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?
(a) `y=2cos(3x)sin(10t)` (b) `y=2sqrt(x-vt)`
(c) `y=3sin(5x-0.5t)+4cos(5x-0.5t)` (d) `y=cosx sint +cos2x sin2t`

Answer

To determine whether the given functions represent a traveling wave, a stationary wave, or none at all, we will analyze each function step by step.

### Step 1: Analyze Function (a) `y = 2cos(3x)sin(10t)`

1. This function can be rewritten using the product-to-sum identities:
\[ y = \frac{1}{2} \left[ 2\sin(10t + 3x) - 2\sin(10t - 3x) \right] \]
This shows that it can be expressed as a sum of two traveling waves moving in opposite directions.

2. **Conclusion**: This represents a stationary wave.

### Step 2: Analyze Function (b) `y = 2sqrt(x - vt)`

1. This function does not have a sinusoidal form and does not repeat. It represents a non-harmonic function.

2. **Conclusion**: This does not represent a wave at all.

### Step 3: Analyze Function (c) `y = 3sin(5x - 0.5t) + 4cos(5x - 0.5t)`

1. This function can be rewritten using the sine addition formula:
\[ y = R\sin(5x - 0.5t + \phi) \]
where \( R \) is the resultant amplitude and \( \phi \) is the phase angle. This indicates that it is a combination of two traveling waves.

2. **Conclusion**: This represents a traveling wave.

### Step 4: Analyze Function (d) `y = cos(x)sin(t) + cos(2x)sin(2t)`

1. Each term \( cos(kx)sin(ωt) \) represents a stationary wave. The first term is a stationary wave with frequency \( ω \) and wave number \( k \), and the second term is another stationary wave but with different parameters.

2. The superposition of two stationary waves results in a stationary wave.

3. **Conclusion**: This represents a stationary wave.

### Final Summary:

- (a) **Stationary Wave**
- (b) **None**
- (c) **Traveling Wave**
- (d) **Stationary Wave**

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