Two waves are described by
`y_1 =0.30 sin [pi(5x-200t)] and y_2=0.30 sin [pi(5x-200t)+pi//3]`
where `y_1,y_2` and x are in meters and t is in seconds. When these two waves are combined, a traveling wave is produced. What are the (a) amplitude, (b) wave speed, and (c) wave length of that traveling wave ?

Answer

To solve the problem, we need to analyze the two waves given by the equations:

1. \( y_1 = 0.30 \sin [\pi(5x - 200t)] \)
2. \( y_2 = 0.30 \sin [\pi(5x - 200t) + \frac{\pi}{3}] \)

We will find the amplitude, wave speed, and wavelength of the resultant wave when these two waves are combined.

### Step 1: Find the Resultant Amplitude

The two waves can be combined using the principle of superposition. The resultant amplitude \( A_r \) can be calculated using the formula for the sum of two waves with the same frequency but different phases:

\[ A_r = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\phi)} \]

Where:
- \( A_1 = 0.30 \, \text{m} \) (amplitude of \( y_1 \))
- \( A_2 = 0.30 \, \text{m} \) (amplitude of \( y_2 \))
- \( \phi = \frac{\pi}{3} \) (phase difference between the two waves)

Now substituting the values:

\[ A_r = \sqrt{(0.30)^2 + (0.30)^2 + 2 \cdot (0.30) \cdot (0.30) \cdot \cos\left(\frac{\pi}{3}\right)} \]

Calculating \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):

\[ A_r = \sqrt{(0.30)^2 + (0.30)^2 + 2 \cdot (0.30) \cdot (0.30) \cdot \frac{1}{2}} \]
\[ = \sqrt{0.09 + 0.09 + 0.09} \]
\[ = \sqrt{0.27} \approx 0.519 \, \text{m} \]

### Step 2: Find the Wave Speed

The wave speed \( v \) can be determined from the angular frequency \( \omega \) and the wave number \( k \).

From the wave equation \( y = A \sin(kx - \omega t) \):
- \( k = 5\pi \)
- \( \omega = 200 \)

The wave speed \( v \) is given by the formula:

\[ v = \frac{\omega}{k} \]

Substituting the values:

\[ v = \frac{200}{5\pi} = \frac{40}{\pi} \approx 12.73 \, \text{m/s} \]

### Step 3: Find the Wavelength

The wavelength \( \lambda \) is related to the wave number \( k \) by the equation:

\[ k = \frac{2\pi}{\lambda} \]

Rearranging gives:

\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{5\pi} = \frac{2}{5} \, \text{m} = 0.4 \, \text{m} \]

### Summary of Results

(a) Amplitude \( A_r \approx 0.519 \, \text{m} \)

(b) Wave speed \( v \approx 12.73 \, \text{m/s} \)

(c) Wavelength \( \lambda = 0.4 \, \text{m} \)

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