The equation of displacement of two waves are given as y 1 = 10 sin ( 3 π t + 3 π ) and y 2 = 5 ( sin 3 π t + 3 cos 3 π t ) . Then the ratio of their amplitudes is

Question

The equation of displacement of two waves are given as y 1 ​ = 10 sin ( 3 π t + 3 π ​ ) and y 2 ​ = 5 ( sin 3 π t + 3 ​ cos 3 π t ) . Then the ratio of their amplitudes is

TestHub · Accepted Answer

The equation of the first wave is given by $y_{1} = 10 sin (3 π t + \frac{π}{3})$

The amplitude of the first wave is $A_{1} = 10$

The equation of the second wave is given by $y_{2} = 5 (sin 3 π t + 3 cos 3 π t)$

Multiplying and dividing the expression inside the bracket by $2$ , we get:

$y_{2} = 5 \times 2 (\frac{1}{2} sin 3 π t + \frac{3}{2} cos 3 π t)$

$y_{2} = 10 (sin 3 π t cos \frac{π}{3} + cos 3 π t sin \frac{π}{3})$

$y_{2} = 10 sin (3 π t + \frac{π}{3})$

The amplitude of the second wave is $A_{2} = 10$

The ratio of their amplitudes is $\frac{A _{1}}{A _{2}} = \frac{10}{10} = 1 : 1$

Answer: $1 : 1$

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