Given below are some functions of x and t to represent the displacement (transverse or longitudinal) of an elastic wave. Some which of these represent (i) a travelling wave, (ii) a stationary wave or (iii) none at all:
(a) `y = 2 cos (3 x) sin (10 t)`
(b) `y = 2 sqrt(x - v t)`
(c ) `y = 3 sin (5 x - 0.5 t) + 4 cos(5 x - 0.5 t)`
(d) `y = cos x sin t + cos 2x sin 2t`

Explanation

(a) It is a product of two harmonic functions, one in terms of x and another in therms of t. Hence, it represents stationry wave.
(b) It does not involve any harmonic function and so it can not represent any elastic wave.
(c ) Here,
`y = 3 sin (5x -0.5t) + 4 cos (5x-0.5t)" "...(1)`
Taking,` 5x-0.5t =theta`
we can write `y = 3 sin theta + 4 cos theta " "...(2)`
Now, we have
`y = A sin (theta + phi)`
`= A (sin theta cos phi + cos theta sin phi)`
`therefore y = (A cos phi) sin theta + (A sin phi) cos theta " "...(3)`
Comparing equation (2) and (3),
`A cos phi =3" "...(4)`
`A sin phi = 4 " "...(5)`
`therefore ` Squaring an dadding we get,
` A ^(2) cos ^(2) phi + A ^(2) sin ^(2) phi =9 + 16`
`therefore A ^(2) (1) = 25 implies A = 5m " "..(6)`
Taking ration of ewuation (5) to equaton (4),
`tan phi = (4)/(3) implies phi tan ^(-1) ((4)/(3)) " "...(7)`
Thus, given wave equation (1), represents harmonic travelling wave with amplitude A=5 m and initial phase (epoch) `phi tan ^(-1) ((4)/(3))`
(d) `y = cos x sin t + cos (2x ) sin (2t)` represent superpositon of two stationary wves given by` y _(1) = cos x sin t and y _(2) = cos (2x) sin (2t).`