Ncert Exercises & Solutions

The displacement of the string is given by,
$y (x, t) = 0.06 \sin (\frac{2 πx}{3}) \cos (120 πt)$
where x and y are in metres and t in seconds. The length of the string is 1.5 m and its mass is $3.0 \times 10^{- 2} kg .$

(a) It represents a progressive wave of frequency 60 Hz.

(b) It represents a stationary wave of frequency 60 Hz.

(c) It is the result of a superposition of two waves of wavelength 3 m, frequency 60 Hz each travelling with a speed of 180 m/s in the opposite direction.

(d) The amplitude of this wave is constant.

Choose the correct alternatives:

1. (b, d)

2. (a, c)

3. (b, c)

4. (c, d)

(2) Hint: Use the standard equation of a stationary wave.

Given equation is,

y (x, t) = 0.06 \sin (\frac{2 πx}{3}) \cos (120 πt)

Step 1: Find if it is a stationary wave.

1. Comparing it with a standard equation of stationary wave,

y (x, t) = asin (kx) \cos (ωt)

Clearly, the given equation belongs to the stationary wave. Hence, option 1 is not correct.

Step 2: Find the frequency of the wave.

2. By comparing;

ω = 120 π

\Rightarrow 2 πf = 120 π \Rightarrow f = 60 Hz

Step 3: Find the wavelength of the wave.

k = \frac{2 π}{3} = \frac{2 π}{λ}

\Rightarrow λ = wavelength = 3 m

Frequency= f = 60 Hz

Speed = v=

fλ

= (60 Hz) x (3m) = 180 m/s

4. In a stationary wave, all particles of the medium execute SHM with varying amplitude nodes.

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