Question
Equation of progressive wave is given by
$$y = 4\sin \left[ {\pi \left( {\frac{t}{5} - \frac{x}{9}} \right) + \frac{\pi }{6}} \right]$$
Then, which of the following is correct ?
A.
$$v = 5\,cm$$
B.
$$\lambda = 18\,cm$$
C.
$$a = 0.04\,cm$$
D.
$$f = 50\,Hz$$
Answer :
$$\lambda = 18\,cm$$
Solution :
Equation of plane progressive simple harmonic wave is
$$y = a\sin \left[ {2\pi \left( {\frac{t}{T} - \frac{x}{\lambda }} \right) + \phi } \right]\,......\left( {\text{i}} \right)$$
The given equation is
$$y = 4\sin \left[ {\pi \left( {\frac{t}{5} - \frac{x}{9}} \right) + \frac{\pi }{6}} \right]$$
Multiplying and dividing $$\left( {\frac{t}{T} - \frac{x}{\lambda }} \right)\,{\text{by}}\,2.$$
It is written as,
$$y = 4\sin \left[ {2\pi \left( {\frac{t}{{10}} - \frac{x}{{18}}} \right) + \frac{\pi }{6}} \right]\,......\left( {{\text{ii}}} \right)$$
Comparing Eqs. (i) and (ii), we find
$$\eqalign{
& a = 4\,cm,\,\,T = 10\,s,\,\,\lambda = 18\,cm \cr
& {\text{and}}\,\,\phi = \frac{\pi }{6} \cr} $$
Hence option (B) is correct.