91Ó°ÊÓ

• The wavelength of the wave,$\mathrm{λ}=20\mathrm{cm}$.
• The scale of the vertical axis is set by$y_s=40cm$.
• The wave equation is to be in the form,$y(x,t)=y_{m}sin(kxÂ\pm \mathrm{Ó\neg }t+\mathrm{Ï•})$ .

By using a general expression for a sinusoidal wave traveling along the +xdirection and corresponding formulas, we can find the amplitude, angular wave numberk, angular frequency$\mathit{Ó\neg }$, the phase constant$\mathbf{Ï•}$, the sign in front of$\mathbf{Ó\neg }$, and the speed of the wave vand the transverse velocity of the particle at x=0when t=5.0 s.

Formula:

A general expression for a sinusoidal wave traveling along the +x direction,

$y(x,t)=y_{m}sin(kxÂ\pm \mathrm{Ó\neg }t+\mathrm{Ï•})$ (i)

The angular wave number,$k=\frac{2\mathrm{π}}{\mathrm{λ}}$ (ii)

The angular frequency,$\mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T}$ (iii)

The frequency,$f=\frac{1}{T}$ (iv)

The speed of the wave,$v=f\mathrm{λ}$ (v)

The transverse velocity of the particle, $u\left(x,t\right)=\frac{∂y}{∂t}$ (vi)

A general expression for a sinusoidal wave traveling along the direction using equation (vi) is given as:

$y(x,t)=y_{m}sin(kxÂ\pm \mathrm{Ó\neg }t+\mathrm{Ï•})$

Figure 16-34 shows that at x=0

$y(0,t)=y_{m}sin(-\mathrm{Ó\neg }t+\mathrm{Ï•})…………………………………………………………………….\left(1\right)$

And it is a positive sine function. That is

$y(0,t)=+y_{m}sin\left(\mathrm{Ó\neg }t\right)$

For the sin function, we can write that

From equation (1) and (2), we can say that the phase constant must be

$\mathrm{Ï•}=\mathrm{Ï€}$

At t = 0, we have

$y(x,0)=y_{m}sin(kx+\mathrm{Ï€})$

Using equation (2), we get the displacement equation as:

$y(x,0)=-y_{m}sin\left(kx\right)$.

which is a negative sine function. A plot of $y\left(x,0\right)$is plotted below.

From the figure we see that the amplitude is

$y_m=4.0\mathrm{cm}$.

Hence, the value of amplitude of the function is 4.0 cm.

Using equation (ii) and the given value of wavelength, the angular wave number is given by:

$\begin{array}{l}k=\frac{2\mathrm{Ï€}}{2\mathrm{Ï€}}\\ =0.31\mathrm{rad}/\mathrm{cm}\end{array}$

Hence, the value of wavenumber is 0.31 rad/cm.

Using equation (iii), the angular frequency is given by:

$$\begin{gathered} \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{10} \\ =0.63\mathrm{rad}/\mathrm{s} \end{gathered}$$

Hence, the value of the angular frequency is 0.63 rad/s.

The figure shows that at x=0,

$y(0,t)=y_{m}sin(-\mathrm{Ó\neg }t+\mathrm{Ï•})$

And it is a positive sine function. That is

$\mathrm{y}(0,\mathrm{t})=+{\mathrm{y}}_{\mathrm{m}}\sin \left(\mathrm{Ó\neg ³Ù}\right)$

Therefore, the phase constant must be$\mathrm{Ï•}=\mathrm{Ï€}$.

Hence, the value of phase constant is $\mathrm{Ï€}$.

The sign is minus since the wave is traveling in the +x direction. Hence, the sign of the angular frequency is negative.

Using equation (iv), the frequency of the wave is given as:

$$\begin{gathered} f=\frac{1}{10} \\ =0.10s^{-1} \end{gathered}$$

Therefore, using equation (v) and the above value f frequency, the speed of the wave is given as:

$$\begin{gathered} v=0.10×20 \\ =2.0\mathrm{cm} \end{gathered}$$

Hence, the value of speed of the wave is 2.0 cm/s.

From the results above, the wave may be expressed as

$$\begin{gathered} y\left(x,t\right)=4.0\sin \left(\frac{\mathrm{Ï€³\ae }}{10}-\frac{\mathrm{Ï€³Ù}}{5}+\mathrm{Ï€}\right) \\ =-4.0\sin \left(\frac{\mathrm{Ï€³\ae }}{10}-\frac{\mathrm{Ï€³Ù}}{5}\right) \end{gathered}$$

Using the equation (vi ) and the above wave equation, the transverse velocity is given as:

$$\begin{gathered} u\left(x,t\right)=\frac{d}{dt}\left(-4.0\sin \left(\frac{\mathrm{Ï€³\ae }}{10}-\frac{\mathrm{Ï€³Ù}}{5}\right)\right) \\ =-4.0\left(\frac{\mathrm{Ï€}}{t}\right)\cos \left(\frac{\mathrm{Ï€³\ae }}{10}-\frac{\mathrm{Ï€³Ù}}{5}\right) \end{gathered}$$

Hence, at the required values, the value of transverses velocity is given by:

$$\begin{gathered} u\left(0,5,0\right)=-4.0\left(\frac{\mathrm{π}}{5.0}\right)\cos \left(-\frac{\mathrm{π}×5.0}{5}\right) \\ =-2.5\mathrm{cm}/\mathrm{s} \end{gathered}$$

Hence, the value of transverse velocity is 2.5 cm/s.