Question

A travelling wave pulse is given by $y\text{ }=\text{ }\dfrac{4}{(3{{x}^{2}}~+\text{ }48{{t}^{2}}~+\text{ }24xt\text{ }+\text{ }2)}$ where s and y are in metre and t in sec. the velocity of wave is

1. 4m/s
2. 2m/s
3. 8m/s
4. 12m/s

Answer 1

A pulse wave, often known as a pulse train, is a non-sinusoidal waveform that comprises square waves with a 50% duty cycle and asymmetrical waves with a comparable duty cycle (duty cycles other than 50 percent ). It's a word used frequently in synthesiser programming, and it's a popular waveform seen on many synthesisers. The duty cycle of the oscillator determines the exact form of the wave. For a more dynamic timbre, the duty cycle can be modified (also known as pulse-width modulation) in several synthesisers. The rectangular wave, or pulse wave, is the periodic form of the rectangular function.

Complete step by step solution:
A displacement is a vector in geometry and mechanics whose length is the smallest distance between the beginning and final positions of a moving point P. It measures the distance and direction of net or total motion along a straight line from the point trajectory's beginning location to its end position. The translation that translates the original position to the end position can be used to identify a displacement.
The rate of change of the displacement as a function of time is the instantaneous velocity of the item when examining movements of objects throughout time. The instantaneous speed, on the other hand, is distinct from velocity, which is the rate at which the distance travelled along a given path changes over time. The velocity may be defined as the rate of change of the position vector over time. A relative velocity is computed with respect to a moving initial position, or equivalently a moving origin, as opposed to an absolute velocity, which is computed with respect to a point that is considered to be 'fixed in space.'
The denominator is 0 for maximum displacement.
Hence
$\text{0 }=\text{ }(3{{x}^{2}}~+\text{ }48{{t}^{2}}~+\text{ }24xt\text{ }+\text{ }2)$
Hence
$3 x^{2}+48 t^{2}+24 x t+2=0$
So
$\left({\sqrt{3} x}\right)^{2}+(4 \sqrt{3} t)^{2}+2 \times \sqrt{3} x \times 4 \sqrt{3}+2=0$
Which becomes
$(\sqrt{3} \mathrm{x}+4 \sqrt{3} \mathrm{t})^{2}+2=0$
For velocity, $(\sqrt{3} \mathrm{x}+4 \sqrt{3} \mathrm{t})=0$
Since we know that $\mathbf{v}=\dfrac{\text{dx}}{\text{d}t}$
Upon differentiating
or $\mathrm{dx} / \mathrm{dt}=\mathrm{v}=-\dfrac{4 \sqrt{3}}{\sqrt{3}}=-4 \mathrm{~m} / \mathrm{s}$
$v=-4m{{s}^{-1}}$
Hence option A is correct

Note:
A pulse wave can be created by subtracting a sawtooth wave from a phase-shifted version of itself. If the sawtooth waves are bandlimited, the resulting pulse wave is bandlimited, too. A single ramp wave (sawtooth or triangle) applied to an input of a comparator produces a pulse wave that is not bandlimited. A voltage applied to the other input of the comparator determines the pulse width.