To tackle the problem of stationary waves in a metallic rod, we first need to understand the setup and the characteristics of the wave. Given that the rod is clamped at both ends, it behaves like a fixed boundary condition, which is essential for forming stationary waves. With six antinodes present, we can derive the equation for the stationary wave and the component wave at a specific point along the rod.
Understanding Stationary Waves
Stationary waves are formed by the interference of two waves traveling in opposite directions. In this case, the rod's length is 1 meter, and with six antinodes, we can determine the wavelength and the wave equation. The presence of six antinodes indicates that there are five nodes in between, which helps us find the wavelength.
Calculating Wavelength
The distance between two consecutive antinodes is half a wavelength (λ/2). Since there are six antinodes, the total length of the rod can be expressed as:
- Number of segments between antinodes = 6 - 1 = 5
- Length of the rod = 1 meter
- Thus, 5 * (λ/2) = 1 meter
From this, we can solve for the wavelength:
λ = (2 * 1 meter) / 5 = 0.4 meters
Formulating the Wave Equation
The general form of a stationary wave can be expressed as:
y(x, t) = A sin(kx) cos(ωt)
Where:
- A is the amplitude (2 × 10^(-6) meters)
- k is the wave number, calculated as k = 2π/λ
- ω is the angular frequency, related to the frequency f by ω = 2πf
First, we calculate k:
k = 2π / 0.4 = 5π rad/m
Angular Frequency
To find ω, we need the frequency. However, since the frequency is not provided, we can express the wave equation in terms of ω:
y(x, t) = (2 × 10^(-6)) sin(5πx) cos(ωt)
Component Wave at 0.1 Meters
To find the component wave at a specific point, we consider the two waves that combine to form the stationary wave. The component waves can be expressed as:
y₁(x, t) = A sin(kx - ωt)
y₂(x, t) = A sin(kx + ωt)
At x = 0.1 meters, we substitute into the wave equations:
y₁(0.1, t) = (2 × 10^(-6)) sin(5π(0.1) - ωt) = (2 × 10^(-6)) sin(0.5π - ωt)
y₂(0.1, t) = (2 × 10^(-6)) sin(5π(0.1) + ωt) = (2 × 10^(-6)) sin(0.5π + ωt)
Final Expressions
Thus, the component waves at 0.1 meters from the end of the rod are:
- y₁(0.1, t) = (2 × 10^(-6)) sin(0.5π - ωt)
- y₂(0.1, t) = (2 × 10^(-6)) sin(0.5π + ωt)
In summary, the stationary wave equation along the rod is:
y(x, t) = (2 × 10^(-6)) sin(5πx) cos(ωt)
And the component waves at 0.1 meters are given by the expressions above. This analysis illustrates how stationary waves form and how to derive their equations based on the properties of the medium and the wave characteristics.
