Question 429: Waves

Question

Equation of two simple harmonic waves are given by $${Y_1} = 2\sin 8\pi \left( {{t \over {0.2}} - {x \over 2}} \right)m$$ and $${Y_2} = 4\sin 8\pi \left( {{t \over {0.16}} - {x \over {1.6}}} \right)m$$ then both waves have

Accepted Answer

The two equations can be written as
$$\mathrm{Y}_1=2 \sin 2 \pi\left(\frac{4 \mathrm{t}}{0.2}-\frac{4 \mathrm{x}}{2}ight)=2 \sin 2 \pi\left(\frac{\mathrm{t}}{0.05}-\frac{\mathrm{x}}{0.5}ight)$$
and $$\mathrm{Y}_2=4 \sin 2 \pi\left(\frac{4 \mathrm{t}}{0.16}-\frac{4 \mathrm{x}}{1.6}ight)=2 \sin 2 \pi\left(\frac{\mathrm{t}}{0.04}-\frac{\mathrm{x}}{0.4}ight)$$
Comparing with standard equation
$$\mathrm{Y}=\mathrm{A} \sin 2 \pi\left(\frac{\mathrm{t}}{\mathrm{T}}-\frac{\mathrm{x}}{\lambda}ight)$$
We get, for the first wave,
$$\mathrm{T}=0.05 \mathrm{~s} 	ext { and } \lambda=0.5 \mathrm{~m}$$
For the second wave,
$$\mathrm{T}=0.04 \mathrm{~s} 	ext { and } \lambda=0.4 \mathrm{~m}$$
Hence their periods (hence frequencies) are not same. Their wavelength are also not same.
For first wave velocity $$=\frac{\lambda}{\mathrm{T}}=\frac{0.5}{0.05}=10 \mathrm{~m} / \mathrm{s}$$
For second wave velocity $$=\frac{0.4}{0.04}=10 \mathrm{~m} / \mathrm{s}$$
Hence velocity in same,

Answer

same period

Answer

same frequency

Answer

same wavelength

Answer

same velocity

Waves

Equation of two simple harmonic waves are given by ${Y_1} = 2\sin 8\pi \left( {{t \over {0.2}} - {x \over 2}} \right)m and {Y_2} = 4\sin 8\pi \left( {{t \over {0.16}} - {x \over {1.6}}} \right)m$ then both waves have

Choose the correct answer: