Question

Two sinusoidal waves travelling in opposite directions interfere to produce a standing wave described by the equation
 $y=\left(1.5m\right)\sin \left(0.400x\right)\cos \left(200t\right)$
Where, $x$ is in meters and $t$ is in seconds. Determine the wavelength, frequency and speed of the interfering waves.

Answer 1

Hint : Compare the given equation with the general equation of standing wave, which is $y=2A\sin kx\cos wt$ and here the expression of $k$ is ${\displaystyle \frac{2\pi }{\lambda }}$ .

Complete step by step answer
From the question, we know that the equation of the standing waves is given as $y=\left(1.5m\right)\sin \left(0.400x\right)\cos \left(200t\right)$ .
We know that the standard equation of stationary waves.
 $y=2A\sin kx\cos wt$ where $k={\displaystyle \frac{2\pi }{\lambda }}$ and $\omega =2\pi f$ .
Here, $\lambda$ is the wavelength, $\omega$ is the angular wavelength, $f$ is the frequency and $k$ is angular wave number.
Now we compare the standard equation of standing wave with the given equation, we get,
 $k=0.4$ and $\omega =200\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ .
Now we substitute the values in the angular wavenumber formula, we get,
$$\begin{gathered} 0.4={\displaystyle \frac{2\pi }{\lambda }} \\ \lambda ={\displaystyle \frac{2\pi }{0.4}} \\ =15.7\mathrm{m} \end{gathered}$$
Thus, the wavelength of the inferring wave is 15.17 m.
Now we substitute the values in the angular velocity formula, we get,
$$\begin{gathered} 200=2\pi f \\ f={\displaystyle \frac{2\pi }{200}} \\ =31.8\mathrm{H}\mathrm{z} \end{gathered}$$
Thus, the frequency of the inferring wave is 31.8 Hz.
The speed of the interfering eaves is expressed as,
 $v={\displaystyle \frac{\omega }{k}}$
Now we substitute the values of $\omega$ as 200 rad/s and $k$ as 0.4 in the above expression. We get,
$$\begin{gathered} v={\displaystyle \frac{200}{0.4}} \\ =500\mathrm{m}/\mathrm{s} \end{gathered}$$
Thus, the velocity of the inferring wave is 500 m/s.

Note
Stationary (standing) waves are produced when two waves having equal speed, frequency, and amplitude are travelling in opposite directions. One of the common examples of stationary waves is plucking of a string of guitar or violin. The general equation of stationary wave is given as $y=2A\cos kx\sin \omega t$ .