Login/Sign Up
Home » The amplitude, frequency and velocity of the component waves producing a stationary wave are and second respectively. Write the equation of stationary wave
The amplitude, frequency and velocity of the component waves producing a stationary wave are 8 \mathrm{~cm}, 30 \mathrm{~Hz} and 180 \mathrm{~cm} / second respectively. Write the equation of stationary wave
Physics | Previous Year Question Papers
The amplitude, frequency and velocity of the component waves producing a stationary wave are 8 \mathrm{~cm}, 30 \mathrm{~Hz} and 180 \mathrm{~cm} / second respectively. Write the equation of stationary wave

Since the waVe equation of a travelling wave:- y=a \sin 2 \pi(t / T-x / \pi)
\mathrm{a}= Amplitude
t= time
\mathrm{T}= Time Period
x= Path difference
\pi= wavelength
Let \mathrm{y}_{1}=\mathrm{a} \operatorname{Sin} 2 \pi\left(\frac{t}{T}-\frac{x}{\pi}\right)
\mathrm{Y}_{2}=\mathrm{a} \operatorname{Sin} 2 \pi\left(\frac{t}{T}+\frac{x}{\pi}\right)
( : : It is travelling in opposite direction)
By principle of superposition, wave equation for the resultant wave =y=y_{1}+y_{2}

    \[\begin{array}{l} y=a \operatorname{Sin} 2 \pi\left(\frac{t}{T}-\frac{x}{\pi}\right)+\mathrm{a} \operatorname{Sin} 2 \pi\left(\frac{t}{T}+\frac{x}{\pi}\right) \\ =a\left(\operatorname{Sin} 2 \pi\left(\frac{t}{T}-\frac{x}{\pi}\right)+\mathrm{a} \operatorname{Sin} 2 \pi\left(\frac{t}{T}+\frac{x}{\pi}\right)\right) \\ \text { Using } \operatorname{Sin} \mathrm{C}+\operatorname{Sin} \mathrm{D}=\mathrm{a} \operatorname{Cos} \frac{C-D}{z} \cdot \frac{\operatorname{Sin} C+D}{z} \\ y=2 a \operatorname{Cos} \frac{2 \pi x}{\pi} \cdot \frac{\operatorname{Sin} 2 \pi \mathrm{t}}{T} \end{array}\]

Here \mathrm{a}=8 \mathrm{~cm} ; \mathrm{f}=30 \mathrm{~Hz}, \mathrm{~V}=180 \mathrm{~cm} / \mathrm{s}

    \[\begin{array}{l} T=\frac{1}{30} \mathrm{sec}, \pi=\text { wavelength }=\frac{v}{f}=V T \\ \pi=180 \times \frac{1}{30}=6 \mathrm{~cm} \end{array}\]

y=2 a \operatorname{Cos} \frac{2 \pi x}{\pi} \operatorname{Sin} \frac{2 \pi t}{T}
y=2 \times 8 \frac{\operatorname{Cos} 2 \pi \times x}{\not 3} \cdot \frac{\operatorname{Sin} 2 \pi t}{\pi \not \beta \varnothing} 15
y=16 \frac{\operatorname{Cos} \pi \mathrm{x}}{3} \cdot \frac{\operatorname{Sin} 60 \pi t}{15}

i

More Material

At a certain location in Africa, a compass points 12^{\circ} west of the geographic north. The north tip of the magnetic needle of a dip circle placed in the plane of magnetic meridian points above the horizontal. The horizontal component of the earth’s field is measured to be 0.16 \mathrm{G} . Specify the direction and magnitude of the earth’s field at the location.At a certain location in Africa, a compass points 12^{\circ} west of the geographic north. The north tip of the magnetic needle of a dip circle placed in the plane of magnetic meridian points above the horizontal. The horizontal component of the earth’s field is measured to be 0.16 \mathrm{G} . Specify the direction and magnitude of the earth’s field at the location.