Question

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string with a speed of10cm/s. If the time interval between instants when the string is flat is0.50s, what is the wavelength of the waves?

Short answer

The wavelength of the waves is 10 m

Step-by-step solution

The given data

i) Velocity of the string, v=10 m/s

ii) Time interval, t =0.50 sec

Understanding the concept of wave motion

The string is flat each time the particle passes through the equilibrium position. The particle may travel towards the positive end or negative end and come back to its equilibrium position. So the time taken during this travel would be half of the period of the wave. Using this we can find the period or frequency of the wave and from that, we can find the wavelength.

Formula:

The frequency of a wave,$f=\frac{1}{t}$ (i)

The wavelength of wave, $\lambda =\frac{v}{f}$ (ii)

Calculation of the wavelength

In the given problem, the total time interval i.e. travelling of particle in positive and negative interval can be given as 2 (0.50s) = 1.0s

Hence, using equation (i) and the total interval, the value of frequency becomes:

$$\begin{gathered} f=\frac{1}{1.0} \\ =1\mathrm{Hz} \end{gathered}$$

Again using the value of frequency in equation (ii), we get the wavelength as follows:
$$\begin{gathered} \mathrm{\lambda }=\frac{10\mathrm{m}/\mathrm{s}}{1\mathrm{Hz}} \\ =10\mathrm{m} \end{gathered}$$

Hence, the value of the wavelength is 10m