Wave Equation

Writing the phase in the convenient shape $y=A\sin \left[2\pi \left(\frac{t}{T}-\frac{x}{\lambda }\right)\right]$ used in NCERT Class 11, Chapter 15 (Waves), the number multiplying $t$ is $1/T$ and the number multiplying $x$ is $1/\lambda$. Comparing this with the given $y=0.02\sin [2\pi (5t-0.25x)]$, we read off $1/T=5$ Hz and $1/\lambda =0.25$ per metre. The wavelength is therefore the reciprocal of the coefficient of $x$: $\lambda =\frac{1}{0.25}=4$ m. Equivalently, expanding gives a wave number $k=2\pi (0.25)=\pi /2$ rad/m, and $\lambda =2\pi /k=2\pi /(\pi /2)=4$ m, which agrees. The option 2 m mistakenly treats the coefficient as $0.5$ rather than $0.25$. The option 0.25 m reports the coefficient itself instead of taking its reciprocal. The option 10 m confuses wavelength with a quantity built from the temporal term. As a further cross-check one can combine the two readings to find the wave speed: with $T=1/5=0.2$ s and $\lambda =4$ m, the speed is $v=\lambda /T=4/0.2=20$ m/s, a perfectly ordinary value that keeps the whole description internally consistent. A plausibility check: the reciprocal of a spatial frequency of $0.25{\text{m}}^{-1}$ has units of metres and gives a physically reasonable crest spacing of a few metres, confirming both the magnitude and the dimensions of the result.