The transverse displacement y(x, t) of a wave on a string is given by y(x, t) = e-(ax² + bt² + 2√(ab) xt). This represents a: (a) wave moving in -x direction with speed √(b/a) (b) standing wave of frequency √(b) (c) standing wave of frequency 1/√(b) (d) wave moving in +x direction with speed √(a/b)

Answer: (a). The exponent is a perfect square: ax² + bt² + 2√(ab) xt = (√(a) x + √(b) t)². So y = f(√(a) x + √(b) t) = f(x + √(b/a) t), a function of (x + vt). A function of (x + vt) is a wave moving in the -x direction with speed v = √(b/a). So the answer is (a).

The transverse displacement y(x, t) of a wave on a string is given by… | JEE Main 2011 (AIEEE)

Q89JEE Main 2011 · AIEEEEasy

Progressive Wave Equation

The transverse displacement $y(x, t)$ of a wave on a string is given by $y(x, t) = e^{-\left(ax^2 + bt^2 + 2\sqrt{ab}\,xt\right)}.$ This represents a:

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Answer: (a)

The exponent is a perfect square: $ax^2 + bt^2 + 2\sqrt{ab}\,xt = (\sqrt{a}\,x + \sqrt{b}\,t)^2$ .
So $y = f(\sqrt{a}\,x + \sqrt{b}\,t) = f\left(x + \sqrt{\frac{b}{a}}\,t\right)$ , a function of $(x + vt)$ .
A function of $(x + vt)$ is a wave moving in the $-x$ direction with speed $v = \sqrt{\frac{b}{a}}$ .
So the answer is (a).

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