Question

Two sinusoidal waves with the same amplitude and wavelength travel through each other along a string that is stretched along an xaxis. Their resultant wave is shown twice in Fig. 16-41, as the antinode Atravels from an extreme upward displacement to an extreme downward displacement in. The tick marks along the axis are separated by 10 cm; height His 1.80 cm. Let the equation for one of the two waves is of the form $\mathit{y}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}{\mathit{y}}_{\mathbf{m}}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{k}\mathit{x}\mathbf{+}\mathit{\omega }\mathit{t}\mathbf{)}$.In the equation for the other wave, what are (a)${\mathit{y}}_{\mathbf{m}}$, (b) k, (c) $\mathit{\omega }$, and (d) the sign in front of$\mathit{\omega }$?

Short answer

a) The value of$y_m$ for the other wave is 4.5 mm

b) The value of k for the other wave .is$16{\mathrm{m}}^{-1}$

c) The value of$\omega$ for the other wave is$5.2\times {10}^2\frac{rad}{s}$

e) The sign of$\omega$ or the other wave is negative.

Step-by-step solution

Given data

Figure for the resultant wave is given.

An antinode A travels from extreme upward to extreme downward in time, t=6.0 ms

Tick marks along the axis are separated by,x=10 cm

Height is, H=1.80 cm

One of the superimposed wave is of the form,$y(x,t)=y_{m}sin(kx+\omega t)$

Understanding the concept of the standing wave

We can find the valueof the amplitude of the standing wave by comparing the given equation with the general equation for the standing wave. The wavelength and period of the standing wave can be predicted from the given figure. The frequency of the wave can be calculated from the period using the corresponding formula. This can be used to find the angular speed. From the equation of the other wave, we can find the sign of angular speed.

Formulae:

The wavenumber of the wave,$k=\frac{2\mathrm{\pi }}{\lambda }.......\left(1\right)$

The frequency of the wave,$f=\frac{1}{T}.....\left(2\right)$

The angular frequency of the wave,$\omega =2\mathrm{\pi f}\dots ..\left(3\right)$

Step 3(a): Calculation for value of ym of  the other wave

To form a standing wave, the equations of waves should be

$y\left(x,t\right)=y_m\sin \left(kx+\omega t\right).....\left(4\right)$

$y\left(x,t\right)=y_m\sin \left(kx-\omega t\right).....\left(5\right)$

Therefore, according to the superposition principle, the equation of the resultant wave is

$y^{\text{'}}=y_{m}sin(kx+\omega t)+y_{m}sin(kx-\omega t)$

$y^{\text{'}}=\left(2y_{m}sin\left(kx\right)cos\omega t\right)$

From the given figure we can write that the amplitude of the standing wave of the two waves is given as:

$$\begin{gathered} y_m^{\text{'}}=\frac{H}{2} \\ =\frac{1.80}{2} \\ =0.9\mathrm{cm} \\ =9.0\mathrm{mm} \end{gathered}$$

The amplitude of one of the waves that superimposes to form the given standing wave is given as:

$$\begin{gathered} y_m=\frac{y_m^{\text{'}}}{2} \\ =\frac{9}{2} \\ =4.5mm \end{gathered}$$

Therefore, the value of$y_m$ for the other wave is 4.5 mm

Step 4(b): Calculation of k for the other wave

From the given figure, we can infer that the wavelength of the standing wave is

$\mathrm{\lambda }=40\mathrm{cm}$

Using equation (1), we get the wavenumber as:

$$\begin{gathered} \mathrm{k}=\frac{2\left(3.142\right)}{\left(40\right)} \\ =0.1571{\mathrm{cm}}^{-1} \\ =15.71{\mathrm{m}}^{-1} \\ ~16{\mathrm{m}}^{-1} \end{gathered}$$

Therefore, the value of k for the other wave is $6{\mathrm{m}}^{-1}$.

Step 5(c): Calculation of value of angular frequency

An antinode A travels from extreme upward to extreme downward in time t=6.0 ms

Therefore period of the wave is given as:

$$\begin{gathered} T=12\mathrm{ms} \\ =12\times {10}^{-3}\mathrm{s} \end{gathered}$$

Using equation (2) and the time, we get the frequency of the other wave as:

$$\begin{gathered} f=\frac{1}{12\times {10}^{-3}} \\ =83.33\mathrm{Hz} \end{gathered}$$

Using equation (3) and the given value of frequency, the angular velocity of the wave is given as:

$$\begin{gathered} \omega =2\left(3.142\right)\left(83.33\right) \\ =523.46 \\ ~5.2\times {10}^2\frac{\mathrm{rad}}{\mathrm{s}} \end{gathered}$$

Therefore, the value of for the other wave is$5.2\times {10}^2\frac{rad}{s}$

Step 6(d): Finding the sign of the angular frequency of the other wave

The other wave equation is

$y(x,t)=y_{m}sin(kx-\omega t)$

Therefore, the sign of role="math" localid="1661162543996" $\omega f$or the other wave is negative.