You have learnt that a travelling wave in a dimension is represented by a funcation `y = f(x,t)` where `x` and `t` must appear in the combination `x-vt` or `x + vt`, i.e `y = f(x+- upsilon t)`. Is the coverse true? (Examine if the following funcations for `y` can possibly represent a travelling wave :
(a) `(x - vt)^(2)`
(b) `log [(x + v)//x_(0)]`
(c) `1//(x + v)`

Explanation

An obvious requirement for an acceptable function for a travelling wave is that it should be finite everywhere and at all times . Only (iii) among the four functions given satisfies this requirement . The remaining functions cannot possibly represent a wave. Note that third function `e^(-(x-Vt)^2)` is finite everywhere at all times because `(x-Vt)^2` is always positive and its maximum value can be `prop` i.e. Minimum value of `e^(-(x-Vt)^2)` can be zero , which is finite .
This last but very important point is that - any function of space and time. [y(x,t)] `(d^2y)/(dx^2)=1/V^2(d^2y)/(dt^2)`
represents a wave ,with speed V, e.g. function Y = A sin `omegat` or y=A sin kx do not satisfy the above equation so do not represent waves, while functions A log (ax+bt) , A sin `(omegat-kx)`, A sin kx sin wt or A sin `(omegat-kx)+B cos (omegat+kx)` satisfy the above equation so represent waves .
It can also be shown that the equation of a travelling wave is of the form : Y =f(at `pm` bx)
Negative sign between at and bx implies that the wave is travelling along positive x-axis and vice-versa .
If a travelling wave is a sin or cos function of (at -bx) or (at + bx) , The wave is said to be harmonic or plane progressive wave
It can also be shown for a plane progressive wave is -
`Y=A sin (omegat-kx)` that
`((dy)/(dt))=-(omega/K)=((dy)/(dx))`
`=-((2pi//T))/((2pi//lambda))((dy)/(dx))`
`=-(eta lambda)(dy)/(dx)=-(V)(dy)/(dx)`
i.e. Particle velocity =-(wave speed ) (slope of wave at that point)
`V_p=-V` (slope )