Chapter 17, Problem 14PQ

(a)

To determine

The wavelength of the wave.

Answer to Problem 14PQ

The wavelength of the wave is $\underset{\_}{7.17\text{m}}$.

Explanation of Solution

Write the equation for the wavelength.

  $\lambda =\frac{2\pi }{k}$                                                                                                                 (I)

Here, $\lambda$ is the wavelength and $k$ is the wave vector.

Write the expression for the wave vector.

  $k=\frac{\omega }{v}$                                                                                                                  (II)

Here, $\omega$ is the angular frequency and $v$ is the speed.

Rewrite the expression for the wavelength from equation (I) by using (II).

  $\lambda =\frac{2\pi }{\left(\omega /v\right)}$                                                                                                          (III)

Conclusion:

Substitute $2.34\text{rad/s}$ for $\omega$ and $2.67\text{m/s}$ for $v$ in Equation (III) to find $\lambda$.

  $\begin{array}{c}\lambda =\frac{2\pi }{\left(\left[2.34\text{rad/s}\right]/\left[2.67\text{m/s}\right]\right)}\\ =\frac{2\pi }{0.876\text{rad/m}}\\ =7.17\text{m}\end{array}$

Thus, the wavelength of the wave is $\underset{\_}{7.17\text{m}}$.

(b)

To determine

The expression for the wave function.

Answer to Problem 14PQ

The expression for the wave function is $\underset{\_}{y\left(x,t\right)=\left(0.346\text{m}\right)\sin \left(0.876x-2.34t\right)}$.

Explanation of Solution

Write the equation of standard equation of the wave function.

  $y\left(x,t\right)=y_{\text{max}}\sin \left(kx-\omega t\right)$                                                                            (IV)

Here, $y\left(x,t\right)$ is the instantaneous displacement, $y_{\max }$ is the amplitude, $x$ is the displacement and $t$ is time.

Rewrite the expression for the wave function from equation (IV) by using (II).

  $y\left(x,t\right)=y_{\text{max}}\sin \left(\left(\frac{\omega }{v}\right)x-\omega t\right)$                                                                        (V)

Conclusion:

Substitute $2.34\text{rad/s}$ for $\omega$, $2.67\text{m/s}$ for $v$ and $34.6\text{cm}$ for $y_{\max }$ in Equation (V) to find $y\left(x,t\right)$.

  $\begin{array}{c}y\left(x,t\right)=\left(34.6\text{cm}\right)\sin \left(\left(\frac{2.34\text{rad/s}}{2.67\text{m/s}}\right)x-\left(2.34\text{rad/s}\right)t\right)\\ =\left[\left(34.6\text{cm}\right)\left(\frac{1\times {10}^{-2}\text{m}}{1\text{cm}}\right)\right]\sin \left(\left(\frac{2.34\text{rad/s}}{2.67\text{m/s}}\right)x-\left(2.34\text{rad/s}\right)t\right)\\ =\left(0.346\text{m}\right)\sin \left(0.876x-2.34t\right)\end{array}$

Thus, the expression for the wave function is $\underset{\_}{y\left(x,t\right)=\left(0.346\text{m}\right)\sin \left(0.876x-2.34t\right)}$.

(c)

To determine

The new wave function if the angular frequency doubled.

Answer to Problem 14PQ

The new wave function if the angular frequency doubled is $\underset{\_}{y\left(x,t\right)=\left(0.346\text{m}\right)\sin \left(1.75x-4.68t\right)}$.

Explanation of Solution

Write the expression for the wave function from equation (IV) by using (II).

  $y\left(x,t\right)=y_{\text{max}}\sin \left(\left(\frac{\omega }{v}\right)x-\omega t\right)$                                                                           (V)

Conclusion:

Substitute $2\left(2.34\text{rad/s}\right)$ for $\omega$, $2.67\text{m/s}$ for $v$ and $34.6\text{cm}$ for $y_{\max }$ in Equation (V) to find $y\left(x,t\right)$.

  $\begin{array}{c}y\left(x,t\right)=\left(34.6\text{cm}\right)\sin \left(\left(\frac{2\left(2.34\text{rad/s}\right)}{2.67\text{m/s}}\right)x-\left[2\left(2.34\text{rad/s}\right)\right]t\right)\\ =\left[\left(34.6\text{cm}\right)\left(\frac{1\times {10}^{-2}\text{m}}{1\text{cm}}\right)\right]\sin \left(\left(\frac{4.68\text{rad/s}}{2.67\text{m/s}}\right)x-\left(4.68\text{rad/s}\right)t\right)\\ =\left(0.346\text{m}\right)\sin \left(1.75x-4.68t\right)\end{array}$

Thus, the new wave function if the angular frequency doubled is $\underset{\_}{y\left(x,t\right)=\left(0.346\text{m}\right)\sin \left(1.75x-4.68t\right)}$.