Question

The equation $y = 4 + 2\sin \left( {6t - 3x} \right)$ represents a wave motion with
(A) Amplitude of 6 units
(B) Amplitude of 4 units
(C) Wace speed of 2 units
(D) Wave speed 1/2 units.

Answer 1

We need to compare the equation with the general equation of a wave motion. Investigate one after the other which option would be correct based on the general equation.

Formula used: In this solution we will be using the following formulae;
$y = A\sin \left( {\omega t - kx} \right) + C$ where $y$ is the vertical displacement of the particles of a wave, $A$ is the amplitude of the wave, $\omega$ is the angular frequency of the wave, $t$ is instantaneous time, $k$ is the wave number of the wave, $x$ is the horizontal position of a particle of the wave, and $C$ is the equilibrium line which divides the wave diagram into two equal halves. This equation is the equation of a transverse wave moving along the positive x axis.
$v = \dfrac{\omega }{k}$ .

Complete Step-by-Step solution:
To answer the above equation correctly, we note that the equation given is the equation of a transverse wave moving along the positive x axis, hence we must recall the general equation of a wave of such wave. It is given as
$y = A\sin \left( {\omega t - kx} \right) + C$ where $y$ is the vertical displacement of the particles of a wave, $A$ is the amplitude of the wave, $\omega$ is the angular frequency of the wave, $t$ is instantaneous time, $k$ is the wave number of the wave, $x$ is the horizontal position of a particle of the wave, and $C$ is the equilibrium line which divides the wave diagram into two equal halves.
The equation given is
$y = 4 + 2\sin \left( {6t - 3x} \right)$ which can be written as
$y = 2\sin \left( {6t - 3x} \right) + 4$
Now, we go through the options one after the other,
For option A, the amplitude is not 6 units. As seen, the amplitude is 2 units. Hence A is incorrect.
This at the same time makes option B incorrect.
For option C, we need to calculate the wave speed. This can be given as
$v = \dfrac{\omega }{k}$ .
Hence, we have
$v = \dfrac{6}{3} = 2$ units.
Hence, the correct option is C.

Note:
For clarity, note that the word unit is used continually, and not their respective units such as metres, or metres per second. This is because no unit was defined in the equation given. When such is given in an essay question, it is safe to give your answer in “units” and not to assume SI units.