To demonstrate standing waves with a spring.
Consider a string is tied to a hook. A pulse or wave starts from free end. When wave meets rigid boundary, it gets reflected with phase change of Pi. Superposition of incident wave and reflected wave happens. It set up steady wave pattern on the string. These wave patterns are called stationary waves.
Consider an incident wave is in direction of positive x axis.
𝑦₁(𝑥,𝑡) = 𝑎 𝑠𝑖𝑛(𝑘𝑥 - ω𝑡)
Reflected wave is in direction of x axis.
𝑦₂(𝑥,𝑡) = 𝑎 𝑠𝑖𝑛(𝑘𝑥 + ω𝑡)
Resultant wave is
𝑦(𝑥,𝑡) = 2𝑎 𝑠𝑖𝑛 𝑘𝑥 𝑐𝑜𝑠 ω𝑡
Amplitude of resultant wave changes from one point to another point. But all parts of string oscillate with same time period. The wave pattern is stationary. These are called stationary waves.
Nodes - These are points in wave where amplitude in wave is minimum.
Antinodes - These are points in wave where amplitude in wave is maximum.
Normal modes - The system vibrates by a set of natural frequencies. These are normal modes. Boundary condition constrain possible frequencies.
1. Both ends of string tied to handle of door. Both ends of the strings are nodes.
Possible wavelength of stationary waves
λ = 2𝐿/𝑛 ; 𝑛 = 1, 2, 3,......
with corresponding frequencies
ν = 𝑛𝑣/2𝐿 ; 𝑛 = 1, 2, 3,......
L - length of string
n - order of harmonic
v - speed of wave
Lowest natural frequency (n=1) is called fundamental frequency.
n = 2 - Second harmonic
n = 3 - Third harmonic.
2. Consider a string tied to thread and thread tied to hook. End of string which tied to thread is not rigid. This end of the string is antinode. Hold free end. The end of string which we hold is node.
Possible wavelength of stationary waves
λ = 4𝐿/𝑛 ; 𝑛 = 1, 3, 5,......
𝑓 = 𝑣/λ = 𝑛𝑣/4𝐿 ; 𝑛 = 1, 3, 5,......
Only odd harmonics can exist when one end is not rigid and other end is rigid.
One end is rigid. Another end is not rigid
