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 `ms^(-1)`
C. the frequency of oscillation is `500xx2pi` Hz
D. two nearest points in the same phase have separation `20pi//7cm`

Answer

To solve the problem, we need to analyze the wave function given by \( y = \cos(500t - 70x) \) and determine the validity of the statements provided. Let's go through the solution step by step.

### Step 1: Identify the Type of Wave
The wave equation is given in the form:
\[ y = \cos(\omega t - kx) \]
where \( \omega = 500 \) and \( k = 70 \).

**Hint:** A wave is classified as a traveling wave if it can be expressed in the form \( y = \cos(\omega t - kx) \) or \( y = \cos(\omega t + kx) \). A standing wave has a different form.

### Step 2: Determine if it is a Standing Wave
Since the wave is expressed in the form \( y = \cos(500t - 70x) \), it is a traveling wave, not a standing wave.

**Conclusion:** The first option is incorrect.

### 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:
\[ v = \frac{500}{70} = \frac{50}{7} \text{ m/s} \]

**Hint:** The speed of a wave can also be calculated using \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength.

### Step 4: Calculate the Frequency of Oscillation
The relationship between angular frequency \( \omega \) and frequency \( f \) is given by:
\[ \omega = 2\pi f \]
Rearranging gives:
\[ f = \frac{\omega}{2\pi} = \frac{500}{2\pi} \text{ Hz} \]

**Hint:** Remember that frequency is not simply \( \omega \) multiplied by \( 2\pi \), but rather divided by \( 2\pi \).

### Step 5: Find the Wavelength
The wavelength \( \lambda \) can be found from the wave number \( k \):
\[ k = \frac{2\pi}{\lambda} \implies \lambda = \frac{2\pi}{k} \]
Substituting \( k = 70 \):
\[ \lambda = \frac{2\pi}{70} \text{ m} \]

**Hint:** The wavelength is the distance between two successive points in the same phase of the wave.

### Step 6: Check the Separation of Two Nearest Points in the Same Phase
The separation of two nearest points in the same phase is equal to the wavelength \( \lambda \):
\[ \lambda = \frac{2\pi}{70} \text{ m} = \frac{20\pi}{7} \text{ cm} \]

**Conclusion:** The fourth option is correct.

### Final Summary of Results
1. The wave is not a standing wave (incorrect).
2. The speed of the wave is \( \frac{50}{7} \) m/s (correct).
3. The frequency of oscillation is \( \frac{500}{2\pi} \) Hz (correct).
4. The separation of two nearest points in the same phase is \( \frac{20\pi}{7} \) cm (correct).

Thus, the correct options are 2 and 4.