you have learnt that a travelling wave in one dimension is represented by a function `y = f(x,t)` where `x` and `t` must appear in the combination `ax +- bt` or `x - vt` or `x + vt`,i.e. `y = f (x +- vt)`. Is the converse true? Examine if the folliwing function for `y` can possibly represent a travelling wave
(a) `(x - vt)^(2)`
(b) `log[(x + vt)//x_(0)]`
(c) `1//(x + vt)`

Explanation

To determine whether the given functions can represent a traveling wave, we need to analyze each function based on the criteria for a traveling wave, which states that the wave function must yield a finite value for all values of \(x\) and \(t\).

### Step-by-Step Solution:

1. **Understanding the Criteria for Traveling Waves**:
- A traveling wave can be represented by a function of the form \(y = f(x \pm vt)\), where \(v\) is the wave speed.
- The function must yield finite values for all \(x\) and \(t\).

2. **Examine the First Function: \(y = (x - vt)^2\)**:
- Substitute \(x = 0\) and \(t = 0\):
\[ y = (0 - v \cdot 0)^2 = 0^2 = 0 \]
- The function yields a finite value (0) at \(x = 0\) and \(t = 0\).
- However, the function represents a point rather than a wave since it does not vary with time in a way that creates a wave-like behavior. Thus, it does not represent a traveling wave.

3. **Examine the Second Function: \(y = \log\left(\frac{x + vt}{x_0}\right)\)**:
- Substitute \(x = 0\) and \(t = 0\):
\[ y = \log\left(\frac{0 + v \cdot 0}{x_0}\right) = \log\left(0\right) \]
- The logarithm of zero is undefined (approaches \(-\infty\)), which means the function does not yield a finite value at \(x = 0\) and \(t = 0\). Therefore, it does not represent a traveling wave.

4. **Examine the Third Function: \(y = \frac{1}{x + vt}\)**:
- Substitute \(x = 0\) and \(t = 0\):
\[ y = \frac{1}{0 + v \cdot 0} = \frac{1}{0} \]
- This expression is undefined (approaches \(\infty\)), indicating that the function does not yield a finite value at \(x = 0\) and \(t = 0\). Thus, it cannot represent a traveling wave.

### Conclusion:
- None of the given functions can represent a traveling wave because they do not yield finite values for all \(x\) and \(t\).