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Two travelling waves produces a standing wave represented by equation.
\( y = 1.0 \, \text{mm} \cos\!\big(1.57 \, \text{cm}^{-1} \big)\,x \sin\!\big(78.5 \, \text{s}^{-1} t\big) \). The node closest to the origin in the region \( x > 0 \) will be at \( x = \) ________ \( \text{cm} \).

A galaxy is moving away from the earth at a speed of \( 286 \, \text{km s}^{-1} \). The shift in the wavelength of a redline at \( 630 \, \text{nm} \) is \( x \times 10^{-10} \, \text{m} \).
[Take the value of speed of light \( c \), as \( 3 \times 10^{8} \, \text{m s}^{-1} \)]

Which of the following equations represents a travelling wave?

The amplitude of wave disturbance propagating in the positive \(x\)-direction is given by \(y = \dfrac{1}{(1+x)^{2}}\) at time \(t = 0\) and \(y = \dfrac{1}{1+(x-2)^{2}}\) at \(t = 1\,\text{s}\), where \(x\) and \(y\) are in metres. The shape of wave does not change during the propagation. The velocity of the wave will be _______________ \(\text{m/s}\).

A closed organ pipe of length \( L \) and an open organ pipe contain gases of densities \( \rho_{1} \) and \( \rho_{2} \) respectively. The compressibility of gases are equal in both the pipes. Both the pipes are vibrating in their first overtone with same frequency. The length of the open pipe is \( \dfrac{x}{3} L \sqrt{\dfrac{\rho_{1}}{\rho_{2}}} \) where \( x \) is ______. (Round off to the Nearest Integer)

A tuning fork is vibrating at \( 250 \, \text{Hz} \). The length of the shortest closed organ pipe that will resonate with the tuning fork will be _______ \( \text{cm} \). (Take speed of sound in air as \( 340 \, \text{m s}^{-1} \))

A wire having a linear mass density \( 9.0 \times 10^{-4} \, \text{kg} / \text{m} \) is stretched between two rigid supports with a tension of \( 900 \, \text{N} \). The wire resonates at a frequency of \( 500 \, \text{Hz} \). The next higher frequency at which the same wire resonates is \( 550 \, \text{Hz} \). The length of the wire is ________ \( \text{m} \).

The frequency of a car horn encountered a change from \(400\,\text{Hz}\) to \(500\,\text{Hz}\), when the car approaches a vertical wall. If the speed of sound is \(330\,\text{m/s}\). Then the speed of car is ____________ \(\text{km/h}\).

A sound wave of frequency \( 245\,\text{Hz} \) travels with the speed of \( 300\,\text{m}\,\text{s}^{-1} \) along the positive \( x \)-axis. Each point of the wave moves to and fro through a total distance of \( 6\,\text{cm} \). What will be the mathematical expression of this traveling wave?

Two cars are approaching each other at an equal speed of \( 7.2 \, \text{km/hr} \). When they see each other, both blow horns having frequency of \( 676 \, \text{Hz} \). The beat frequency heard by each driver will be ________ \( \text{Hz} \). \([\text{velocity of sound in air is } 340 \, \text{m/s}]\)

The cars X and Y are approaching each other with velocities \( 36 \, \text{km/h} \) and \( 72 \, \text{km/h} \) respectively. The frequency of a whistle sound as emitted by a passenger in car X, heard by the passenger in car Y is \( 1320 \, \text{Hz} \). If the velocity of sound in air is \( 340 \, \text{m/s} \), the actual frequency of the whistle sound produced is ______ \( \text{Hz} \).

Two waves are simultaneously passing through a string and their equations are: \( y_1 = A_1 \sin k (x - v t) \), \( y_2 = A_2 \sin k (x - v t + x_0) \). Given amplitudes \( A_1 = 12 \, \text{mm} \) and \( A_2 = 5 \, \text{mm} \), \( x_0 = 3.5 \, \text{cm} \) and wave number \( k = 6.28 \, \text{cm}^{-1} \). The amplitude of resulting wave will be __________ mm.

A source and a detector move away from each other in absence of wind with a speed of \( 20 \, \text{m /s}^{} \) with respect to the ground. If the detector detects a frequency of \( 1800 \, \text{Hz} \) of the sound coming from the source, then the original frequency of source considering speed of sound in air \( 340 \, \text{m /s}^{} \) will be ______ \( \text{Hz} \).

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JEE Main 2021 24th Feb Evening Shift

Which of the following equations represents a travelling wave?