Question

A wave is represented by the equation $y = A\sin \left( {10\pi x + 15\pi t + \dfrac{\pi }{3}} \right)$, where x is in meters and t in seconds. The expression represents
(multiple choice correct)
A. wave is traveling in the positive x-direction with a velocity 1.5m/s
B. wave is traveling in the negative x-direction with a velocity 1.5m/s
C. wave is traveling in the positive x-direction with a wavelength 2m
D. wave is traveling in the negative x-direction with a wavelength 2m

Answer 1

The direction of motion of a wave is determined by the signs of x and t. The velocity of a wave is determined by the frequency times the wavelength. The wavelength is related to the coefficient of t in the wave equation.

Complete step by step answer:
In this question a wave question is given, and we are asked to comment on the direction in which it is moving, its velocity, and its wavelength.
Before that, let us understand what the wave equation signifies. In the wave equation given
y is the displacement of the wave about its axis
A is the amplitude of the wave (maximum displacement about the axis i.e. maximum value of y)
x is the position of the wave along the direction of the motion (we assume the wave to be moving in x direction)
t is the time
Now that we know what all the terms in the wave equation signifies, we can compare it to the general wave equation to find out what the constant terms mean. The general wave equation is
$y = A\sin \left( {kx + \omega t + \phi } \right)$
On comparing we find out
ω is the angular frequency of the wave (ω = 15π)
k is the angular wavelength of the wave (k = 10π)
ϕ is the initial angular phase of the wave $\left( {\phi = \dfrac{\pi }{3}} \right)$
To determine the direction in which the wave is moving, we must check the signs of kx and ωt. If their signs are the same, then it moves in the negative x-direction, otherwise if their signs are opposite, then it moves in the positive x-direction. This is because t is always positive, but x can be positive and negative depending upon the direction of motion.
The velocity of the wave is defined as frequency times wavelength. Therefore, it is given by
\omega = 2\pi \nu ,k = \dfrac{{2\pi }}{\lambda } \\\ \nu = frequency \\\ \lambda = wavelength \\\
{V_{wave}} = \nu \lambda \\\ V = velocity \\\ \therefore {V_{wave}} = \dfrac{\omega }{{2\pi }} \times \dfrac{{2\pi }}{k} = \dfrac{\omega }{k} = \dfrac{{15\pi }}{{10\pi }} = 1.5m{s^{ - 1}} \\\
And wavelength is given by
k = \dfrac{{2\pi }}{\lambda } \\\ \therefore \lambda = \dfrac{{2\pi }}{k} = \dfrac{{2\pi }}{{10\pi }} = 0.2m \\\
Comparing the options, we find that B is correct.

So, the correct answer is “Option B”.

Note:
While comparing the equation of wave with the general wave equation, always keep in mind to check the signs of all the coefficients carefully, otherwise it may lead to wrong answers.