Chapter 17, Problem 73CP

To determine

The wave intensity at a distance $r$ in front of a point source with power ${\left(Power\right)}_{\text{avg}}$ moving with constant speed ${\upsilon }_s$ is $\underset{\_}{I=\frac{{\left(Power\right)}_{\text{avg}}}{4\pi r^2}\left(\frac{\upsilon -{\upsilon }_s}{\upsilon }\right)}$.

Answer to Problem 73CP

It is proved that the wave intensity at a distance $r$ in front of a point source with power ${\left(Power\right)}_{\text{avg}}$ moving with constant speed ${\upsilon }_s$ is $\underset{\_}{I=\frac{{\left(Power\right)}_{\text{avg}}}{4\pi r^2}\left(\frac{\upsilon -{\upsilon }_s}{\upsilon }\right)}$.

Explanation of Solution

The wave front that passes the observer is spherical. Consider $T$ as the source of vibration, and $T_{\text{MW}}$ is the energy put into each wave front during each vibration.

Write the expression for average power.

  ${\left(Power\right)}_{\text{avg}}=\frac{T_{\text{MW}}}{T}$                                                                                                      (I)

The radius of the wave front received by the observer at a distance $r$.

  $R_w=\upsilon \Delta t$                                                                                                                  (II)

Here, $\upsilon$ is the speed of the wave, and $\Delta t$ is the time interval to travel a distance $r$.

The forward distance travelled by the source due to the radiation form the source.

  $d_s={\upsilon }_s\Delta t$                                                                                                                 (III)

Here, ${\upsilon }_s$ is the speed of the source, $d_s$ is the forward distance moved by the source.

The total distance traveled by the wave front is the sum of the distance $r$, and $d_s$.

Write the expression for the total distance traveled by the wave front, and obtain an expression for $\Delta t$.

  $\begin{array}{c}{R}_w=r+d_s\\ \upsilon \Delta t=r+{\upsilon }_s\Delta t\\ \Delta t=\frac{r}{\upsilon -{\upsilon }_s}\end{array}$

The energy per unit area during one cycle gives the intensity of the wave.

Write the expression for the intensity of the wave.

  $I=\frac{T_{\text{MW}}}{A}$                                                                                                                 (IV)

Here, $A$ is the area of cross section.

Write the expression for the area of a spherical wave front.

  $A=4\pi R_w^2$                                                                                                                (V)

Substitute, $\upsilon \Delta t$ for $R_w$, and $\frac{r}{\upsilon -{\upsilon }_s}$ for $\Delta t$ in equation (V).

  $\begin{array}{c}A=4\pi {\left(\upsilon \Delta t\right)}^2\\ =4\pi {\left(\upsilon \times \frac{r}{\upsilon -{\upsilon }_s}\right)}^2\\ =\frac{4\pi {\upsilon }^2{r}^2}{{\left(\upsilon -{\upsilon }_s\right)}^2}\end{array}$

Substitute, ${\left(Power\right)}_{\text{avg}}T$ for $T_{\text{MW}}$, and $\frac{4\pi {\upsilon }^2{r}^2}{{\left(\upsilon -{\upsilon }_s\right)}^2}$ for $A$ in equation (IV).

  $\begin{array}{c}I=\frac{{\left(Power\right)}_{\text{avg}}T}{\left(\frac{4\pi {\upsilon }^2{r}^2}{{\left(\upsilon -{\upsilon }_s\right)}^2}\right)}\\ =\frac{{\left(Power\right)}_{\text{avg}}T{\left(\upsilon -{\upsilon }_s\right)}^2}{4\pi {\upsilon }^2{r}^2}\end{array}$

Write an expression for the time taken by the wave front to pass the observer.

  $T^{\prime }=\frac{1}{f^{\prime }}$                                                                                                                   (VI)

Here, $f^{\prime }$ is the Doppler shifted frequency.

Write the expression for the Doppler shifted frequency.

  $f^{\prime }=\frac{\upsilon }{T\left(\upsilon -{\upsilon }_s\right)}$                                                                                                     (VII)

The intensity of the wave received by the observer is.

  $\begin{array}{c}I=\left(\frac{T_{\text{MW}}}{A}\right)\frac{1}{T^{\prime }}\\ =\left(\frac{T_{\text{MW}}}{A}\right)f^{\prime }\\ =\frac{{\left(Power\right)}_{\text{avg}}T{\left(\upsilon -{\upsilon }_s\right)}^2}{4\pi {\upsilon }^2{r}^2}{f}^{\prime }\end{array}$                                                                               (VIII)

Substitute, $\frac{\upsilon }{T\left(\upsilon -{\upsilon }_s\right)}$ for $f^{\prime }$ in equation (VIII).

  $\begin{array}{c}I=\frac{{\left(Power\right)}_{\text{avg}}T{\left(\upsilon -{\upsilon }_s\right)}^2}{4\pi {\upsilon }^2{r}^2}\left(\frac{\upsilon }{T\left(\upsilon -{\upsilon }_s\right)}\right)\\ =\frac{{\left(Power\right)}_{\text{avg}}\upsilon \left(\upsilon -{\upsilon }_s\right)}{4\pi {\upsilon }^2{r}^2}\end{array}$

Conclusion:

Therefore, it is proved that the wave intensity at a distance $r$ in front of a point source with power ${\left(Power\right)}_{\text{avg}}$ moving with constant speed ${\upsilon }_s$ is $\underset{\_}{I=\frac{{\left(Power\right)}_{\text{avg}}}{4\pi r^2}\left(\frac{\upsilon -{\upsilon }_s}{\upsilon }\right)}$.