Question

A sinusoidal wave is travelling along a rope. The oscillator that generates the wave completes $60$ vibrations in $30{\text{ s}}$ . Also a given maximum travels $425{\text{ cm}}$ along the rope in $10.0{\text{ s}}$ . What is the wavelength of the wave?

Answer 1

The velocity is the distance travelled along the rope by the time taken to travel that distance. The frequency is the number of vibrations by the total time taken to generate those vibrations. The wavelength is velocity by frequency. Frequency is defined as the number of repetitions per second and is expressed in Hertz. The velocity of a wave is the speed with which a specific part of the wave passes a given point.

Complete step by step solution:
Wavelength of a wave is defined as the distance between two successive troughs or crests of the wave. It is generally measured in the direction of propagation of the wave. It is inversely proportional to the frequency. This means the wavelength is longer if the frequency is less.
Since velocity can be calculated as distance over time so,
Velocity of the wave is calculated as:
$v = \dfrac{{425}}{{10}} = 42.5cm{s^{ - 1}}$
Since, frequency is the number of vibrations per second so,
Frequency of the wave is calculated as: ${\text{f = }}\dfrac{{60}}{{30}} = 2{\text{Hz}}$
and, wavelength is just the ratio of velocity and frequency of the wave so,
Wavelength of the wave is calculated as: $\lambda = \dfrac{{\text{v}}}{{\text{f}}} = \dfrac{{42.5}}{2} = 21.24{\text{cm}}$
Therefore, the wavelength of the wave is $21.24{\text{cm}}$

Note: The sinusoidal wave is a curve which describes a smooth repetitive oscillation. It can be defined as a wave form in which the amplitude is always proportional to the sine of the displacement angle of the wave at every point of time. Sine wave is called a sinusoidal wave because of the waveform shape that is produced by a simple single loop generator. These types of waves are called sine waves because they are based on the trigonometric sine functions.