Question

A sinusoidal wave travels along a string. The time for a particular point to move from maximum displacement to zero is 0.170s. (a)What are the period and (b)What is the frequency? (c)What if the wavelength is 1.40m; what is the wave speed?

Short answer

1. The time period of the wave is 0.680 s
2. The frequency of the wave is 1.47 Hz
3. The speed of the wave is 2.06 m/s

Step-by-step solution

The given data

• The time for a particular point to move from a maximum to zero, t=0.170 s
• The wavelength of the wave,$\lambda =1.40\mathrm{m}$

Understanding the concept of the wave equation

$v=\frac{\lambda }{T}$The sinusoidal wave traveling along a string can be described using the standard equation. The period is the inverse of frequency. These quantities can be calculated by using their defining equations.

Formula:

The frequency of the wave equation,$f=\frac{1}{T}$ (i)

The speed of the wave in terms of wavelength, $v=\frac{\lambda }{T}$ (ii)

(a) Calculation for the period of the wave

The period is defined as the time taken by the particle to move from the 1^st maximum to the next maximum. The time taken by the time to move from a maximum to zero will be one-fourth of this period. So we can calculate the time period by substituting the value of time.

$$\begin{gathered} t=\frac{T}{4} \\ T=4\times t \\ =4\times 0.170\mathrm{s} \\ 0.680\mathrm{s} \end{gathered}$$

Hence, the value of period is 0.680 s

Step       4:     (b)       Calculation      for       frequency      of       the          wave

Using equation (i) and the derived time period, the frequency of wave is calculated. Substitute the value of the time period in equation (i).

$$\begin{gathered} f=\frac{1}{0.680\mathrm{s}} \\ =1.47\mathrm{Hz} \end{gathered}$$

Hence, the value of the frequency is 1.47Hz

(c) Calculation of the speed of the wave

Using equation (ii) and the given value of wavelength, the wavespeed is calculated. Substitute the value of wavelength and time period in equation (ii).

$$\begin{gathered} v=\frac{1.40\mathrm{m}}{0.680\mathrm{s}} \\ =2.06\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of speed is 2.06 m/s