MHT CET · Physics · Waves and Sound

Find the wrong statement form the following about the equation of stationary wave given by $Y = 0.04 \cos (π x) \sin (50 π t)$ m where t is in second. Then for the stationary wave.

A Time Period = 0.02s
B Wavelength = 2 m
C Velocity = 50 m/s
D Amplitude = 0.02 m

Verified Solution

Answer & Solution

Correct Answer

(A) Time Period = 0.02s

Step-by-step Solution

Detailed explanation

Key Idea – The displacement of a wave in term of time period is given by

y = A s i n [2 π (\frac{t}{T} - \frac{x}{λ})]

This equation in terms of speed of wave (v),
Becomes

y = A s i n [\frac{2 π}{λ} (v t - x)]

The given equation of wave is

y = 0.04 \cos (π x) \sin (50 π t)

= 0.02 \sin (50 π t + π x) + 0.02 \sin (50 π t - π x)

[∵ 2 s i n A . c o s B = \sin (A + B) . \cos (A - B)]

Thus, the given wave is the combination of two waves,

y_{1} = 0.02 \sin (50 π t + π x) (i n - v e x - d i r e c t i o n)

and

y_{2} = 0.02 \sin (50 π t - π x)

(in +

v e

x-direction)
Comparing them with the general equation of wave

asin (ω t + k x)

, we get
Amplitude, a = 0.02m
Time period, T =

\frac{2 π}{50 π} = \frac{1}{25} = 0.04 s

Wavelength,

λ = \frac{2 π}{π} = 2 m

Velocity, v =

\frac{50 π \times λ}{2 π}

= \frac{100}{2} = 50 m s^{- 1}

So, option (a) is wrong.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Find the wrong statement form the following about the equation of stationary wave given by Y=0.04cos⁡πxsin⁡50πt m where t is in second. Then for the stationary wave.

Step-by-step Solution

Find the wrong statement form the following about the equation of stationary wave given by $Y = 0.04 \cos (π x) \sin (50 π t)$ m where t is in second. Then for the stationary wave.