Consider a wave represented by `y=cos(500t-70x)` where y is in millimetres, x in metres and t in second. Which of following are true?

Options

A. The wave is a standing wave
B. the speed of the wave is `50//7 m//s`
C. The frequency of oscillation is `500xx2pi` Hz.
D. Two nearest points in the same phase have separation `20pi//7 cm`

Explanation

To solve the problem step by step, we will analyze the wave equation given and determine the truth of each statement provided.

### Given Wave Equation:
The wave is represented by the equation:
\[ y = \cos(500t - 70x) \]
where \( y \) is in millimeters, \( x \) is in meters, and \( t \) is in seconds.

### Step 1: Identify the parameters
From the wave equation, we can identify:
- Angular frequency \( \omega = 500 \, \text{rad/s} \)
- Wave number \( k = 70 \, \text{rad/m} \)

### Step 2: Check if the wave is a standing wave
A standing wave has the general form:
\[ y = A \cos(kx) \cos(\omega t) \]
The given wave equation does not fit this form, as it is a traveling wave (not a standing wave). Therefore, the first statement is **false**.

### Step 3: Calculate the speed of the wave
The speed \( v \) of the wave can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{500}{70} = \frac{50}{7} \, \text{m/s} \]
Thus, the second statement is **true**.

### Step 4: Calculate the frequency of oscillation
The frequency \( f \) can be calculated using the relationship:
\[ \omega = 2\pi f \]
Rearranging gives:
\[ f = \frac{\omega}{2\pi} = \frac{500}{2\pi} \, \text{Hz} \]
The third statement claims the frequency is \( 500 \times 2\pi \, \text{Hz} \), which is incorrect. Therefore, the third statement is **false**.

### Step 5: Calculate the separation of two nearest points in the same phase
The separation between two points in the same phase is equal to the wavelength \( \lambda \). The wavelength can be calculated using:
\[ k = \frac{2\pi}{\lambda} \]
Thus,
\[ \lambda = \frac{2\pi}{k} = \frac{2\pi}{70} \, \text{m} \]
To convert this to centimeters:
\[ \lambda = \frac{2\pi}{70} \times 100 = \frac{200\pi}{70} = \frac{20\pi}{7} \, \text{cm} \]
The fourth statement claims the separation is \( \frac{20\pi}{7} \, \text{cm} \), which is correct. Therefore, the fourth statement is **true**.

### Conclusion:
The statements that are true are:
- Statement 2: The speed of the wave is \( \frac{50}{7} \, \text{m/s} \) (True)
- Statement 4: The two nearest points in the same phase have separation \( \frac{20\pi}{7} \, \text{cm} \) (True)

### Final Answers:
- **True Statements**: 2 and 4
- **False Statements**: 1 and 3