Question

(Optional) Consider a long whip antenna of the type used on some automobiles and trucks. Show that the natural frequencies of oscillation for the antenna are $f_{\mathrm{m}}=mv/4L$, where $m=1,3,5,\dots ,v$ is the wave speed, and $L$ is the length of the antenna. (Hint: The boundary conditions are a node and an antinode.)

Short answer

The natural frequencies for the antenna are given by $f_m=\frac{mv}{4L}$, where $m=1,3,5,\dots$.

Step-by-step solution

Understanding the Physical System

The long whip antenna can be modeled similar to a vibrating string with nodes and antinodes, where it is fixed at the base (a node) and free at the top (an antinode). This is akin to having one end fixed and one end free as seen in harmonic oscillators or waves in strings.

Analyzing the Boundary Conditions

For the boundary conditions of a node at the fixed end and an antinode at the free end, the harmonic modes of vibration are characterized by odd harmonics (1st, 3rd, 5th, etc.). This is because the free end must have a point of maximum displacement (antinode), and the fixed end must have a point of no displacement (node).

Derive the Expression for Wavelength

Given the boundary condition of node and antinode, the length of the antenna corresponds to one-quarter of the wavelength of the fundamental frequency because the shortest wave fitting the condition is a quarter-wave. For the mth odd harmonic, the length corresponds to $$L=\frac{(2m-1)\lambda }{4}$$ where $\lambda$ is the wavelength.

Relate Wavelength to Wave Speed

The wave speed $v$ relates to frequency $f$ and wavelength $\lambda$ by the equation $v=f\lambda$. Solving for frequency gives us $f=\frac{v}{\lambda }$.

Substitute Wavelength in Frequency Expression

Substitute the expression for $\lambda$ from step 3 into the expression for $f$ from step 4: $$f=\frac{v}{(4L/(2m-1))}=\frac{(2m-1)v}{4L}$$ Since $m$ is an odd integer, we can rewrite $(2m-1)$ as $m$ if $m$ represents odd numbers, giving:$$f_m=\frac{mv}{4L},\text{ where }m=1,3,5,\dots$$

Conclusion

The natural frequencies of oscillation for a long whip antenna with a node at one end and an antinode at the other are given by $$f_m=\frac{mv}{4L}$$ for odd integers $m=1,3,5,\dots$.

Key concepts

Boundary Conditions

Boundary conditions are essential when studying wave oscillations in systems like the whip antenna. These conditions dictate how waves behave at the boundaries of a physical system. For the whip antenna:

• The base is fixed, making it a node. A node is a point where the medium doesn't move, showcasing no displacement.
• The tip of the antenna is free, characterized as an antinode. An antinode, in contrast, is where maximum displacement occurs.

These boundary conditions reveal the relationship between the antenna's physical structure and its oscillation patterns. When one end is fixed (node) and the other free (antinode), it forces the system into specific natural frequencies that determine how the antenna vibrates. Recognizing these conditions helps us identify that only certain vibrational patterns, or harmonics, will be present - specifically, only the ones that fit the node-to-antinode design.

Harmonic Modes

In wave systems like antennas, harmonic modes are the possible natural oscillation patterns. With the whip antenna's node-antinode boundary condition, only odd harmonics manifest.

• These odd harmonics include the 1st, 3rd, 5th, etc., as the natural frequencies of the antenna.
• The reason for skipping even harmonics is the lack of a second antinode needed for such patterns.

For each harmonic, $m$ describes the mode number, representing the specific pattern the wave takes to fit within these boundaries. Odd harmonics can be seen as different stair steps where each step involves fitting more wave sections into the same physical space of the antenna, corresponding to higher frequencies. Understanding harmonic modes means knowing how a specific setup affects which frequency modes occur, providing insights into designing systems with desired resonant properties.

Wave Speed

Wave speed is a vital parameter linking the characteristics of a medium to the oscillation frequencies. It’s controlled by the inherent properties of the material the wave travels through. For the whip antenna, wave speed $v$ helps relate frequency $f$ and wavelength $\lambda$. This connection is:$$v=f\lambda$$When rearranging to solve for frequency, it becomes:$$f=\frac{v}{\lambda }$$The formula links the physical length of the antenna, wave speed, and natural frequencies. Particularly:

• In the antenna's context, this dictates how wave energy travels along its length.
• Knowing the wave speed, one can calculate frequencies using the derived expression $f_m=\frac{mv}{4L}$, allowing predictions based on measurable quantities.

Wave speed, therefore, defines how quickly oscillations propagate through the antenna, making it central to predicting how the system naturally wants to move.