A standing wave has a wavenumber 200 rad/m. What is the distance between two adjacent nodes?
1/200 = 0.005 meters / rad
pi rad / node x 0.005 meters / rad = 0.0157 m / node
k =200 rad/m
k=2π/λ
=> λ=2π/k=2π/200 =π/100=0.0314 m
x= λ/2=0.0314/2=0.0157 m
To find the distance between two adjacent nodes in a standing wave, you first need to understand the concept of nodes and wavelength.
In a standing wave, nodes are points that remain stationary, with zero displacement. They correspond to the crests and troughs of a wave. Wavelength, on the other hand, represents the distance between two consecutive points that are in phase or have the same displacement.
The wavenumber (k) of a wave is defined as the number of wavelengths per unit distance. Mathematically, it is given by:
k = 2π / λ
where k is the wavenumber and λ (lambda) is the wavelength.
Now, in this scenario, the wavenumber (k) is given as 200 rad/m. We can use this information to find the wavelength (λ) by rearranging the equation:
λ = 2π / k
Substituting the given value of k:
λ = 2π / 200
Simplifying further:
λ = π / 100
Therefore, the wavelength of the standing wave is π/100 radians per meter.
Now, to find the distance between two adjacent nodes, we can recall that in a standing wave, there is half a wavelength between two adjacent nodes. So, the distance between two adjacent nodes is:
Distance between adjacent nodes = 0.5 * λ
Substituting the value of λ:
Distance between adjacent nodes = 0.5 * (π / 100)
Simplifying:
Distance between adjacent nodes = π / 200
Hence, the distance between two adjacent nodes in this standing wave is π/200 meters.