frac{\pi}{3} - \frac{\pi x}{2}\right) \]

### Step 2: Differentiate the wave equation with respect to \( x \)
To find the slope of the wave, we need to calculate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left[2 \sin\left(\pi t + \frac{\pi}{3} - \frac{\pi x}{2}\right)\right] \]
Using the chain rule:
\[ \frac{dy}{dx} = 2 \cos\left(\pi t + \frac{\pi}{3} - \frac{\pi x}{2}\right) \cdot \left(-\frac{\pi}{2}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = -\pi \cos\left(\pi t + \frac{\pi}{3} - \frac{\pi x}{2}\right) \]

### Step 3: Substitute \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \)
Now we substitute \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \) into the slope equation:
\[ \frac{dy}{dx} = -\pi \cos\left(\pi(1) + \frac{\pi}{3} - \frac{\pi(3)}{2}\right) \]
Calculating the argument of the cosine:
\[ \pi(1) + \frac{\pi}{3} - \frac{3\pi}{2} = \pi + \frac{\pi}{3} - \frac{3\pi}{2} \]
Finding a common denominator (which is 6):
\[ \pi = \frac{6\pi}{6}, \quad \frac{\pi}{3} = \frac{2\pi}{6}, \quad \frac{3\pi}{2} = \frac{9\pi}{6} \]
Thus:
\[ \frac{dy}{dx} = -\pi \cos\left(\frac{6\pi}{6} + \frac{2\pi}{6} - \frac{9\pi}{6}\right) = -\pi \cos\left(-\frac{1\pi}{6}\right) \]
Using the property of cosine:
\[ \cos(-\theta) = \cos(\theta) \Rightarrow \cos\left(-\frac{\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]
So:
\[ \frac{dy}{dx} = -\pi \cdot \frac{\sqrt{3}}{2} = -\frac{\pi \sqrt{3}}{2} \]

### Final Answer
The slope of the wave at \( x = 3 \, \text{cm} \) and \( t = 1 \, \text{s} \) is:
\[ \frac{dy}{dx} = -\frac{\pi \sqrt{3}}{2} \]

---