Question

The equation of a stationary wave is: $y=4sin(\dfrac{\pi x}{15})cos(96\pi t)$. The distance between a node and its next antinode is
A. 7.5 units
B. 1.5 units
C. 22.5 units
D. 30 units

Answer 1

Hint: We will first compare the general equation of a standing or stationary wave with the given wave equation, which will give us the wavelength of the wave. Then, we will subtract the location of a node from its next antinode or vice versa to get the required result.

Complete step-by-step answer:
The type of wave whose peak amplitude profile does not move in space but which oscillates with time is called a stationary wave.
The generalized equation of stationary wave is given by $y=2Asin(\dfrac{2\pi x}{\lambda})cos(\omega t)$, where A is the amplitude, $\omega$ is the angular frequency, $\lambda$ is the wavelength, is the longitudinal position of wave and t is time.
A node appears where the amplitude is always zero and occurs at locations with even multiples of quarter wavelength. That is $x=…, -\dfrac{\lambda}{2}, 0, \dfrac{\lambda}{2}, \lambda, …$
And antinodes appear where the wavelength is always maximum and occurs at locations with odd multiples of quarter wavelength. That is $x=…, -\dfrac{\lambda}{4}, \dfrac{\lambda}{4}, \dfrac{3\lambda}{4}, …$
Comparing this equation with the given one, we get wavelength, $\lambda =30$ units and frequency f = 48 Hz.
Now, according to question, we need to find out the distance between a consecutive node and an antinode, which will always be $\dfrac{\lambda}{4}=\dfrac{30}{4}\; units=7.5$ units.
Hence, option a is the correct answer.

Note: It has been asked in the question to find out the distance between a node and its next antinode and one may commit a mistake by not subtracting the location of a node with its next antinode or vice versa and that may get a larger distance.