To find the wavelength of a wave in a metal rod when it is vibrating in its fundamental mode, you simply measure the length of the rod.
In the fundamental mode of vibration, the wave pattern in the rod is such that there are displacement antinodes at both ends and a displacement node in the center. This corresponds to half a wavelength fitting within the length of the rod.
According to the provided information:
- In the fundamental mode of vibration, the wavelength of the sound emitted will be twice the length of the rod.
Therefore, the formula for the wavelength ($\lambda$) in the fundamental mode is:
$\lambda = 2 \times L$
Where $L$ is the length of the metal rod.
Understanding the Fundamental Mode
The fundamental mode is the simplest vibration pattern a rod can sustain. Imagine the rod ends are free to move (like an air column open at both ends for sound waves). When the rod is struck or excited, it vibrates. The fundamental mode (also called the first harmonic) has the lowest possible frequency and the longest wavelength.
- Displacement: The ends of the rod experience maximum displacement (movement), acting as displacement antinodes. The center of the rod has minimum displacement, acting as a displacement node.
- Pressure/Stress: Conversely, the ends of the rod experience minimum pressure/stress changes (pressure nodes), while the center experiences maximum pressure/stress changes (pressure antinode).
This fundamental mode wave pattern looks like half a full wavelength fitting neatly within the rod's boundaries.
Practical Steps to Find the Wavelength
Finding the wavelength in the fundamental mode is quite straightforward:
- Measure the Rod's Length: Use a ruler, tape measure, or caliper to accurately determine the length ($L$) of the metal rod. Ensure you measure the entire vibrating length.
- Calculate the Wavelength: Multiply the measured length by two.
Example
Consider a metal rod with a length of 0.5 meters.
- Length of the rod ($L$) = 0.5 m
- Wavelength ($\lambda$) in the fundamental mode = $2 \times L$
- $\lambda = 2 \times 0.5 \text{ m} = 1.0 \text{ m}$
So, the wavelength of the fundamental wave in this rod is 1.0 meter.
Relationship with Frequency and Wave Speed
While the wavelength in the fundamental mode is determined solely by the rod's length, the wave's frequency and speed are also related. The fundamental frequency ($f_1$) is the lowest frequency at which the rod naturally vibrates. The speed of the wave ($v$) in the metal is a property of the material and its temperature.
These are related by the fundamental wave equation:
$v = f \times \lambda$
In the fundamental mode:
$v = f_1 \times (2L)$
The reference mentions that frequency can be measured using instruments like a microphone, oscilloscope, or frequency counter. If you know the frequency and the wavelength (calculated from the rod's length), you can also determine the speed of sound in that specific metal.
| Property | Symbol | How to Find (Fundamental Mode) | Relationship to others |
|---|---|---|---|
| Length of Rod | $L$ | Measure directly | $\lambda = 2L$ |
| Wavelength | $\lambda$ | Calculate: $\lambda = 2L$ | $\lambda = v/f$ |
| Frequency | $f$ | Measure (e.g., microphone) | $f = v/\lambda$ |
| Wave Speed | $v$ | Property of material | $v = f \times \lambda$ |
Understanding the fundamental mode and its direct relationship to the rod's length is the key to finding the wavelength in this specific vibration state.
