Given below are some functions of `x` and `t` to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent `(i)` a travelling wave, `(ii)` a stationary wave or `(iii)` none at all `y = 2 cos (3x) sin (10t)`

Answer

To determine the type of wave represented by the given function \( y = 2 \cos(3x) \sin(10t) \), we will analyze its structure step by step.

### Step 1: Identify the form of the wave equation
The general forms of wave equations are:
- **Traveling Wave**: \( y = A \sin(\omega t - kx) \) or \( y = A \cos(\omega t - kx) \)
- **Stationary Wave**: \( y = A \cos(kx) \sin(\omega t) \)

### Step 2: Analyze the given equation
The given equation is:
\[ y = 2 \cos(3x) \sin(10t) \]

### Step 3: Compare with the standard forms
In the equation \( y = 2 \cos(3x) \sin(10t) \):
- The term \( \cos(3x) \) depends only on \( x \).
- The term \( \sin(10t) \) depends only on \( t \).

This means that the equation is a product of two functions, one depending solely on position \( x \) and the other depending solely on time \( t \).

### Step 4: Conclusion
Since the equation can be expressed as a product of two separate functions (one in \( x \) and one in \( t \)), it matches the form of a stationary wave.

Thus, the function \( y = 2 \cos(3x) \sin(10t) \) represents a **stationary wave**.

### Final Answer
- (i) Traveling wave: None
- (ii) Stationary wave: \( y = 2 \cos(3x) \sin(10t) \)
- (iii) None at all: None

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